Download Answers/Rubrics

PreCalculus
3rd and 4th Nine-Weeks
Scope and Sequence
Topic 4: Trigonometry and Trigonometric Functions (45 – 50 days)
A) Uses radian and degree angle measure to solve problems and perform conversions as needed.
B) Uses the unit circle to explain the circular properties and periodic nature of trigonometric
functions and to find the trigonometric ratios of any angle.
C) Describes and compares the characteristics of the trigonometric functions (with and without
the use of technology) for sine, cosine, tangent, cotangent, cosecant, and secant.
D) Determines solutions to trigonometric equations.
E) Describes how a change in the value of any constant in a general-form trigonometric
equation such as y = a sin (b-x) + c affects the graph of the equation.
F) Represents the inverse of a trigonometric function symbolically and graphically
G) Creates a scatterplot of bivariate data, identifies a trigonometric function to model the data,
and uses that model to identify patterns and make predictions.
H) Derives and applies the basic trigonometric identities; i.e. angle addition, angle subtraction,
and double-angle.
I) Uses trigonometric relationships to determine lengths and angle measures; i.e., Law of Sines
and Law of Cosines.
J) Models and solves problems using trigonometry.
Topic 5: Noncartesian Representations (35 – 40 days)
A)
B)
C)
D)
Uses vectors to model and solve application problems.
Uses parametric equations to model and solve application problems.
Uses polar coordinates.
Expresses complex numbers in trigonometric form and computes sums, differences,
products, quotients, powers, and roots of complex numbers in trigonometric form.
COLUMBUS PUBLIC SCHOOLS
MATHEMATICS CURRICULUM GUIDE
SUBJECT
PreCalculus
STATE STANDARD 3 and 4
Patterns, Functions, and Algebra,
Geometry and Spatial Sense
TIME RANGE
45-50 days
GRADING
PERIOD
3-4
MATHEMATICAL TOPIC 4
Trigonometry and Trigonometric Functions
A)
B)
C)
D)
E)
F)
G)
H)
I)
J)
CPS LEARNING GOALS
Uses radian and degree angle measure to solve problems and perform conversions as needed.
Uses the unit circle to explain the circular properties and periodic nature of trigonometric
functions and to find the trigonometric ratios of any angle.
Describes and compares the characteristics of the trigonometric functions (with and without
the use of technology) for sine, cosine, tangent, cotangent, cosecant, and secant.
Determines solutions to trigonometric equations.
Describes how a change in the value of any constant in a general-form trigonometric
equation such as y = a sin (b-x) + c affects the graph of the equation.
Represents the inverse of a trigonometric function symbolically and graphically
Creates a scatterplot of bivariate data, identifies a trigonometric function to model the data,
and uses that model to identify patterns and make predictions.
Derives and applies the basic trigonometric identities; i.e. angle addition, angle subtraction,
and double-angle.
Uses trigonometric relationships to determine lengths and angle measures; i.e., Law of Sines
and Law of Cosines.
Models and solves problems using trigonometry.
COURSE LEVEL INDICATORS
Course Level (i.e., How does a student demonstrate mastery?):
9 Converts between degrees and radians and can explain the appropriate use of each.
Math M:11-B:02
9 Sets up and solves angular velocity and arc-length problems by using radian measure.
Math G:12-D:02
9 Describes and compares the characteristics of the trigonometric functions; e.g., general
shape, number of roots, domain and range, even or odd asymptotic and global behavior for
sine, cosine, tangent, cotangent, cosecant, and secant both algebraically and graphically.
Math A:12-A:03
9 Relates a given sine or cosine graph to its equivalent other in terms of phase shift.
Math A:12-A:03
9 Determines all zeros for a trigonometric function algebraically and also gives all zeros within
a given range (such as between 0 and 2π radians). Math A:12-A:03
9 Identifies the amplitude, frequency/period, phase shift, vertical shift, etc. of a given
trigonometric function and uses these to sketch a graph of the function. Math A:12-A:03
9 Identifies the extrema of trigonometric functions with and without technology.
Math A:12-A:03
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 1 of 163
Columbus Public Schools 1/5/06
9 Uses the unit circle to explain the circular properties and periodic nature of trigonometric
functions and to find the trigonometric ratios of any angle. Math G:11-A:04
9 Uses the unit circle to explain the periodic nature of the sine and cosine functions, the nature
of reference angles, and the range of possible values for sine and cosine. Math A:12-A:03
9 Represents the inverse of a trigonometric function symbolically and graphically.
Math A:12-A:04
9 Plots bivariate data and determines the trigonometric function that best fits the data both
analytically and by regression. Math D:11-A:04
9 Determines phase shift, vertical shift, amplitude and frequency to be able to create the
trigonometric function equation best fitting the data. Math A:12-A:03
9 Collects real world motion data and models it using trigonometric equations.
Math D:11-C:04
9 Verifies identities analytically by applying fundamental trigonometric identities to re-write
and combine expressions. Math G:12-A:02
9 Verifies trigonometric identities graphically. Math G:12-A:02
9 Uses the double-angle, half-angle, and angle-addition formulas to determine specified
trigonometric values. Math G:11-A:04
9 Uses trigonometric relationships to determine lengths and angle measures; i.e., Law of Sines
and Law of Cosines. Math G:11-A:04
9 Uses the Law of Sines and the Law of Cosines to determine the missing angles or sides of
triangles. Math G:11-A:04
9 Uses Heron’s formula to find the area of a triangle when the sides are known but base and/or
height are not given. Math G:11-A:04
9 Determines general solutions to trigonometric equations and specific solutions within a given
interval. Math G:11-A:04
Previous Level:
9 Defines the basic trigonometric ratios in right triangles: sine, cosine, and tangent.
Math G:09-I:01
9 Uses right triangle trigonometric relationships to determine lengths and angle measures.
Math G:09-I:02
9 Evaluates expressions containing square roots. Math N:09-I:04
9 Sketches a basic sine and cosine graph by hand. Math A:10-D:02
9 Sketches the graph of a function by means of technology. Math A:08-D:09
Next Level:
9 Determines the average rate of change for specific trigonometric functions.
Math A:12-A:10
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 2 of 163
Columbus Public Schools 1/5/06
The description from the state for the Measurement Standard says:
Students estimate and measure to a required degree of accuracy and precision by selecting and
using appropriate units, tools and technologies.
The grade-band benchmark from the state, for this topic in the grade band 11 – 12 is:
B. Apply various measurement scales to describe phenomena and solve problems.
The description from the state, for the Geometry and Spatial Sense Standard says:
Students identify, classify, compare and analyze characteristics, properties and relationships of
one-, two-, and three-dimensional geometric figures and objects. Students use spatial reasoning,
properties of geometric objects and transformations to analyze mathematical situations and solve
problems.
The grade-band benchmarks from the state, for this topic in the grade band 11 – 12 are:
A. Use trigonometric relationships to verify and determine solutions in problem situations.
D. Use coordinate geometry to represent and examine the properties of geometric figures.*
The description from the state, for the Patterns, Functions, and Algebra Standard says:
Students use patterns, relations, and functions to model, represent and analyze problem situations
that involve variable quantities. Students analyze, model and solve problems using various
representations such as tables, graphs and equations.
The grade-band benchmark from the state, for this topic in the grade band 11 – 12 is:
A. Analyze functions by investigating rates of change, intercepts, zeros, asymptotes and local
and global behavior.
The description from the state for the Data Analysis Standard says:
Students pose questions and collect, organize, represent, interpret and analyze data to answer
those questions. Students develop and evaluate inferences, predictions and arguments that are
based on data.
The grade-band benchmarks from the state, for this topic in the grade band 11 – 12 are:
A. Create and analyze tabular and graphical displays of data using appropriate tools, including
spreadsheets and graphing calculators.
C. Design and perform a statistical experiment, simulation or study; collect and interpret data;
and use descriptive statistics to communicate and support predictions and conclusions.
The description from the state, for the Mathematical Processes Standard says:
Students use mathematical processes and knowledge to solve problems. Students apply
problem-solving and decision-making techniques, and communicate mathematical ideas.
The grade-band benchmark from the state, for this topic in the grade band 11 – 12 is:
J. Apply mathematical modeling to workplace and consumer situations including problem
formulation, identification of a mathematical model, interpretation of solution within the
model, and validation to original problem situation.
*This is an extension of the benchmarks in grades 8-10 for more complex figures.
PreCalculus Standard 4 and 5
Columbus Public Schools 1/5/06
Page 3 of 163
Trigonometry and Trigonometric Functions
PRACTICE ASSESSMENT ITEMS
Trigonometry - A
A wheel rotating at 50 revolutions per minute rotates at
A. 50π radians/minute
B. 100π radians/minute
C. 150π radians/minute
D. 200π radians/minute
Which of the following gives the measures of two angles, one positive and one negative, that are
coterminal with a 73 degree angle?
A. 433o, -287o
B. 433o, -107o
C. 163o, -17o
D. 253o, -107o
Which of the following expressions can be used to calculate the length of an arc that subtends a
central angle of measure 65º on a circle of diameter 30 meters?
A. 65D ×
B. 65D ×
π
180º
180D
π
× 30 meters
× 30 meters
C. 65D × 30 meters
D. 65 meters
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 4 of 163
Columbus Public Schools 1/5/06
PRACTICE ASSESSMENT ITEMS
Trigonometry - A
Answers/Rubrics
Low Complexity
A wheel rotating at 50 revolutions per minute rotates at
A. 50π radians/minute
B. 100π radians/minute
C. 150π radians/minute
D. 200π radians/minute
Answer: B
Which of the following gives the measures of two angles, one positive and one negative, that are
coterminal with a 73 degree angle?
A. 433o, -287o
B. 433o, -107o
C. 163o, -17o
D. 253o, -107o
Answer: A
Moderate Complexity
Which of the following expressions can be used to calculate the length of an arc that subtends a
central angle of measure 65º on a circle of diameter 30 meters?
A. 65D ×
B. 65D ×
π
180º
180D
π
× 30 meters
× 30 meters
C. 65D × 30 meters
D. 65 meters
Answer: A
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 5 of 163
Columbus Public Schools 1/5/06
PRACTICE ASSESSMENT ITEMS
Trigonometry - A
A neighborhood carnival has a Ferris wheel whose radius is 30 feet. You measure that it takes
70 seconds to complete one revolution. Which of the following correctly determines the angular
speed ω and the linear speed v of the Ferris wheel?
A. ω =
B. ω =
C. ω =
D. ω =
θ
t
θ
t
θ
t
θ
t
=
π rad
70sec
= .045rad / sec , v =
s 2π × 30 ft
=
= 2.7 ft / sec
70sec
t
=
2π rad
s 30 ft
= .09rad / sec , v = =
= 0.43 ft / sec
70sec
t 70sec
=
30 ft
= 0.43 ft / sec ,
70sec
=
s 2π × 30 ft
2π rad
= .09rad / sec , v = =
= 2.7 ft / sec
t
70sec
70sec
v=
s 30 ft
=
= 0.43 ft / sec
t 70sec
A water sprinkler sprays water over a distance of 30 feet while rotating through an angle of 135º
a. Determine the area of the lawn that receives water.
b. If we wanted the area of coverage to be exactly 1500 square feet, determine we should
change the radius to, keeping the angle the same.
It takes ten identical pieces to form a circular track for a pair of toy racing cars. If the inside arc
of each piece is 5.8 inches shorter than the outside arc, determine the width of the track.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 6 of 163
Columbus Public Schools 1/5/06
PRACTICE ASSESSMENT ITEMS
Trigonometry - A
Answers/Rubrics
High Complexity
A neighborhood carnival has a Ferris wheel whose radius is 30 feet. You measure that it takes
70 seconds to complete one revolution. Which of the following correctly determines the angular
speed ω and the linear speed v of the Ferris wheel?
A. ω =
B. ω =
C. ω =
D. ω =
θ
t
θ
t
θ
t
θ
t
=
π rad
70sec
= .045rad / sec , v =
s 2π × 30 ft
=
= 2.7 ft / sec
70sec
t
=
2π rad
s 30 ft
= .09rad / sec , v = =
= 0.43 ft / sec
70sec
t 70sec
=
30 ft
= 0.43 ft / sec ,
70sec
=
2π rad
s 2π × 30 ft
= .09rad / sec , v = =
= 2.7 ft / sec
70sec
t
70sec
v=
s 30 ft
=
= 0.43 ft / sec
t 70sec
Answer: D
Short Answer/Extended Response
A water sprinkler sprays water over a distance of 30 feet while rotating through an angle of
135º
a. Determine the area of the lawn that receives water.
b. If we wanted the area of coverage to be exactly 1500 square feet, determine we
should change the radius to, keeping the angle the same.
Solution
3π
radians
180
4
A = 12 r 2θ = 12 302 ⋅ 34π = 1060.29 ft 2
a. 135D ×
π radians
b. A = 12 r 2θ
D
=
1500 = 12 r 2
3π
4
r 2 = 1273.24
r = 35.68 ft
A 4 point response: shows work and gets correct answers to both parts.
A 3 point response gets one solution correct, and the other has a single error in
substitution or calculation.
A 2 point response gets the first part right, but does not get the proper set-up for the
second part.
A 1 point response has a major conceptual error on the first part, and does not show
any understanding of the second part.
A 0 point response shows no understanding of the problem.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 7 of 163
Columbus Public Schools 1/5/06
PRACTICE ASSESSMENT ITEMS
Trigonometry - A
Answers/Rubrics
Short Answer/Extended Response
It takes ten identical pieces to form a circular track for a pair of toy racing cars. If the inside
arc of each piece is 5.8 inches shorter than the outside arc, determine the width of the track.
Solution:
x+5.8
x
r
w
⎛π ⎞
⎛π ⎞
Each piece of track has an angle measure of 2 ⎜ ⎟ (or ⎜ ⎟ ) radians.
⎝ 10 ⎠
⎝5⎠
⎛π ⎞
The arc length of the inside of one piece we’re taking as x, so x = ⎜ ⎟ r
⎝5⎠
⎛π ⎞
The arc length of the outside of one piece will then be: x + 5.8 = ⎜ ⎟ (r + w)
⎝5⎠
To solve both of these equations simultaneously, we multiply both equations by 5 and get:
5x = πr and 5x + 29 = π(r + w)
⎛ 29 ⎞
Solving these, we get w = ⎜ ⎟ = 9.23 inches
⎝π ⎠
A 4-point response uses the arc-length formulas to set-up and solve simultaneous equations
and gets the correct width.
A 3-point response has appropriate set-up for the two arc-length formulas but doesn’t
reach a proper solution for the width.
A 2-point response specifies the arc-length formula but doesn’t properly represent the 5.8inch difference and does not determine the width.
A 1-point response illustrates all necessary components of the problem but goes no further.
A 0-point response shows no understanding of the problem.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 8 of 163
Columbus Public Schools 1/5/06
PRACTICE ASSESSMENT ITEMS
Trigonometry - B
Which of the following gives the range of the function f(x) = (sin x)2+(cos x)2 ?
A. {0}
B. {1}
C. [0, 1]
D. [–1, 1]
E. [0, 2]
A sinusoid with amplitude 6 has a maximum value of 3. What is its minimum value?
A. -12
B -10
C. -9
D. 0
E. 3
Which of the following functions have identical graphs?
(i) y = sin 2x + π6
( )
(ii) y = cos (2x − )
(iii) y = cos (2x − )
π
6
π
3
A) i and iii
B) ii and iii
C) i and ii
D) i, ii and iii
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 9 of 163
Columbus Public Schools 1/5/06
PRACTICE ASSESSMENT ITEMS
Trigonometry - B
Answers/Rubrics
Low Complexity
Which of the following gives the range of the function f(x) = (sin x)2+(cos x)2 ?
A. {0}
B. {1}
C. [0, 1]
D. [–1, 1]
E. [0, 2]
Answer: B
A sinusoid with amplitude 6 has a maximum value of 3. What is its minimum value?
A. -12
B. -10
C. -9
D. 0
E. 3
Answer: C
Moderate Complexity
Which of the following functions have identical graphs?
(i) y = sin 2x + π6
( )
(ii) y = cos (2x − )
(iii) y = cos (2x − )
π
6
π
3
A) i and iii
B) ii and iii
C) i and ii
D) i, ii and iii
Answer: A
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 10 of 163
Columbus Public Schools 1/5/06
PRACTICE ASSESSMENT ITEMS
Trigonometry - B
Which of the following trigonometric functions are odd?
i. y = sin(x)
ii. y = cos(x)
iii. y = tan(x)
A. i and ii only
B. ii and iii only
C. i and iii only
D. i, ii, and iii
Based on years of weather data in a certain city, the expected low temperature T (in ºF) can be
⎡ 2π ⎛ 381⎞ ⎤
approximated by T = 40sin ⎢
⎜ t − 4 ⎟⎠ ⎥ + 15 where t is in days with t = 0 corresponding to
⎣ 365 ⎝
⎦
January 1. Predict the date when the coldest day of the year will occur and give the temperature
for that day. Show your solution.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 11 of 163
Columbus Public Schools 1/5/06
PRACTICE ASSESSMENT ITEMS
Trigonometry - B
Answers/Rubrics
High Complexity
Which of the following trigonometric functions are odd?
i. y = sin(x)
ii. y = cos(x)
iii. y = tan(x)
A. i and ii only
B. ii and iii only
C. i and iii only
D. i, ii, and iii
Answer: C
Short Answer/Extended Response
Based on years of weather data in a certain city, the expected low temperature T (in ºF) can be
⎡ 2π ⎛ 381⎞ ⎤
approximated by T = 40sin ⎢
⎜ t − 4 ⎟⎠ ⎥ + 15 where t is in days with t = 0 corresponding to
⎣ 365 ⎝
⎦
January 1. Predict the date when the coldest day of the year will occur and give the temperature
for that day. Show your solution.
Solution: By using the minimum function on the graphing calculator, we get minimum
temperatures of -25o on day 4 (January 4).
A 2-point response discusses proper use of the graph to get the correct answer.
A 1-point response uses a graph to get an incorrect answer.
A 0-point response shows no mathematical understanding of the task.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 12 of 163
Columbus Public Schools 1/5/06
PRACTICE ASSESSMENT ITEMS
π
6
A)
Trigonometry - C
is the reference angle for which one of the following non-acute angles?
π
3
2π
B)
3
5π
C)
6
4π
D)
3
Which of the following must have the same value as cos(68º)?
A. sin(22º)
B. cos(22º)
C. tan(22º)
D. none of the above
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 13 of 163
Columbus Public Schools 1/5/06
PRACTICE ASSESSMENT ITEMS
Trigonometry - C
Answers/Rubrics
Low Complexity
π
6
A)
is the reference angle for which one of the following non-acute angles?
π
3
2π
B)
3
5π
C)
6
4π
D)
3
Answer: C
Moderate Complexity
Which of the following must have the same value as cos(68º)?
A. sin(22º)
B. cos(22º)
C. tan(22º)
D. none of the above
Answer: A
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 14 of 163
Columbus Public Schools 1/5/06
PRACTICE ASSESSMENT ITEMS
If cosθ =
A. −
12
and tan θ < 0 , then sin θ = ___?
13
12
13
B.
−5
12
C.
5
13
D. −
Trigonometry - C
5
13
What would be the coordinates of the point P where the terminal ray of an angle θ intersects the
unit circle?
A. (sin θ ,cosθ )
B. (cosθ ,sin θ )
P
C. (r, r2)
r=1
D. (r 2 ,θ )
θ
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 15 of 163
Columbus Public Schools 1/5/06
PRACTICE ASSESSMENT ITEMS
Trigonometry - C
Answers/Rubrics
High Complexity
If cosθ =
A. −
12
13
B.
−5
12
C.
5
13
D. −
12
and tan θ < 0 , then sin θ = ___?
13
5
13
Answer: D
High Complexity
What would be the coordinates of the point P where the terminal ray of an angle θ intersects the
unit circle?
A. (sin θ ,cosθ )
B. (cosθ ,sin θ )
C. (r, r2)
P
D. (r 2 ,θ )
r=1
θ
Answer: B
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 16 of 163
Columbus Public Schools 1/5/06
PRACTICE ASSESSMENT ITEMS
Trigonometry - C
7π
without using a calculator by using ratios and the relevant reference triangle.
6
Show all work.
Evalute cos
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 17 of 163
Columbus Public Schools 1/5/06
PRACTICE ASSESSMENT ITEMS
Trigonometry - C
Answers/Rubrics
Short Answer/Extended Response
7π
without using a calculator by using ratios and the relevant reference triangle.
6
Show all work.
Evalute cos
Solution:
7π
is in the third quadrant where cosine is negative. The reference angle for the third
6
7π
π
π
3
quadrant is
− π , or . Thus we need − cos which equals −
6
6
6
2
A 2-point response properly explains where we get the sign of the answer and how we
determine the reference angle, and gets the correct answer.
A 1-point response makes at most one error (sign, reference angle, or result of trig
function).
A 0-point response shows no mathematical understanding of the task.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 18 of 163
Columbus Public Schools 1/5/06
PRACTICE ASSESSMENT ITEMS
Trigonometry - D
Which one of the following trigonometric functions has its graph symmetric about the line
y = - 6?
A) y = −6 sin(3x )
B) y = 8sin( x − 6) + 1
C) y = 4 cos(2 x ) − 6
⎛ x ⎞
D) y = 2sin ⎜ ⎟
⎝ −6 ⎠
What is the phase shift of y = 5sin(2x − 3π ) ?
A) 3π to the left
B) 3π to the right
C)
3π
to the left
2
D)
3π
to the right
2
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 19 of 163
Columbus Public Schools 1/5/06
PRACTICE ASSESSMENT ITEMS
Trigonometry - D
Answers/Rubrics
Low Complexity
Which one of the following trigonometric functions has its graph symmetric about the line
y = - 6?
A) y = −6 sin(3x)
B) y = 8sin(x − 6) + 1
C) y = 4 cos(2x) − 6
⎛ x ⎞
D) y = 2sin ⎜ ⎟
⎝ −6 ⎠
Answer: C
Moderate Complexity
What is the phase shift of y = 5sin(2x − 3π ) ?
A) 3π to the left
B) 3π to the right
C)
3π
to the left
2
D)
3π
to the right
2
Answer: C (Trick: This is really y = 5sin2(x PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 20 of 163
3π
)
2
Columbus Public Schools 1/5/06
PRACTICE ASSESSMENT ITEMS
Trigonometry - D
Which equation matches the graph below?
4
3
2
1
-6
-5
-4
-3
-2
-1
1
2
3
A) y = −2cos 4 x − π
)
4
5
6
-1
-2
-3
-4
(
(
B) y = −2cos 4x − π
(
C) y = −2cos 4 x + π
(
D) y = −2cos 4x + π
)
)
)
State the amplitude and period of the sinusoid and (relative to the parent function) the phase shift
⎛ x + 4⎞
and vertical shift: y = 6cos ⎜
− 3.
⎝ 2 ⎟⎠
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 21 of 163
Columbus Public Schools 1/5/06
PRACTICE ASSESSMENT ITEMS
Trigonometry - D
Answers/Rubrics
High Complexity
Which equation matches the graph below?
4
3
2
1
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
-1
-2
-3
-4
(
A) y = −2cos 4 x − π
(
B) y = −2cos 4x − π
(
C) y = −2cos 4 x + π
(
D) y = −2cos 4x + π
)
)
)
)
Answer: D
Short Answer/Extended Response
State the amplitude and period of the sinusoid and (relative to the parent function) the phase shift
⎛ x + 4⎞
and vertical shift: y = 6cos ⎜
− 3.
⎝ 2 ⎟⎠
Solution:
Cosine graph, shifted left 4, multiply period by 2 (horizontal stretch, yielding a period of
2π
or 4π , multiply amplitude by 6, and shift down by 3.
1
2
A 2 point response correctly determines all information asked for.
A 1 point response gets at least 2 pieces of the question correct but shows a lack of
understanding for the other two.
A 0 point response shows no mathematical understanding of the problem.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
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The exact value of arctan
Trigonometry - E
3 is:
A. 0
B.
C.
D.
π
6
π
4
π
3
The range of the function f ( x ) = arcsin x is:
A. ( −∞, ∞ )
B. ( −1,1)
C. [−1,1]
D. [0, π ]
E.
[ −π / 2, π / 2]
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
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PRACTICE ASSESSMENT ITEMS
Trigonometry - E
Answers/Rubrics
Low Complexity
The exact value of arctan
3 is:
A. 0
B.
C.
D.
π
6
π
4
π
3
Answer: D
Moderate Complexity
The range of the function f ( x ) = arcsin x is:
A. ( −∞, ∞ )
B. ( −1,1)
C. [−1,1]
D. [0, π ]
E.
[ −π / 2, π / 2]
Answer: E
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
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Trigonometry - E
sec(tan −1 (x)) =
A. x
B. csc x
C.
1 + x2
D.
1 − x2
E.
sin x
cos 2 x
You rent a cottage on the ocean for a week one summer and notice that the tide comes in twice
daily with approximate regularity. Remembering that the trigonometric functions model
repetitive behavior, you place a meter stick in the water to measure water height every hour
between 6:00 AM to midnight. At low tide the height of the water is 0 centimeters and at high
tide the height is 80 centimeters. Determine a reasonable defining equation for this function and
explain how you determined your answer.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
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Trigonometry - E
Answers/Rubrics
High Complexity
sec(tan −1 (x)) =
A. x
B. csc x
C.
1 + x2
D.
1 − x2
E.
sin x
cos 2 x
Answer: C
Short Answer/Extended Response
You rent a cottage on the ocean for a week one summer and notice that the tide comes in twice
daily with approximate regularity. Remembering that the trigonometric functions model
repetitive behavior, you place a meter stick in the water to measure water height every hour
between 6:00 AM to midnight. At low tide the height of the water is 0 centimeters and at high
tide the height is 80 centimeters. Determine a reasonable defining equation for this function
and explain how you determined your answer.
Student Answers Will Vary. A sample solution is:
1
(80 − 0) = 40.
2
−c
−π
Displacement: 3 =
, therefore c =
.
2
b
π⎞
⎛π
y = 40 sin ⎜ x − ⎟ + 40
2⎠
⎝6
Amplitude: a =
Period:
2π
π
= 12 , therefore b = .
b
6
Vertical Shift = d = 40.
A 2 point response correctly determines all information asked for.
A 1 point response gets at least 2 pieces of the question correct but shows a lack of
understanding for the other two.
A 0 point response shows no mathematical understanding of the problem.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
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Trigonometry - F
Which statement is always true for the inverse of every trigonometric function?
A. The inverse of every trigonometric function has a range of [-1, 1].
B. The inverse of every trigonometric function passes the vertical line test.
C. The inverse of every trigonometric function has a domain of [0, 2π].
D. The inverse of every trigonometric function is not a function.
Which graph represent the equation y = cos-1x?
A.
-6.28319
B.
4
C.
4
D.
4
4
3
3
3
3
2
2
2
2
1
1
1
-3.14159
3.14159
6.28319 -6.28319
-3.14159
3.14159
6.28319 -6.28319
-3.14159
1
3.14159
6.28319 -6.28319
-3.14159
3.14159
-1
-1
-1
-1
-2
-2
-2
-2
-3
-3
-3
-3
-4
-4
-4
-4
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 27 of 163
6.28319
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PRACTICE ASSESSMENT ITEMS
Trigonometry - F
Answers/Rubrics
Low Complexity
Which statement is always true for the inverse of every trigonometric function?
A. The inverse of every trigonometric function has a range of [-1, 1].
B. The inverse of every trigonometric function passes the vertical line test.
C. The inverse of every trigonometric function has a domain of [0, 2π].
D. The inverse of every trigonometric function is not a function.
Answer: D
Moderate Complexity
Which graph represent the equation y = cos-1x?
4
A.
-6.28319
4
B.
3
4
C.
3
4
D.
3
3
2
2
2
2
1
1
1
1
-3.14159
3.14159
6.28319 -6.28319
-3.14159
3.14159
6.28319 -6.28319
-3.14159
3.14159
6.28319 -6.28319
-3.14159
3.14159
-1
-1
-1
-1
-2
-2
-2
-2
-3
-3
-3
-3
-4
-4
-4
-4
6.28319
Answer: C
PreCalculus Standard 4 and 5
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PRACTICE ASSESSMENT ITEMS
Trigonometry - F
Suppose that the average monthly low temperatures (rounded to the nearest degree) for a small
town are shown in the table.
Month 1
2
3
4
5
6
7
8
9
10
11
12
13
Temp 59
63.4 65
63.4 59
53
47
42.6 41
42.6 47
53
59
(oF)
Model this data using f(x) = a cos(b(x-c)) + d
A. f ( x ) = 16cos
( ( x − 3))+ 53
B. f ( x ) = 12cos
( ( x − 3))+ 53
C. f ( x ) = 16cos
( x )+ 53
D. f ( x ) = 16cos
( ( x + 3))+ 53
π
6
π
6
π
6
π
6
Tides go up and down in a 12.2-hour period. The average depth of a certain river is 14m and
ranges from 9 to 19m. The variation can be approximated by a sine curve. Write an equation
that gives the approximate variation y, if x is the number of hours after midnight with high tide
occurring at 8 am. Justify each part of your answer using relevant terminology.
PreCalculus Standard 4 and 5
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PRACTICE ASSESSMENT ITEMS
Trigonometry - F
Answers/Rubrics
High Complexity
Suppose that the average monthly low temperatures (rounded to the nearest degree) for a small
town are shown in the table.
Month 1
2
3
4
5
6
7
8
9
10
11
12
13
Temp 59
63.4 65
63.4 59
53
47
42.6 41
42.6 47
53
59
(oF)
Model this data using f(x) = a cos(b(x-c)) + d
A. f ( x ) = 16cos
( ( x − 3))+ 53
B. f ( x ) = 12cos
( ( x − 3))+ 53
C. f ( x ) = 16cos
( x )+ 53
D. f ( x ) = 16cos
( ( x + 3))+ 53
π
6
π
6
π
6
π
6
Answer: A
PreCalculus Standard 4 and 5
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PRACTICE ASSESSMENT ITEMS
Trigonometry - F
Answers/Rubrics
Short Answer/Extended Response
Tides go up and down in a 12.2-hour period. The average depth of a certain river is 14m and
ranges from 9 to 19m. The variation can be approximated by a sine curve. Write an equation
that gives the approximate variation y, if x is the number of hours after midnight with high tide
occurring at 8 am. Justify each part of your answer using relevant terminology.
Answer:
2π
π
, or
.
12.2
6.1
The average depth of 14 gives our vertical shift of 14.
The starting point being hour 8 means that 8 must be subtracted from x in the equation
(phase shift of 8 hours to the left from the parent graph)
Half the difference between our maximum, 19m, and our minimum, 9 m, gives us our
amplitude of 5.
The period is 12.2, so the coefficient for x would be
⎛ π
y = 5sin ⎜
( x − 8 ) ⎞⎟ + 14
⎝ 6.1
⎠
A 4 point response includes proper explanations for the period, vertical shift, period, phase
shift, and amplitude, and puts them together into the equation given above.
A 3 point response has one mistake in one of the four properties described above but shows
a proper understanding of the other three.
A 2 point response has flaws in two of the four modifications to the parent function, but
properly represents the other two in the equation.
A 1 point response shows understanding of one aspect of the data in terms of modifying the
parent function.
A 0 point response shows no mathematical understanding of the problem.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
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4
and
5
24
sin(2 x) = ,
25
12
sin(2 x) = ,
25
8
sin(2 x) = ,
5
16
sin(2 x) = ,
25
If sin( x) =
A.
B.
C.
D.
Trigonometry - G
3
cos( x) = , then sin(2x) and cos(2x) would be:
5
7
cos(2 x) = −
25
1
cos(2 x) = −
5
6
cos(2 x) =
5
9
cos(2 x) =
25
Which one of the following is equivalent to tan θ + sec θ ?
A. sin θ + cos θ
B. tan θ + csc θ
C.
sin θ + 1
cos θ
D.
cos θ
1 + sin θ
E. cos θ − cot θ
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PRACTICE ASSESSMENT ITEMS
Trigonometry - G
Answers/Rubrics
Low Complexity
4
and
5
24
sin(2 x) = ,
25
12
sin(2 x) = ,
25
8
sin(2 x) = ,
5
16
sin(2 x) = ,
25
If sin( x) =
A.
B.
C.
D.
3
cos( x) = , then sin(2x) and cos(2x) would be:
5
7
cos(2 x) =
25
1
cos(2 x) = −
5
6
cos(2 x) =
5
9
cos(2 x) =
25
Answer: A
Moderate Complexity
Which one of the following is equivalent to tan θ + sec θ ?
A. sin θ + cos θ
B. tan θ + csc θ
C.
sin θ + 1
cos θ
D.
cos θ
1 + sin θ
E. cos θ − cot θ
Answer: C
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
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PRACTICE ASSESSMENT ITEMS
Which of the following gives the expression
Trigonometry - G
cos x
cos x
−
in completely simplified form?
1 + cos x 1 − cos x
A. 2 csc2x
B. -2cot2x
C. -2tan2x
D. 2sec2x
Prove the identity: (cos x)(tan x + sin x cot x) = sin x + cos 2 x
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PRACTICE ASSESSMENT ITEMS
Trigonometry - G
Answers/Rubrics
High Complexity
Which of the following gives the expression
cos x
cos x
−
in completely simplified form?
1 + cos x 1 − cos x
A. 2 csc2x
B. -2cot2x
C. -2tan2x
D. 2sec2x
Answer: B
Short Answer/Extended Response
Prove the identity: (cos x)(tan x + sin x cot x) = sin x + cos 2 x
Answers Will Vary.
A sample solution is:
cos x ⎞
⎛ sin x
(cos x ) ⎜
+ sin x •
= sin x + cos 2 x
⎟
sin x ⎠
⎝ cos x
⎛ sin x
⎞
(cos x ) ⎜
+ cos x ⎟ = sin x + cos 2 x
⎝ cos x
⎠
2
sin x + cos x = sin x + cos 2 x
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
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PRACTICE ASSESSMENT ITEMS
Trigonometry - H
Two boats starting at the same place and time, speed away along courses that form a 150º angle.
If one boat travels at 54 miles per hour and the other boat travels at 30 miles per hour, determine
how far apart the boats are after 20 minutes?
A. 15.62 mi
B. 18.43 mi
C. 24.08 mi
D. 27.13 mi
E. 579.88 mi
Given a triangle with the following information provided: a = 38, b = 19, C = 122º, which of the
following would we use first in order to solve the triangle:
A. The Pythagorean Theorem
B. The Law of Sines
C. The Law of Cosines
D. The Quadratic Formula
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
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PRACTICE ASSESSMENT ITEMS
Trigonometry - H
Answers/Rubrics
Low Complexity
Two boats starting at the same place and time, speed away along courses that form a 150º angle.
If one boat travels at 54 miles per hour and the other boat travels at 30 miles per hour, determine
how far apart the boats are after 20 minutes?
A. 15.62 mi
B. 18.43 mi
C. 24.08 mi
D. 27.13 mi
E. 579.88 mi
Answer: D
Moderate Complexity
Given a triangle with the following information provided: a = 38, b = 19, C = 122º, which of the
following would we use first in order to solve the triangle:
A. The Pythagorean Theorem
B. The Law of Sines
C. The Law of Cosines
D. The Quadratic Formula
Answer: C
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
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PRACTICE ASSESSMENT ITEMS
Trigonometry - H
Given the following information about a triangle:
a = 29, b = 25, c = 10, and the measure of angle B = 93o
Which one of the following is true?
A. Side-lengths a, b, and c don’t satisfy the Triangle Inequality.
B. We can use the Law of Sines to determine angles A or C.
C. We cannot have a triangle since the longest side is not opposite the largest angle.
D. The three side-lengths form a Pythagorean triple.
Given a triangle where a = 55, c = 80, and A = 35º, find two possible values for angle C.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
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PRACTICE ASSESSMENT ITEMS
Trigonometry - H
Answers/Rubrics
High Complexity
Given the following information about a triangle:
a = 29, b = 25, c = 10, and the measure of angle B = 93o
Which one of the following is true?
A. Side-lengths a, b, and c don’t satisfy the Triangle Inequality.
B. We can use the Law of Sines to determine angles A or C.
C. We cannot have a triangle since the longest side is not opposite the largest angle.
D. The three side-lengths form a Pythagorean triple.
Answer: C
Short Answer/Extended Response
Given a triangle where a = 55, c = 80, and A = 35º, find two possible values for angle C.
sin 35º sin C
=
55
80
⎛ 80sin 35° ⎞
C = sin −1 ⎜
⎟
55
⎝
⎠
=56.54º,123.46º
A 4-point answer clearly shows correct use of the Law of Sines to get correct values.
A 3-point answer shows a correct solution for the 1st angle but makes a minor computation
error in determining the second angle.
A 2-point answer shows correct set-up and work to find only the first angle.
A 1-point answer sets up the proper Law of Sines proportion but fails to get a single correct
answer.
A 0-point answer shows no understanding of the problem.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
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PRACTICE ASSESSMENT ITEMS
Trigonometry - I
A ramp leading to a freeway overpass is 540 ft long and rises 38 ft. What is the average angle of
inclination of the ramp to the nearest tenth of a degree?
A. 0.07o
B. 4.0o
C. 33.3o
D. 96.0o
To approximate the height of a radio tower, Mark counts off 72 feet from the base of the tower
and then measures the angle of elevation from the ground to the top of the tower from that point
to be 40o. Approximately how tall is the tower?
A. 46.3 ft
B. 55.2 ft
C. 60.4 ft
D. 72 ft
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
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PRACTICE ASSESSMENT ITEMS
Trigonometry - I
Answers/Rubrics
Low Complexity
A ramp leading to a freeway overpass is 540 ft long and rises 38 ft. What is the average angle of
inclination of the ramp to the nearest tenth of a degree?
A. 0.07o
B. 4.0o
C. 33.3o
D. 96.0o
Answer: B
Moderate Complexity
To approximate the height of a radio tower, Mark counts off 72 feet from the base of the tower
and then measures the angle of elevation from the ground to the top of the tower from that point
to be 40o. Approximately how tall is the tower?
A. 46.3 ft
B. 55.2 ft
C. 60.4 ft
D. 72 ft
Answer: C
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
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PRACTICE ASSESSMENT ITEMS
Trigonometry - I
A boat leaves harbor and travels at 30 knots on a bearing of 83o. After four hours, it changes
course to a bearing of 138o and continues at 30 knots for three hours. After the entire seven-hour
trip, how far is the boat from its starting point?
A. 93.4 nautical miles
B. 140.5 nautical miles
C. 186.8 nautical miles
D. 373.6 nautical miles
From the top of a 225 ft. building, a man observes a car moving towards the building. If the
angle of depression from the man to the car changes from 18º to 39º during the period of
observation, determine how far the car travels. Include a diagram to back up your work.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
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PRACTICE ASSESSMENT ITEMS
Trigonometry - I
Answers/Rubrics
High Complexity
A boat leaves harbor and travels at 30 knots on a bearing of 83o. After four hours, it changes
course to a bearing of 138o and continues at 30 knots for three hours. After the entire seven-hour
trip, how far is the boat from its starting point?
A. 93.4 nautical miles
B. 140.5 nautical miles
C. 186.8 nautical miles
D. 373.6 nautical miles
Answer: C
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
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PRACTICE ASSESSMENT ITEMS
Trigonometry - I
Answers/Rubrics
Short Answer/Extended Response
From the top of a 225 ft. building, a man observes a car moving towards the building. If the
angle of depression from the man to the car changes from 18º to 39º during the period of
observation, determine how far the car travels. Include a diagram to back up your work.
From the top of a 225 ft. building, a man observes a car moving towards the building. If the
angle of depression from the man to the car changes from 18º to 39º during the period of
observation, determine how far the car travels. Include a diagram to back up your work.
18o
21o
39o
225 ft
time 1
time 2
Time 2
Time1
51o
72o
225 ft
225 ft
y
x
x
225
x = 225 tan 72° = 692.41 ft
y
225
y = 225 ⋅ tan 51° = 277.85 ft
tan 72° =
note: you could also solve tan18° =
tan 51° =
225
x
note: you could also solve tan 39° =
225
y
Change in Distance = 692.41 ft – 277.85 ft = 414.56 ft.
A 4-point answer shows both tangent problems set-up and solved properly and gets the
correct difference.
A 3-point answer gets both the setups and substitutions correct, but makes one mistake in
the calculations somewhere along the way.
A 2-point answer sets up the diagram(s) properly and shows the need to find a difference
but is unable to use an appropriate trig ratio to determine these distances.
A 1-point answer attempts to set up the diagram(s) properly so as to get the necessary angle
measures, but is missing crucial pieces.
A 0-point answer shows no mathematical understanding of the problem.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
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PRACTICE ASSESSMENT ITEMS
Trigonometry - J
What is the factored form for the expression 1 – cos3x?
A. (1 – cos x) (1 + cos x + cos2x)
B. (1 – cos x)3
C. (1 – cos x) (sin x + cos2x)
D. (1 – cos x) (1 – 2 cos x + cos 2x)
Find all solutions to the equation sin 2 x + 2sin x + 1 = 0 in the interval[0, 2π ] .
A. x =
B. x =
π 3π
2
π
C. x =
D. x =
2
,
2
,π,
3π
2
π
2
3π
2
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
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PRACTICE ASSESSMENT ITEMS
Trigonometry - J
Answers/Rubrics
Low Complexity
What is the factored form for the expression 1 – cos3x?
A. (1 – cos x) (1 + cos x + cos2x)
B. (1 – cos x)3
C. (1 – cos x) (sin x + cos2x)
D. (1 – cos x) (1 – 2 cos x + cos 2x)
Answer: A
Moderate Complexity
Find all solutions to the equation sin 2 x + 2sin x + 1 = 0 in the interval[0, 2π ] .
A. x =
B. x =
π 3π
2
π
C. x =
D. x =
2
,
2
,π,
3π
2
π
2
3π
2
Answer: D
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 46 of 163
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PRACTICE ASSESSMENT ITEMS
Trigonometry - J
Which values of t on [−2π , 2π ] satisfy cos t < − 12 ?
⎛ 4π 2π
A. ⎜ −
,−
3
⎝ 3
⎞ ⎛ 2π 4π ⎞
,
⎟∪⎜
⎟
⎠ ⎝ 3 3 ⎠
⎡ 4π 2π ⎤ ⎡ 2π 4π ⎤
B. ⎢ −
,− ⎥ ∪ ⎢ , ⎥
3 ⎦ ⎣ 3 3 ⎦
⎣ 3
⎛ 4π 4π ⎞
C. ⎜ −
,
⎟
⎝ 3 3 ⎠
⎡ 2π 2π ⎤
D. ⎢ −
,
⎣ 3 3 ⎥⎦
⎛ x ⎞ 1 + cos x
Use a graphing calculator to find all solutions to the equation tan ⎜ ⎟ =
in the
⎝ 2 ⎠ 1 − cos x
interval[0, 2π ] . Include a sketch of your graphs as well as an explanation of how you reached
your answer.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
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PRACTICE ASSESSMENT ITEMS
Trigonometry - J
Answers/Rubrics
High Complexity
Which values of t on [−2π , 2π ] satisfy cos t < − 12 ?
⎛ 4π 2π ⎞ ⎛ 2π 4π ⎞
A. ⎜ −
,−
,
⎟∪⎜
⎟
3 ⎠ ⎝ 3 3 ⎠
⎝ 3
⎡ 4π 2π ⎤ ⎡ 2π 4π ⎤
B. ⎢ −
,− ⎥ ∪ ⎢ , ⎥
3 ⎦ ⎣ 3 3 ⎦
⎣ 3
⎛ 4π 4π ⎞
C. ⎜ −
,
⎟
⎝ 3 3 ⎠
⎡ 2π 2π ⎤
D. ⎢ −
,
⎣ 3 3 ⎥⎦
Answer: A
Short Answer/Extended Response
⎛ x ⎞ 1 + cos x
Use a graphing calculator to find all solutions to the equation tan ⎜ ⎟ =
in the
⎝ 2 ⎠ 1 − cos x
interval[0, 2π ] . Include a sketch of your graphs as well as an explanation of how you reached
your answer.
(1.5708, 1.)
1
1.5708
3.14159
4.71239
6.28319
A 2-point response gives accurate sketches of the two graphs for the given interval and
indicates the correct point where the two intersect.
A 1-point response has one or more errors in one or both of the two graphs but uses
otherwise valid reasoning to create an answer to the equation.
A 0-point response shows no mathematical understanding of the problem.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
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Teacher Introduction
Trigonometry
Periodic Functions
Periodic functions are functions where the values repeat themselves at regular intervals. There is
a c such that f(t + c) = f(t) for every t in the functions domain, where c is the period length. The
length can be found by determining the distance between two maxima (or 2 minima values). If
the periodic function is shifted horizontally by its period, in either direction, the resulting graph
will be the same as the original graph. The midline of a periodic function is a horizontal line
halfway between the minimum and maximum values. The amplitude is the vertical distance, i.e.
height difference, between the functions maximum and its midline.
For example: F(x) = x – int(x) with Δx = 0.1 should be graphed by the student. The student
should look not only at the graph, but at the table of values. This function is periodic with a
period of 1.
Sine and Cosine Functions
The sine and cosine functions should be studied using a unit circle with a radius of 1 and
centered at the origin. Each point on the circle can be located by its angle of rotation, θ, where
y = sin θ and x = cos θ. The pictures of these are included in the strategies for Learning Goal B.
Unit Circle
(0, 1)
(x, y)
(1, 0)
θ
(1, 0)
(0, -1)
To form triangles, you extend a ray from the center to the side of the circle and draw an altitude
to the x-axis. The altitude and x-axis form a right angle and the ray becomes the hypotenuse of
the right triangle, with length = 1.
The angle θ is the angle the ray makes with the x-axis. The adjacent side is on the x-axis with
length equal to the value of the x-coordinate. The opposite side is the altitude with length equal
to the value of the y-coordinate.
This can be generalized for non-unit circles of radius r: x = rcosθ and y = rsinθ.
If θ is the angle measured from the positive x-axis, and P(x, y) is the point on the circle that
intercepts the terminal ray, then a right triangle is formed with the hypotenuse as part of the
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terminal ray. By using the Pythagorean Theorem, cos2θ + sin2θ = 1 (Pythagorean Identity) can
be proved.
On the unit circle, we represent angles as rotations of a ray counter-clockwise from the positive
x-axis. The x-axis is the initial side of the angle and the ray is the terminal side of the angle. An
angle formed in this way is said to be in standard position.
Terminal Side
(0, 1)
(-1, 0)
θ
(1, 0)
(0, -1)
Initial Side
Example: Use the height of a Ferris wheel by generating the points for 0o to 360o for the
equation y = 225 + 225sinx and creating a scatterplot to illustrate this concept.
Radians
1 radian is the angle at the center of a unit circle which spans an arc of length one. Radians are
commonly used in analytical trigonometry and in calculus. The formula for Arc length is:
S = rθ .
π radians = 180o
Students will need to be familiar with changing the mode on the graphing calculator from
degrees to radians and vice versa.
Graphs of Sine and Cosine
Relate the unit circle to the graphs of the sine and cosine functions with special emphasis on the
2nd, 3rd, and 4th quadrants.
When 0° < θ < 90°; cos θ, sin θ, and tan θ have positive values because x and y are both positive
in the first quadrant.
When 90° < θ < 180°; cos θ and tan θ have negative values because x is negative in the second
quadrant. Sin θ has a positive value because y is positive in the second quadrant.
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When 180° < θ < 270°; cos θ and sin θ have negative values because x and y are negative in the
third quadrant. Tan θ has a positive value.
When 270° < θ < 360°; sin θ and tan θ have negative values because y is negative in the fourth
quadrant. Cos θ has a positive value because x is positive in the fourth quadrant.
If students have trouble remembering the exact values of the sine and cosine functions for
π
4
, and
π
3
π
6
,
, remind them of the relationships in a 30, 60, 90 and a 45, 45, 90 triangle.
π
π
3
4
o
60
2
1
30o
45o
2
1
π
45o
6
3
π
4
1
As students compare the two graphs, they should recognize that both have a period of 2π, an
amplitude of 1, a domain of (-∞, ∞) and a range of [-1, 1]. They should also note that the sine
function is odd while the cosine function is an even. Like other functions, sine and cosine
functions can be shifted horizontally, vertically, inverted, compressed, stretched, or a
combination of those shifts.
Sinusoidal Functions
Any transformation of the sine and cosine function is called a sinusoidal function.
Students will be working from the general form y = a sin b( x + c) + d where the different
variables have the following properties (note this is slightly different than what the book does):
a is called the amplitude; it is the distance from the highest point of the sine curve to the
mid-line.
2π
b does not have a name itself but is used to determine the period: p =
. The period is
b
how many times the graph completes a cycle in a 2π interval.
d is the vertical shift; it tells how far up or down the entire sine curve is shifted.
c gives the phase shift; how far left or right the graph is shifted (a positive c means shift
to the left; a negative c means shift to the right).
One of the big challenges in analyzing sinusoidal functions is to turn the given function into
something that looks like the above form so it can be analyzed easily. For example, the given
equation y = 4 + 2sin(3x − π ) is usually approached by factoring the 3 out, to give something
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like y = 4 + 2sin 3 ( x − π3 ) , which allows us to see that the period length is
is
π
3
2π
and the phase shift
3
units to the right.
One thing that will help when students learn to convert equations into this standard form is that
they can put both into their graphing calculator. If they are correct, both graphs will wind up on
top of each other.
Solving Trigonometric Equations
Remind students that it is better to remember the Pythagorean Theorem as leg2 + leg2 =
hypotenuse2 rather than a2 + b2 = c2, since there is no guarantee that c is always the hypotenuse.
There are two special right triangles. The first is a 45-45-90 triangle. The special ratio is
1:1: 2 . The second is a 30-60-90 triangle. The special ratio is 1: 3 : 2.
Trigonometry is based on similar right triangles. The sine (sin) of an angle is the ratio of the
opposite side to the hypotenuse. The cosine (cos) of an angle is the ratio of the adjacent side to
the hypotenuse. The cosine of an angle is the ratio of the adjacent side to the hypotenuse. The
tangent (tan) of an angle is the ratio of the opposite side to the adjacent side.
There are many different ways to help your students remember the sine, cosine, and tangent
functions.
•
Use the old Indian chief SOH CAH TOA. Tell a story of how a great Indian chief
was also a great mathematician. And he developed sine, cosine, and tangent to match
his name.
SOH (sin = opp / hyp)
CAH (cos = adj / hyp)
TOA (tan = opp / adj)
• The following phrase could also be used.
Some
Caught
Taking
Old
Horse
Another Horse
Oats
Away
The angle of elevation is the angle between the line of sight and the horizontal when looking up.
The angle of depression is the angle between the line of sight and the horizontal when looking
down. It is helpful to remember that the angle of elevation and the angle of depression are
alternate interior angles to each other.
Look down to person
depression
elevation
Look up to bird
Real life applications are architecture and engineering.
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Non-Right-Angle Trigonometry (Law of Sines/Law of Cosines)
We use the Law of Sines and Law of Cosines when we need to solve a triangle that is not a right
triangle. By solving, we mean determining the missing side lengths and angle measures, given a
minimum of three of these. Students are already familiar with the Pythagorean Theorem which
relates the three side-lengths of a right triangle to each other. These two new laws are also about
relating information to other information.
To use either law, we must use the convention that the three sides of a triangle are labeled a, b,
and c, and that their opposite angles are labeled A, B, and C, respectively.
The Law of Sines states that for any triangle:
sin A sin B sin C
=
=
a
b
c
What we need to know in order to use this law is one side length and its opposite angle. In other
words, we must know either A and a, B and b, or C and c. The other piece of information we are
given determines which other fraction we will pick out. For example, if we are given the values
sin B sin C
of C, c, and B, then we will set up the proportion,
, allowing us to find the length of
=
b
c
side b. We cannot use the Law of Sines if we do not have three pieces of information, and they
must meet the requirement we just mentioned. In this last example, if we were given C, c, and b,
then we would use the same proportion, but that would leave us with a value for the sin of B, and
we would still have to take the inverse sine in order to determine the angle.
There are a few situations where the Law of Sines yields unexpected results. First, consider a
case where a = 7, b = 8, and A = 100º. Attempting to use the Law of Sines to find the measure of
B will give no answer (try it yourself). This is exactly as we expect and want, since in any
triangle, the longest side must lie opposite the biggest angle. This cannot be, because there
cannot exist any angle bigger than 100º in this triangle, but the 100º is opposite the 7 which isn’t
the longest side.
Another interesting type of result is what is called the ambiguous case. Here, the Law of Sines
may give two possible (and viable) results for a missing angle.
The Law of Cosines reads:
c 2 = a 2 + b 2 − 2ab cos C
This law relates the three side-lengths to the measure of one angle.
The two most common uses of this law are:
1. To find the missing side when the given information is arranged in side angle side formation.
2. To find a missing angle when all three side-lengths are known. Here, what we are really doing
is solving the above equation as if it were in this form:
⎛ c 2 − a 2 − b2 ⎞
C = cos −1 ⎜
⎟
⎝ −2ab ⎠
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One important property of the Law of Cosines that should be pointed out to students is that when
angle C is 90º (when we have a right triangle), then the equation degenerates into the familiar
Pythagorean Theorem.
In solving a triangle, you will not have to use the Law of Cosines more than once. Once you
know all three side-lengths and an angle, you can use the Law of Sines to determine another
angle.
Also, don’t forget such geometry basics as the fact that the angle measures of a triangle must add
up to 180º. If you have 2 of them, it’s a simple computation to get the third.
Trigonometric Identities
The section on trigonometric identities should emphasize their use in verifying trigonometric
formulas, such as the double angle formula and the reductions rules, and their application in
solving trigonometric equations. Although students should have some experience in verifying
identities algebraically, this is not the focus of the section. Students should know that, although
identities can be verified graphically, this does not constitute a proof.
Trigonometric Regression
Trigonometric Regression is expected to be done using technology. Please refer to your
Graphing Calculator Resource Manual.
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TEACHING STRATEGIES/ACTIVITIES
Vocabulary: Mathematical model, domain, range, function, continuous, jump
discontinuity, increasing, decreasing, constant, lower bound, upper bound,
boundedness, local extrema, absolute extrema, odd function, even function,
asymptote, sine function, cosine function, tangent function, secant function, cosecant
function, inverse sine function, inverse cosine function, inverse tangent function,
inverse, relation, transformation, translation, reflection, stretch, shrink, regression,
angle, central angle, degree, minute, second, bearing, radian, arc length, sector,
similar, standard position, sine, cosine, tangent, cosecant, secant, cotangent, solving a
triangle, right triangle, vertex, positive angle, negative angle, terminal ray, initial ray,
standard position, coterminal angles, reference angle, quadrantal angle, periodic
function, period, sinusoid, amplitude, frequency, phase shift, vertical shift, damped
oscillation, damping factor, angle of elevation, angle of depression, simple harmonic
motion, trigonometric identity, Pythagorean identities, trigonometric equation,
double-angle identity, half-angle identity, Law of Sines, Law of Cosines,
semiperimeter, Heron’s Formula.
Core:
Learning Goal A: Uses radian and degree angle measure to solve problems and perform
conversions as needed.
1. A blank “Unit Circle” (included in this Curriculum Guide) and a “Circle With Radian
Measure” (included in this Curriculum Guide) have been provided for the teacher to use at
his/her discretion.
2. “The Radian Snow.....” (included in this Curriculum Guide) can also be used to help students
learn the radian measurements. Students may turn it into a snowman, a snowcat, etc.
Learning Goal B: Uses the unit circle to explain the circular properties and periodic nature of
trigonometric functions and to find the trigonometric ratios of any angle.
1. Use the Unit Circle Investigation (included in this Curriculum Guide) to help the students
discover the relationships of the unit circle. Provide students with the handout, scissors,
tracing paper, and the unit circle.
2. As a way to reinforce the learning of the angle measures in radians the corresponding
coordinates on the unit circle, the students will play the game, “The Radian Walk” (included
in this Curriculum Guide).
3. A “Circle With Sine and Cosine Coordinates” (included in this Curriculum Guide) has been
provided for the teacher to use at his/her discretion.
Learning Goal C: Describes and compares the characteristics of the trigonometric functions
(with and without the use of technology) for sine, cosine, tangent, cotangent, cosecant, and
secant.
1. The “Sine Cosine Game” (included in this Curriculum Guide) can be used to reinforce the
cyclic nature of the trigonometric function. There are four different games which increase in
level of difficulty.
2. As students increase their mathematics learning level, they often learn that Mathematics is
not the dreaded subject that they have previously encountered. “The Story of Joe S____.”
(included in this Curriculum Guide) is a play on mathematical words which students may
find enjoyable.
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Learning Goal D: Determines solutions to trigonometric equations.
1. Student will use their knowledge of unit circle as a way to solve simple trigonometric
equations both graphically and with technology in “Solving Trigonometric Equations”
(included in this Curriculum Guide).
2. Have students complete the activity “Problem Solving: Trigonometric Ratios” (included in
Curriculum Guide) to apply their knowledge of trigonometric ratios.
Learning Goal E: Describes how a change in the value of any constant in a general-form
trigonometric equation such as y = a sin (b-x) + c affects the graph of the equation.
1. Reinforce the properties of the unit circle and introduce students to periodic functions using
the “Pasta Waves” activity (included in this Curriculum Guide). Students will need the
handout “Pasta Waves”, a copy of the “Unit Circle” (included in this Curriculum Guide), a
copy of the “S(x) Graph” (included in this Curriculum Guide), a copy of the “C(x) Graph”
(included in this Curriculum Guide), a copy of the “Observations and Predictions” sheet
(included in this Curriculum Guide), spaghetti, and glue.
2. In “Getting in Shape” (included in this Curriculum Guide), the students will explore the
concept of amplitude.
3. “Speeding Up” (included in this Curriculum Guide), is an extension to “Getting in Shape”
which deals with changing the period of a function.
4. The third activity in this series is “Running With a Friend” (included in the Curriculum
Guide. This activity addresses the period of a function.
5. Students will used the “Sine Curve Equation” (included in this Curriculum Guide) to show
their understanding of the amplitude, period, and phase shift of the sine curve.
Learning Goal F: Represents the inverse of a trigonometric function symbolically and
graphically
1. Once the student understands the unit circle and an inverse function, they should complete
“Finding the Inverse Without a Calculator” (included in this Curriculum Guide).
2. In “Match the Inverse” (included in this Curriculum Guide), the student will match the graph
of the inverse with its equation.
Learning Goal G: Creates a scatterplot of bivariate data, identify a trigonometric function to
model the data., and use that model to identify patterns and make predictions.
1. In “Daylight Hours” (included in this Curriculum Guide), students will explore the use of
real life data to create a trigonometric equation to model the data.
Learning Goal H: Derives and applies the basic trigonometric identities; i.e. angle addition,
angle subtraction, and double-angle.
1. Students will play the game “Tic-Tac Trig – 4 in a Row” (included in this Curriculum Guide)
as a way to solidify their understanding of the simplification of the multiplication of
trigonometric functions.
Learning Goal I: Uses trigonometric relationships to determine lengths and angle measures;
i.e., Law of Sines and Law of Cosines.
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1. In the “Sine Spaghetti Investigation” (included in this Curriculum Guide), students will
investigate the Law of Sines.
2. Use the “Law of Sines: Real Life Applications” (included in this Curriculum Guide) to show
some applications of the Law of Sines.
3. “Sines Just Aren’t Enough” will give the students is using the Law of Cosines when the Law
of Sines can not be used.
Learning Goal J: Models and solve problems using trigonometry.
1. The Textbook contains a wide selection of problems throughout this entire section. In “A
Variety of Problems” (included in this Curriculum Guide), the students will use various
trigonometric functions and properties to find the solutions.
2. “The Big Wheel” (included in this Curriculum Guide) is another application problem that the
students should solve.
Reteach:
1. Have students practice finding the missing side or angle of a right triangle with the “Find the
Missing Side or Angle” activity (included in this Curriculum Guide). Students will need a
copy of the activity and a scientific or graphing calculator.
2. Have students create their own unit circle to have as a reference with the “Simple Circle”
activity (included in this Curriculum Guide) or the “Paper Plate” activity (included in this
Curriculum Guide).
3. Students practice locating the special right triangles on the unit circle by completing the
“Crop Circles” activity (included in this Curriculum Guide). Have students discuss with a
small group or the whole class their strategies for determining how much of each circle to
shade.
4. Have students complete the activity “Memory Match” (included in this Curriculum Guide) to
reinforce right triangle terminology. Students can work in groups of three or four. First
place all cards facedown and then have each student take turns drawing two cards. If the two
cards drawn go together as a pair the student will keep it as a match. Students take turns
drawing. The student with the most pairs or matches wins. Students will need a scientific
calculator.
Extension:
1. “When Good Trig Goes Bad” requires the student to use many of the ideas to prove
trigonometric ideas.
RESOURCES
Learning Goal A:
Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 351-371
Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004):
Resource Manual pp. 77-80
Learning Goal B:
Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 372-385
Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004):
Resource Manual pp. 81-82
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Learning Goal C:
Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 372-385
Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004):
Resource Manual pp. 81-82
Learning Goal D:
Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 426-437
Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004):
Resource Manual pp. 91-92
Learning Goal E:
Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 386-415
Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004):
Resource Manual pp. 83-88
Learning Goal F:
Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 416-425
Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004):
Resource Manual pp. 89-90
Learning Goal G:
Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 426-442
Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004):
Resource Manual pp. 91-92
Learning Goal H:
Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 443-477
Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004):
Resource Manual pp. 99-106
Learning Goal I:
Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 478-496
Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004):
Resource Manual pp. 107-110
Learning Goal J:
Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 351-496
Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004):
Resource Manual pp. 77-115
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
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Unit Circle
Trigonometry - A
Name
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
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Trigonometry - A
Circle With Radian Measure
Name
π
3π
4
2π
3
2
π
3
π
4
5π
6
π
6
π
0
2π
7π
6
11π
6
5π
4
4π
3
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
3π
2
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5π
3
7π
4
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The Radian Snow.....
Trigonometry - A
Name
The snow..... picture has a unit circle for its base. You are to label the sixteen points on the unit
circle with the radian measure inside the circle and the coordinates of the points outside the
circle. Then color and decorate the snow.....
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
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The Radian Snow.....
Answer Key
Trigonometry - A
These are some examples of student work.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
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Unit Circle Investigation
Trigonometry - B
Name
Assume that the radius of the outlined circle is 1 unit.
1. Label the ordered pairs on the x and y-axes. Draw a perpendicular line segment from the
π
point on the outlined circle where the ray crosses the outlined circle. Trace the right triangle
6
π
defined by the x-axis, the perpendicular segment, and the ray. Cut it out. Label the lengths of
6
the sides and the angles of the triangle.
π
ray crosses
4
the circle. Place the 30º-60º triangle on its outline. Label the ordered pair where the vertex
π
π
crosses the circle. What is the cos (30º)? What is the sin (30º)? How do you know?
6
6
2. Make another triangle by dropping a perpendicular from the point where the
3. Align the triangle between the 60º angle and the x-axis. Label the ordered pair where the
π
π
vertex crosses the circle. What is the cos (60º)? What is the sin (60º)? How are these
3
3
π
related to the sine and cosine of (30º). Explain this relationship in terms of the right triangles.
6
2π
ray. Label the ordered pair
3
5π
where the ray crosses the circle. Move the triangle so that it is aligned under the
ray. Label
6
the ordered pair where the ray crosses the circle. These triangles are called the reference
triangles and allow you to find the trigonometric function of angles outside the first quadrant.
What are the sine and cosine of these angles?
4. Align the right triangle along the negative x-axis under the
5. How are the sine and cosine of angles in the second quadrant related to the angles in the first
quadrant?
6. Align the triangle with the x-axis in the 3rd and 4th quadrants and find the sine and cosine of
each of the related angles in these quadrants.
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7. Find the angles and ordered pairs associated with
π
around the circle.
4
Trigonometry - B
π
from the calculator. Give the degree measure of that angle.
12
Mark all the angles whose sines and cosines can be determined from this. Give their measures in
degrees and radians. Give the sine and cosine of each. Check your answers with the calculator.
8. Find the sine and cosine of
9. Pick another angle θ between 0º and 45 º. Use your calculator to evaluate sin θ and cos θ. On
your circle, draw in the arc that approximates this measure. Give the measures of all the angles
whose sines and cosines can be determined from this, and give their sines and cosines. Check
your answers with the calculator.
10. In what quadrants are sine and cosine positive and negative?
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
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Trigonometry - B
11. Suppose you know the measure of an angle θ and 0< θ <45º. Draw in all the other angles on
the circle whose sine and cosine can be determined from the cos θ and the sin θ. Write
expressions for each of these angles in terms of θ. Write the ordered pairs for each of these
angles in terms of cos θ and sin θ.
(cos θ, sin θ)
θ
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
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Unit Circle
Trigonometry - B
Name
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
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Unit Circle Investigation
Answer Key
Trigonometry - B
Assume that the radius of the outlined circle is 1 unit.
1. Label the ordered pairs on the x and y-axes. Draw a perpendicular line segment from the
π
point on the outlined circle where the ray crosses the outlined circle. Trace the right triangle
6
π
defined by the x-axis, the perpendicular segment, and the ray. Cut it out. Label the lengths of
6
the sides and the angles of the triangle.
π
ray crosses
4
the circle. Place the 30º-60º triangle on its outline. Label the ordered pair where the vertex
π
π
crosses the circle. What is the cos (30º)? What is the sin (30º)? How do you know?
6
6
3
π
π
1
cos
=
and sin
=
2
6
6
2
They are the x and y coordinates of the terminal ray.
2. Make another triangle by dropping a perpendicular from the point where the
3. Align the triangle between the 60º angle and the x-axis. Label the ordered pair where the
π
π
vertex crosses the circle. What is the cos (60º)? What is the sin (60º)? How are these
3
3
π
related to the sine and cosine of (30º). Explain this relationship in terms of the right triangles.
6
π 1
π
3
cos = and sin =
3
2
3 2
The values of the functions are interchanged. This relationship can be seen in terms of a
30-60-90 triangle.
2π
4. Align the right triangle along the negative x-axis under the
ray. Label the ordered pair
3
5π
where the ray crosses the circle. Move the triangle so that it is aligned under the
ray. Label
6
the ordered pair where the ray crosses the circle. These triangles are called the reference
triangles and allow you to find the trigonometric function of angles outside the first quadrant.
What are the sine and cosine of these angles?
2π
3
2π
1
5π 1
5π
3
sin
, cos
= − , sin
= , cos
=−
=
3
2
6
2
6
3
2
2
5. How are the sine and cosine of angles in the second quadrant related to the angles in the first
quadrant?
Sin values are the same as the ones in the first quadrant.
Cosine values are the opposite of the ones in the first quadrant.
6. Align the triangle with the x-axis in the 3rd and 4th quadrants and find the sine and cosine of
each of the related angles in these quadrants.
See student answers on the unit circle.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 67 of 163
Columbus Public Schools 1/5/06
7. Find the angles and ordered pairs associated with
π
around the circle.
4
Trigonometry - B
See student answers on the unit circle.
π
from the calculator. Give the degree measure of that angle.
12
Mark all the angles whose sines and cosines can be determined from this. Give their measures in
degrees and radians. Give the sine and cosine of each. Check your answers with the calculator.
8. Find the sine and cosine of
π
12
= 15o,
5π
5π
7π
= 0.9659, cos
= 0.2588, sin
=0.9659
12
12
12
12
12
7π
11π
11π
13π
13π
= -0.2588, sin
= 0.2588, cos
= -0.9659, sin
= -0.2588, cos
= -0.9659
cos
12
12
12
12
12
17π
17π
19π
19π
23π
= -0.9659, cos
= -0.2588, sin
= -0.9659, cos
= 0.2588, sin
= -0.2588
sin
12
12
12
12
12
23π
= 0.9659
cos
12
sin
π
5π
7π
11π
13π
17π
19π
23π
= 75o,
= 105o,
= 165o,
= 195o,
= 255o,
= 285o,
= 345o
12
12
12
12
12
12
12
= 0.2588, cos
π
= 0.9659, sin
9. Pick another angle θ between 0º and 45 º. Use your calculator to evaluate sin θ and cos θ. On
your circle, draw in the arc that approximates this measure. Give the measures of all the angles
whose sines and cosines can be determined from this, and give their sines and cosines. Check
your answers with the calculator.
Student answers will vary based on the value of θ that they choose. Check the students unit
circle.
10. In what quadrants are sine and cosine positive and negative?
Sine is positive in the first and second quadrants.
Sine is negative in the third and fourth quadrants.
Cosine is positive in the first and fourth quadrants.
Cosine is negative in the second and third quadrants.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 68 of 163
Columbus Public Schools 1/5/06
Trigonometry - B
11. Suppose you know the measure of an angle θ and 0< θ <45º. Draw in all the other angles on
the circle whose sine and cosine can be determined from the cos θ and the sin θ. Write
expressions for each of these angles in terms of θ. Write the ordered pairs for each of these
angles in terms of cos θ and sin θ.
(-sinθ, cosθ)
(sin θ, cos θ)
(cos θ, sin θ)
(-cos θ,sin θ)
θ
(-cos θ, -sin θ)
(-sin θ, -cos θ)
(cos θ, -sin θ)
(sin θ, -cos θ)
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 69 of 163
Columbus Public Schools 1/5/06
The Radian Walk
Trigonometry - B
The teacher will make a unit circle with diameter about 12 feet on the floor in the middle of the
classroom. Make the x- and y-axes to mark the 90 degree angles. Label them so that students
know the location of 0 radians. Mark the 30 degree, 45 degree, and 60 degree angles in each
quadrant.
Place a spinner at the origin. Turn on a tape or CD player and have the students walk around the
circle until the music stops. At that point each student must be on one of the marked angles of
the unit circle. Spin the spinner. The spinner indicates the student who must name her or his
coordinates and place on the unit circle.
If that student makes a mistake, she or he is eliminated from the game. The person remaining on
the circle after all others have dropped out is the winner. If your class size is larger than fifteen
or sixteen students, you may want to use two unit circles.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 70 of 163
Columbus Public Schools 1/5/06
Trigonometry - B
Circle With Sine and Cosine Coordinates
Name
(0, 1)
⎛ 1 3⎞
⎜⎜ − ,
⎟⎟
⎝ 2 2 ⎠
⎛
2 2⎞
,
⎜⎜ −
⎟⎟
⎝ 2 2 ⎠
⎛1 3⎞
⎜⎜ ,
⎟⎟
⎝2 2 ⎠
⎛ 2 2⎞
,
⎜⎜
⎟⎟
⎝ 2 2 ⎠
⎛
3 1⎞
, ⎟⎟
⎜⎜ −
2
2⎠
⎝
⎛ 3 1⎞
, ⎟⎟
⎜⎜
2
2⎠
⎝
(-1, 0)
(1, 0)
⎛
3 1⎞
, − ⎟⎟
⎜⎜ −
2
2⎠
⎝
⎛
2
2⎞
,−
⎜⎜ −
⎟
2 ⎟⎠
⎝ 2
⎛ 1
3⎞
⎜⎜ − , −
⎟
2 ⎟⎠
⎝ 2
⎛
⎜⎜
⎝
⎛1
3⎞
⎜⎜ , −
⎟
2 ⎟⎠
⎝2
⎛ 3 1⎞
, − ⎟⎟
⎜⎜
2
2⎠
⎝
2
2⎞
,−
⎟
2
2 ⎟⎠
(0, -1)
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 71 of 163
Columbus Public Schools 1/5/06
Sine Cosine Game
Trigonometry - C
After students have been introduced to the sine and cosine functions, begin a class period with
this game. Divide the class into pairs and distribute single sheets of paper to each pair. On the
sheet are two columns. The left-hand column contains expression involving either the sine or
cosine function. In the right-hand column are actual decimal approximations of the expression in
the left-hand column, but in a random order. The values are chosen such that only one of the
five decimal values is plausible for each function.
At the signal from the teacher, the pairs of students have 3 minutes to match each expression in
the left-hand column with its proper decimal representation in the right-hand column.
Calculators are not permitted. Any pair is a winner if they obtain all five correct matches.
After the introduction of all six trigonometric functions, the game can be played with mixing the
inverse functions or mixing all of the functions.
Four different matching games are provided for your use.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 72 of 163
Columbus Public Schools 1/5/06
Sine Cosine Game
Trigonometry - C
Names:
Game # 1
_______
cos 35
A.
-0.9961
_______
cos 98
B.
0.8192
_______
cos 4
C.
0.0349
_______
cos 175
D.
-0.1391
_______
cos 272
E.
0.9976
------------------------------------------------------------------Names:
Game # 2
_______
sin 224
A.
0.1908
_______
sin 52
B.
0.7888
_______
sin 290
C.
-0.0871
_______
sin 355
D.
-0.6946
_______
sin 11
E.
-0.9396
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 73 of 163
Columbus Public Schools 1/5/06
Trigonometry - C
Names:
Game # 3
_______
cos 7
A.
-1.2360
_______
sec 98
B.
1.0456
_______
cos 304
C.
0.5591
_______
sec 17
D.
-7.1852
_______
sec 144
E.
0.9925
------------------------------------------------------------------Names:
Game # 4
_______
csc 130
A.
-28.653
_______
sin 289
B.
1.3054
_______
sin 185
C.
-1.0402
_______
csc 254
D.
-0.9455
_______
csc 358
E.
-0.0871
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 74 of 163
Columbus Public Schools 1/5/06
Sine Cosine Game
Answer Key
Trigonometry - C
Names:
Game # 1
__B____
cos 35
A.
-0.9961
__D____
cos 98
B.
0.8192
__E____
cos 4
C.
0.0349
__A____
cos 175
D.
-0.1391
__C____
cos 272
E.
0.9976
------------------------------------------------------------------Names:
Game # 2
__D____
sin 224
A.
0.1908
__B____
sin 52
B.
0.7888
__E____
sin 290
C.
-0.0871
__C____
sin 355
D.
-0.6946
__A____
sin 11
E.
-0.9396
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 75 of 163
Columbus Public Schools 1/5/06
Trigonometry - C
Names:
Game # 3
__E____
cos 7
A.
-1.2360
__D____
sec 98
B.
1.0456
__C____
cos 304
C.
0.5591
__B____
sec 17
D.
-7.1852
__A____
sec 144
E.
0.9925
------------------------------------------------------------------Names:
Game # 4
__B____
csc 130
A.
-28.653
__D____
sin 289
B.
1.3054
__E____
sin 185
C.
-1.0402
__C____
csc 254
D.
-0.9455
__A____
csc 358
E.
-0.0871
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 76 of 163
Columbus Public Schools 1/5/06
Trigonometry - C
The Story of Joe S _ _ _ .
Name
Fill in the following blanks with letters to make math words.
One day Joe S _ _ _ went to call on his new neighbors who lived in the a _ _ _ _ _ _ _ house.
He was a handsome t _ _ _ _ _ _ , a confirmed bachelor. Joe was met at the door by two
sisters who had anything but c _ _ _ _ _ _ _ _
f _ _ _ _ _ _ . The first, Deca Gon, had real
c _ _ _ _ _ _ _ _ _ _ _ problems. Her d _ _ _ _ _ _ _ _ _ _ _ _ c _ _ _ _ _ were intersected at
various a _ _ _ _ _ by p _ _ _ _ _ _ _ l _ _ _ _ .
in a pretty c _ _ _ _ _ _ _ _ _
i________
The second sister, Polly Gon, was dressed
s _ _ and it was obvious that her natural c _ _ _ _ _ ran into
n _ _ _ _ _ _ . Just looking from the first to the second, Joe found his
i _ _ _ _ _ _ _ c _ _ _ _ _ _ _ _ _ _ rapidly. What poor Joe did not know was that Polly knew
all the a _ _ _ _ _ and was an expert at taking s _ _ _ _ _ _ . Joe was invited to come in and sit
down. Deca proved to be as s _ _ _ _ _ as she looked and just sat there like a l _ _ . Polly, at
a given s _ _ _ , sent Deca out to find some r _ _ _ _ to make tea. While she was gone, Polly
served Joe s _ _
p _ . Then she used the c _ _ _ _ _ _ _ _ _ _ _ _
soon r _ _ _ _ _ _ to z _ _ _
a _ _ _ _ and Joe was
p _ _ _ _ . Next, she introduced an "I _ ... T _ _ _
p _ _ _ _ _ _ _ _ _ _ ." That is, if Joe would marry her, then ... Well, Joe said, "yes," but then
began to consider the possibility of spending the rest of his life a _ _ _ _ _ _ _ to her s _ _ _ .
He began to get very nervous and his head was going in c _ _ _ _ _ _ . He knew she was a
c _ _ _ _ _ _ n _ _ _ _ _ and he was really afraid of her. So, he tried to make her think that he
was o _ _ by asking, "Do u _ _ _ ?", pretending that he did. When that didn't work, he stood
up and announced that he was going to the bus t _ _ _ _ _ _ _ to m _ _ _ _ _ , his true
beloved.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 77 of 163
Columbus Public Schools 1/5/06
Trigonometry - C
The Story of Joe Sine.
Answer Key
Fill in the following blanks with letters to make math words.
One day Joe S ine went to call on his new neighbors who lived in the adjacent house.
He was a handsome tangent , a confirmed bachelor. Joe was met at the door by two
sisters who had anything but congruent
figures . The first, Deca Gon, had real
construction problems. Her discontinuous curves were intersected at
various angles by parallel lines . The second sister, Polly Gon, was dressed
in a pretty coordinate
set and it was obvious that her natural curves ran into
imaginary numbers . Just looking from the first to the second, Joe found his
interest compounding rapidly. What poor Joe did not know was that Polly knew
all the angles and was an expert at taking squares. Joe was invited to come in and sit
down. Deca proved to be as square as she looked and just sat there like a log . Polly, at
a given sign , sent Deca out to find some roots to make tea. While she was gone, Polly
served Joe sum
pi . Then she used the corresponding
soon reduced to zero
angle and Joe was
power . Next, she introduced an "If ... Then
proposition ." That is, if Joe would marry her, then ... Well, Joe said, "yes," but then
began to consider the possibility of spending the rest of his life adjacent to her side .
He began to get very nervous and his head was going in circles . He knew she was a
complex number and he was really afraid of her. So, he tried to make her think that he
was odd by asking, "Do unit ?", pretending that he did. When that didn't work, he stood
up and announced that he was going to the bus terminal to median , his true beloved.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 78 of 163
Columbus Public Schools 1/5/06
Solving Trigonometric Equations
Trigonometry - D
Name
1. Use the unit circle to find all the solutions to the following equations in the interval [0,2π].
3
2
A. sin x =
B. cos x = −
C. tan x = − 3
D. csc x = −2 E. cos x = 0
2
2
Solve each of the equations graphically. Sketch your graphs below.
All of these equations have more than one solution. Explain why in terms of the unit circle and
in terms of the graph.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 79 of 163
Columbus Public Schools 1/5/06
Trigonometry - D
2. Use your graphing calculator to find all the solutions to the following equations in the interval
[0,2π]. Sketch your graphs below.
A. sin2x =
3
2
B. cos 4 x = −
2
2
C. tan2x = − 3
D. csc 3x = −2 E. cos6x = 0
3. For each equation, give the number of solutions and explain the number of solutions in terms
of the period of the trigonometric function.
3
2
A. sin2x =
B. cos 4 x = −
2
2
C. tan2x = − 3
D. csc 3x = −2
E. cos6x = 0
4. Give the period of the trigonometric function and then predict the number of solutions of each
equation on the interval [0,2π]. Find the solutions.
A. sin 4 x = −.9903
B. tan.25x = 4
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 80 of 163
Columbus Public Schools 1/5/06
Trigonometry - D
Solving Trigonometric Equations
Answer Key
1. Use the unit circle to find all the solutions to the following equations in the interval [0,2π].
3
2
A. sin x =
B. cos x = −
C. tan x = − 3
D. csc x = −2 E. cos x = 0
2
2
Solve each of the equations graphically. Sketch your graphs below.
y = cos(x)
y = sin(x)
1
1
π
2π
3
3
1.5708
3.14159
4.71239
1.5708
6.28319
-1
-1
3π
4
y = tan (x)
2
3.14159
4.71239
6.28319
5π
4
y = csc (x)
3
2
1
1
1.5708
-1
3.14159
4.71239
1.5708
6.28319
5π
3
2π
3
3.14159
-1
7π
6
-2
4.71239
6.28319
11π
6
-3
-2
y = cos (x)
1
1.5708
3.14159
π
4.71239
6.28319
3π
2
2
-1
All of these equations have more than one solution. Explain why in terms of the unit circle and
in terms of the graph.
The trigonometric functions are each positive in two quadrants and negative in two
quadrants. In terms of the graph, the graph is divided into halves, the part that is positive
and above the x axis and the part that is negative and below the x axis.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 81 of 163
Columbus Public Schools 1/5/06
Trigonometry - D
2. Use your graphing calculator to find all the solutions to the following equations in the interval
[0,2π]. Sketch your graphs below.
3
2
A. sin2x =
B. cos 4 x = −
C. tan2x = − 3 D. csc 3x = −2 E. cos6x = 0
2
2
B
A
(.523599, .866025)
1
(3.66519, .866025)
(1.0472, .866025)
1.5708
1
(4.18879, .866025)
3.14159
4.71239
6.28319
1.5708
-1
-1
3.14159
(.982, -.707)
6.28319
(3.731, -.707)
(2.553, -.707)
(.589, -.707)
(5.301, -.707)
(4.123, -.707)
(5.694, -.707)
(2.160, -.707)
C
4.71239
2
D
2
1
1
1.5708
3.14159
4.71239
1.5708
6.28319
3.14159
4.71239
6.28319
(4.01426, -2.)
(1.91986, -2.)
-1
(5.41052, -2.)
-1
(1.0472, -1.73205)
-2
-2
(1.22173, -2.)
(4.18879, -1.73205)
(2.61799, -1.73205)
(5.75959, -1.73205)
-3
(3.31613, -2.)
E
(6.10865, -2.)
1
1.5708
3.14159
4.71239
6.28319
-1
3. For each equation, give the number of solutions and explain the number of solutions in terms
of the period of the trigonometric function.
3
2
A. sin2x =
B. cos 4 x = −
2
2
4 solutions, 2 periods
8 solutions, 4 periods
D. csc 3x = −2
6 solutions, 3 periods
C. tan2x = − 3
4 solutions, 2 periods
E. cos6x = 0
12 solutions, 6 periods
The number of periods is equal to the coefficient in front of the variable. The number of
solutions is twice the number of periods.
4. Give the period of the trigonometric function and then predict the number of solutions of each
equation on the interval [0,2π]. Find the solutions.
A. sin 4 x = −.9903
B. tan.25x = 4
Period is
π
2
, there will be 8 solutions.
Period is 8π, there will be 1 solution in the given
interval.
1
5
4
(5.30327, 4.)
3
1.5708
3.14159
4.71239
2
6.28319
1
1.5708
3.14159
4.71239
6.28319
-1
-1
(1.14325, -.9903),(1.21295, -.9903),(2.71404, -.9903),(2.78374, -.9903)
(4.28484, -.9903),(4.35454, -.9903),(5.85564, -.9903),(5.92534, -.9903)
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 82 of 163
Columbus Public Schools 1/5/06
Problem Solving: Trigonometric Ratios
Trigonometry - D
Name ____________________________
Materials: scientific calculator
1. Use the information given in the figure below to determine the sine, cosine, and tangent of
Sin θ = _______ Cos θ = _______ Tan θ = _______
∠θ. Explain your answer.
(0,5)
B
(3,4)
θ
A
C
(5,0)
2. Use the information given in the figure below to determine the perimeter of rectangle ABCD.
Support your answer by showing your work.
B
C
100 cm
35°
A
D
Perimeter = ____________
3. Which of the following trigonometric ratios: sine, cosine, or tangent of an angle can have a
value greater than 1? Why is it that the values of the other two trigonometric ratios can never
be greater than 1? Explain.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 83 of 163
Columbus Public Schools 1/5/06
Trigonometry - D
4. John, an employee of the U.S. Forestry Service has been asked to determine the height of a
tall tree in Wayne National Forest. He uses an angle measuring device to determine the
angle of elevation (angle formed by the line of sight to the top of the tree and a horizontal) to
be about 33°. He walks off 40 paces to the base of the tree. If each pace is .6 meters, how
tall is the tree to the nearest meter? Support your answer by showing your work and
including a diagram.
5. Determine the perimeter to the nearest centimeter and the area to the nearest square
centimeter of the triangle shown below. Support your answer by showing your work and
giving an explanation.
B
10 cm
A
C
6. Use what you know about the side lengths of special right triangles to complete the following
table. Express your answers in simplified radical form.
30°
45°
45°
60°
30°
45°
60°
Sin
Cos
Tan
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 84 of 163
Columbus Public Schools 1/5/06
Problem Solving: Trigonometric Ratios
Answer Key
Trigonometry - D
1. Use the information given in the figure below to determine the sine, cosine, and tangent of
4
3
4
Sin θ =
Cos θ =
Tan θ =
∠θ. Explain your answer.
5
5
3
Solution: The lengths of AC and BC can be
determined by using the coordinates of point
B(3,4). The length of AB can be determined
by using the fact that it is a radius of a circle.
AB = 5, BC = 4, and AC = 3. By definition:
(0,5)
B
(3,4)
θ
A
C
(5,0)
BC 4
AC 3
= ; cos θ =
= ; and
AB 5
AB 5
BC 4
tan θ =
=
AC 3
sin θ =
2. Use the information given in the figure below to determine the perimeter of rectangle ABCD.
Support your answer by showing your work.
Solution: By definition:
57.36 cm C
B
BC
BC
sin 35° =
; .5736 ≈
; BC ≈ 57.36
100
100
CD
CD
81.92 cm
cos 35° =
; .8192 ≈
; CD ≈ 81.92
100
100
100 cm
35°
The perimeter of the rectangle = 2(57.36) +
2(81.92) = 278.56 cm.
A
D
Perimeter = 278.56 cm
3. Which of the following trigonometric ratios: sine, cosine, or tangent of an angle can have a
value greater than 1? Why is it that the values of the other two trigonometric ratios can never
be greater than 1? Explain.
Solution: The tangent of an angle can be greater than 1. The sine of an acute angle of a
length of leg opposite the angle
right triangle is defined as
and the cosine of an acute
length of hypotenuse
length of leg adjacent to the angle
angle of a right triangle is defined as
. The length of a
length of hypotenuse
leg of a right triangle will always be less than the length of the hypotenuse. If the
numerator of a fraction is less than the denominator, the fraction is always less than 1.
Therefore, the sine and cosine of an angle will never be greater than 1 by definition of
the sine and cosine ratios.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 85 of 163
Columbus Public Schools 1/5/06
Trigonometry - D
4. John, an employee of the U.S. Forestry Service has been asked to determine the height of a
tall tree in Wayne National Forest. He uses an angle measuring device to determine the
angle of elevation (angle formed by the line of sight to the top of the tree and a horizontal) to
be about 33°. He walks off 40 paces to the base of the tree. If each pace is .6 meters, how
tall is the tree to the nearest meter? Support your answer by showing your work and
including a diagram.
Solution:
h
tan 33° =
24
h
h
.6494 ≈
24
33°
h ≈ 16 m
24 m
5. Determine the perimeter to the nearest centimeter and the area to the nearest square
centimeter of the triangle shown below. Support your answer by showing your work and
giving an explanation.
Sample Solution:
Triangle ABC is an isosceles right triangle. The legs have
B
equal lengths, therefore the acute angles each have a measure
of 45°. The ratio of the sides of a 45°- 45°- 90° triangle is
45°
10 10 2
10 cm
1:1: 2 . The length of each leg is
=
= 5 2.
2
2
The perimeter of the triangle is
45°
10 + 5 2 + 5 2 = 10 + 10 2 ≈ 24 cm.
A
C
The area of the triangle is
1
1
• 5 2 • 5 2 = • 25 • 2 ≈ 25 cm 2 .
2
2
B
45°
10 cm
45°
A
C
Sample Solution:
Triangle ABC is an isosceles right triangle. The legs have
equal lengths, therefore the acute angles each have a measure
of 45°. The lengths of the legs can be found by using the sine
and cosine ratios.
AC
sin∠B = sin 45° =
10
AC = sin 45° •10 ≈ .707 •10 ≈ 7.07 cm
AB
cos∠B = cos 45° =
10
AB = cos 45° •10 ≈ .707 •10 ≈ 7.07 cm
The perimeter of the triangle is
7.07 + 7.07 + 10 = 24.14 ≈ 24 cm.
The area of the triangle is
.5 • 7.07 • 7.07 = 24.99 ≈ 25 cm 2 .
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 86 of 163
Columbus Public Schools 1/5/06
Trigonometry - D
6. Use what you know about the side lengths of special right triangles to complete the following
table. Express your answers in simplified radical form.
30°
45°
45°
60°
Sin
Cos
Tan
30°
45°
60°
1
2
1
2
=
2
2
1
2
=
2
2
1
=1
1
3
2
1
2
3
2
1
3
=
3
3
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 87 of 163
3
= 3
1
Columbus Public Schools 1/5/06
Pasta Waves
Trigonometry - E
Name
What graphs are created from the distances of the x-axis and y-axis to points on the unit circle?
Instructions:
1. Using the unit circle on the “Unit Circle” page, break a piece of pasta so that it is equal to the
vertical distance from the x-axis to the point at 30°.
2. Glue the pasta on the “S(x) Graph” page at the 30°.
3. Repeat this process for each angle on the unit circle.
4. With your pencil, draw a smooth curve that is formed by the ends of the pasta. This is the
graph of S(x).
5. Using the unit circle, break a piece of pasta so that it is equal to the horizontal distance from
the y-axis to the point at 0°.
6. Glue the pasta on the “C(x) Graph” page at 0°.
7. Repeat this process for each angle from the unit circle.
8. With your pencil, draw a smooth curve that is formed by the ends of the pasta. This is the
graph of C(x).
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 88 of 163
Columbus Public Schools 1/5/06
Unit Circle
Trigonometry - E
90º
120º
60º
135º
45º
150º
30º
180º
0º, 360º
330º
210º
315º
225º
240º
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
300º
270º
Page 89 of 163
Columbus Public Schools 1/5/06
S(x) Graph
Trigonometry - E
1
0.5
-0.5
-1
30º 45º
60º
90º
120º 135º
150º
180º
210º
225º 240º
270º
300º
315º 330º
360º
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 90 of 163
Columbus Public Schools 1/5/06
C(x) Graph
Trigonometry - E
1
0.5
-0.5
-1
30º 45º
60º
90º
120º 135º
150º
180º
210º
225º 240º
270º
300º
315º 330º
360º
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 91 of 163
Columbus Public Schools 1/5/06
Observations and Predictions
Trigonometry - E
1. In the space provided below, list as many similarities as you can between the graphs
of S(x) and C(x) that you created.
2. In the space provided below, list as many differences as you can between the graphs
of S(x) and C(x) that you created.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 92 of 163
Columbus Public Schools 1/5/06
Trigonometry - E
3. Predict what would happen if you continued to measure around the unit circle another
revolution and then pasted your results to the graphs of S(x) and C(x). Draw a sketch of your
prediction.
4. Predict what would happen if you continued to measure around the unit circle in a clockwise
fashion and then pasted your results to the graphs of S(x) and C(x). Draw a sketch of your
prediction.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 93 of 163
Columbus Public Schools 1/5/06
Trigonometry - E
Getting in Shape
Name
You decide to try jogging to shape up. You are fortunate to have a large neighborhood park
nearby that has a circular track with a radius of 100 meters.
1. If you run one lap around the track, how may meters have you traveled? Explain how you
determined your answer.
2. You start off averaging a relatively slow rate of approximately 100 meters per minute. How
long does it take you to complete one lap? Explain how you determined your answer.
You want to improve your speed. Gathering data describing our position on the track as a
function of time may be useful. You sketch the track on a coordinate system with the center at
the origin. Assume that the starting line has coordinates (100, 0) and that you run
counterclockwise.
(100, 0)
3. You begin to gather data about your position at various times. Using the preceding diagram
and the results from number 2, complete the following table to locate your coordinates at
selected times along your path.
t (minutes) x-coordinate y-coordinate
0
100
0
π
2
π
3π
2
2π
5π
2
3π
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 94 of 163
Columbus Public Schools 1/5/06
Trigonometry - E
4. What patterns do you notice about the numerical data in number 3? Predict your coordinates
for the next two time intervals.
5. To better analyze your position at times other than those listed in the previous table. You
decide to make some educated guesses. Consider t =
the coordinates of your position P when t =
π
4
π
4
minutes. Use the graph to approximate
minutes, and label these coordinates on the graph.
(100, 0)
6. Compute the angle θ given in the graph of number 5 and explain how you arrived at your
answer.
7. Gather more information about your position by completing the following table.
t (minutes) x-coordinate y-coordinate
π
4
3π
4
5π
4
7π
4
9π
4
8. On the following grid, use the data from number 3 and 7 to plot (t, y). Connect your data
pairs to make a smooth graph.
-1.5708
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 95 of 163
1.5708
3.14159
4.71239
6.28319
Columbus Public Schools 1/5/06
Trigonometry - E
9. Use your graphing calculator to plot y = 100 sin x. Make sure your calculator is in radian
mode. How does it compare with the graph in number 8?
10. What are the maximum and minimum values of the sine function in problem 9?
The amplitude of a periodic function is defined by:
Amplitude =
1
( M − m ) , where M is the
2
maximum output value and m is the minimum output value.
11. What is the amplitude of the sine function in problem 9? How does this relate to equation?
12. Determine the amplitude of each of the following:
a. y = 1.5 sin x
b. y = 15 sin (x + π)
c. y = 3 sin (2x)
d. y = -2 sin x
13. Is amplitude ever negative? Why or Why not?
14. Does this same idea of amplitude apply to other trigonometric functions? Which ones?
Why or Why not?
15. Write a general statement about the amplitude of the trigonometric functions.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 96 of 163
Columbus Public Schools 1/5/06
Trigonometry - E
Getting in Shape
Answer Key
You decide to try jogging to shape up. You are fortunate to have a large neighborhood park
nearby that has a circular track with a radius of 100 meters.
1. If you run one lap around the track, how may meters have you traveled? Explain how you
determined your answer.
1 lap = the circumference of the circle
C = 2πr = 2π(100 m) = 628 m
2. You start off averaging a relatively slow rate of approximately 100 meters per minute. How
long does it take you to complete one lap? Explain how you determined your answer.
628
t=
= 6.28 min.
100
You want to improve your speed. Gathering data describing our position on the track as a
function of time may be useful. You sketch the track on a coordinate system with the center at
the origin. Assume that the starting line has coordinates (100, 0) and that you run
counterclockwise.
(100, 0)
3. You begin to gather data about your position at various times. Using the preceding diagram
and the results from number 2, complete the following table to locate your coordinates at
selected times along your path.
t (minutes) x-coordinate y-coordinate
0
100
0
π
2
π
3π
2
2π
5π
2
3π
0
100
-100
0
0
-100
100
0
0
100
-100
0
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 97 of 163
Columbus Public Schools 1/5/06
Trigonometry - E
4. What patterns do you notice about the numerical data in number 3? Predict your coordinates
for the next two time intervals.
7π
The data are cyclic. t =
= (0, -100) and t = 4π = (100, 0)
2
5. To better analyze your position at times other than those listed in the previous table. You
decide to make some educated guesses. Consider t =
the coordinates of your position P when t =
π
4
π
4
minutes. Use the graph to approximate
minutes, and label these coordinates on the graph.
(70.7, 70.7)
(-70.7, 70.7)
(100, 0)
(-70.7, -70.7)
25π
1
= lap
4
4
200π 8
6. Compute the angle θ given in the graph of number 5 and explain how you arrived at your
answer.
π
1
π
1
θ=
= 45o
(2π) =
(360) = 45o
4
4
8
8
7. Gather more information about your position by completing the following table.
t=
π
; d = 100(
π
(70.7, -70.7)
) = 25π ;
t (minutes) x-coordinate y-coordinate
70.71
70.71
π
4
-70.71
70.71
3π
4
-70.71
-70.71
5π
4
70.71
-70.71
7π
4
70.71
70.71
9π
4
8. On the following grid, use the data from number 3 and 7 to plot (t, y). Connect your data
pairs to make a smooth graph.
-1.5708
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 98 of 163
1.5708
3.14159
4.71239
6.28319
Columbus Public Schools 1/5/06
Trigonometry - E
9. Use your graphing calculator to plot y = 100 sin x. Make sure your calculator is in radian
mode. How does it compare with the graph in number 8?
The graphs are the same.
10. What are the maximum and minimum values of the sine function in problem 9?
100 is the maximum value and -100 is the minimum value
The amplitude of a periodic function is defined by:
Amplitude =
1
( M − m ) , where M is the
2
maximum output value and m is the minimum output value.
11. What is the amplitude of the sine function in problem 9? How does this relate to equation?
Amplitude = 100
It is the coefficient in the equation.
12. Determine the amplitude of each of the following:
a. y = 1.5 sin x
b. y = 15 sin (x + π)
c. y = 3 sin (2x)
1.5
15
d. y = -2 sin x
3
2
13. Is amplitude ever negative? Why or Why not?
Amplitude is never negative. Amplitude is one half of the maximum minus the minimum.
14. Does this same idea of amplitude apply to other trigonometric functions? Which ones?
Why or Why not?
Amplitude also applies to the cosine function. The other functions do not have an absolute
maximum and minimum.
15. Write a general statement about the amplitude of the trigonometric functions.
Amplitude = a = M − m
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 99 of 163
Columbus Public Schools 1/5/06
Trigonometry - E
Speeding Up
Name
After a lot of practice, you finally speed up on your circular track of radius 100 meters. You
achieve your personal goal of 200 meters per minute.
1. Running at 200 meters per minute counterclockwise, how long does it take you to complete
1
1
one lap? of a lap?
of a lap?
2
4
2. Use the results of number 1 to complete the following table that indicates coordinates at
selected special points along your path.
t (minutes) x-coordinate y-coordinate
0
100
0
π
4
π
2
3π
4
π
5π
4
3π
2
7π
4
2π
3. On the following grid, use the data from number 2 to plot (t, y). Connect your data pairs to
make a smooth graph.
-1.5708
1.5708
3.14159
4.71239
6.28319
4. What is the amount of time it takes for the graph of problem 3 to complete one full cycle?
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 100 of 163
Columbus Public Schools 1/5/06
Trigonometry - E
5. When you doubled your speed, what effect did that have on the amount of time to complete
one cycle? Explain how you determined your answer.
6. Use your graphing calculator to plot y = 100 sin 2x. How does this compare to the graph in
number 3?
7. The period of a trigonometric function is the shortest time (distance) to complete one full
cycle. Determine the period of each of the following:
a. y = 100 sin x
b. y = 100 sin (2x)
c. y = 100 sin (3x)
8. On the following grid, use the data from number 2 to plot (t, x). Connect your data pairs to
make a smooth graph.
-1.5708
1.5708
3.14159
4.71239
6.28319
9. What equation should you put in the calculator to produce the graph from number 8?
10. If you tripled your speed on the track from the original 100 meters per minute, what effect
would this have on the amount of time to compete one lap? What equations would describe the x
and y coordinates of your position?
11. Determine the period of each of the following functions:
a. y = 100 cos x
b. y = 100 cos (2x)
c. y = 100 cos (3x)
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 101 of 163
Columbus Public Schools 1/5/06
Trigonometry - E
Speeding Up
Answer Key
After a lot of practice, you finally speed up on your circular track of radius 100 meters. You
achieve your personal goal of 200 meters per minute.
1. Running at 200 meters per minute counterclockwise, how long does it take you to complete
1
1
one lap? of a lap?
of a lap?
2
4
200π
1
π
1
π
One lap = 200π meters; t =
= π min;
min;
min
lap ; t =
lap; t =
2
4
200
2
4
2. Use the results of number 1 to complete the following table that indicates coordinates at
selected special points along your path.
t (minutes) x-coordinate y-coordinate
0
100
0
0
100
π
4
-100
0
π
2
0
-100
3π
4
π
100
0
0
100
5π
4
-100
0
3π
2
0
-100
7π
4
2π
100
0
3. On the following grid, use the data from number 2 to plot (t, y). Connect your data pairs to
make a smooth graph.
-1.5708
1.5708
3.14159
4.71239
6.28319
4. What is the amount of time it takes for the graph of problem 3 to complete one full cycle?
π minutes
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 102 of 163
Columbus Public Schools 1/5/06
Trigonometry - E
5. When you doubled your speed, what effect did that have on the amount of time to complete
one cycle? Explain how you determined your answer.
Doubling the speed cuts the time in half.
6. Use your graphing calculator to plot y = 100 sin 2x. How does this compare to the graph in
number 3?
The graphs are the same.
7. The period of a trigonometric function is the shortest time (distance) to complete one full
cycle. Determine the period of each of the following:
a. y = 100 sin x
b. y = 100 sin (2x)
c. y = 100 sin (3x)
2π
2π
3
π
8. On the following grid, use the data from number 2 to plot (t, x). Connect your data pairs to
make a smooth graph.
-1.5708
1.5708
3.14159
4.71239
6.28319
9. What equation should you put in the calculator to produce the graph from number 8?
y = 100 cos (2x)
10. If you tripled your speed on the track from the original 100 meters per minute, what effect
would this have on the amount of time to compete one lap? What equations would describe the x
and y coordinates of your position?
2π
The time would be reduced by a factor of 3 from 2π min to
min.
3
x = 100 cos (3t) and y = 100 sin (3t)
11. Determine the period of each of the following functions:
a. y = 100 cos x
b. y = 100 cos (2x)
c. y = 100 cos (3x)
2π
2π
π
3
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 103 of 163
Columbus Public Schools 1/5/06
Trigonometry - E
Running With a Friend
Name
While jogging, you become good friends with another runner, who started out like you, with a
speed of 100 meters per minute on the 100 meter track. You enjoy running together but prefer to
keep a healthy distance between each other during the run. You start at (100, 0) and when you
arrive at (0, 100), your friend starts at (100,0).
1. Assume that you both maintain the same speed of 100 meters per minute. How far ahead of
your friend are you? How long after you start does your friend wait before starting.
2. Complete the following table to give coordinates for both you and your friend at selected
points along the track.
t (minutes)
0
π
Your
Your
Friend
Friend
x-coordinate y-coordinate x-coordinate y-coordinate
100
0
------100
2
π
3π
2
2π
5π
2
0
3. On the following grid, use the data from number 2 to plot (t, y) for both you and your friend.
Connect the points to make a smooth curve.
3.14159
6.28319
9.42478
4. What is the relationship between the two graphs?
The displacement, or phase shift, of a graph is the smallest movement (left or right) necessary for
the first graph to match the second graph.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 104 of 163
Columbus Public Schools 1/5/06
Trigonometry - E
5. Which graph is the displaced graph in number 3? What is the displacement?
6. Would you expect the same type of relationship between the two graphs representing the xcoordinates? Why or Why Not?
Displacement is defined for the cosine function in the same manner.
7. If your x-coordinate is given by 100 cos t, predict the defining equation of your friend’s xcoordinate.
In general, in the following functions defined by y = a sin(bx + c) and y = a cos(bx + c), the
values for b and c affect the displacement, or phase shift, of the function. The phase shift is
c
given by − .
b
c
c
is negative, then the shift is to the __________________. When − is positive, then
b
b
the shift is to the __________________.
8. If −
9. In this activity, your friend’s y-coordinate is given by y = a sin(bx + c), where a = 100 and
b = 1. Calculate c.
10. For the following functions, identify the amplitude, period, and displacment:
a. y = 0.7 cos(2x +
π
2
)
b. y = 3 sin(x – 1)
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 105 of 163
c. y = - 2.5 sin(0.4x +
π
3
)
Columbus Public Schools 1/5/06
Trigonometry - E
Running With a Friend
Answer Key
While jogging, you become good friends with another runner, who started out like you, with a
speed of 100 meters per minute on the 100 meter track. You enjoy running together but prefer to
keep a healthy distance between each other during the run. You start at (100, 0) and when you
arrive at (0, 100), your friend starts at (100,0).
1. Assume that you both maintain the same speed of 100 meters per minute. How far ahead of
your friend are you? How long after you start does your friend wait before starting.
You are
1
lap ahead.
4
m = 50π
m = 157 min
2. Complete the following table to give coordinates for both you and your friend at selected
points along the track.
t (minutes)
0
π
2
π
3π
2
2π
5π
2
Your
Friend
Friend
Your
x-coordinate y-coordinate x-coordinate y-coordinate
100
0
-------
0
100
100
0
-100
0
0
100
0
-100
-100
0
100
0
0
-100
0
100
100
0
3. On the following grid, use the data from number 2 to plot (t, y) for both you and your friend.
Connect the points to make a smooth curve.
-1.5708
1.5708
3.14159
4.71239
6.28319
7.85398
9.42478
4. What is the relationship between the two graphs?
The graph for the friend is shifted
π
units to the right.
2
The displacement, or phase shift, of a graph is the smallest movement (left or right) necessary for
the first graph to match the second graph.
PreCalculus Standard 4 and 5
Columbus Public Schools 1/5/06
Page 106 of 163
Trigonometry and Trigonometric Functions
Trigonometry - E
5. Which graph is the displaced graph in number 3? What is the displacement?
The friend’s graph is displaced by
π
2
minutes.
6. Would you expect the same type of relationship between the two graphs representing the xcoordinates? Why or Why Not?
Yes. By looking at the table, you can see that the x-coordinates are the same for values of t
that differ by
π
2
minutes.
Displacement is defined for the cosine function in the same manner.
7. If your x-coordinate is given by 100 cos t, predict the defining equation of your friend’s xcoordinate.
x = 100 cos(t -
π
2
)
In general, in the following functions defined by y = a sin(bx + c) and y = a cos(bx + c), the
values for b and c affect the displacement, or phase shift, of the function. The phase shift is
c
given by − .
b
c
c
is negative, then the shift is to the __left______________. When − is positive,
b
b
then the shift is to the ___right___________.
8. If −
9. In this activity, your friend’s y-coordinate is given by y = a sin(bx + c), where a = 100 and
b = 1. Calculate c.
c=-
π
2
10. For the following functions, identify the amplitude, period, and displacment:
a. y = 0.7 cos(2x +
π
2
Amplitude = 0.7
Period = π
Displacement = −
π
4
)
b. y = 3 sin(x – 1)
c. y = - 2.5 sin(0.4x +
Amplitude = 3
Period = 2π
Amplitude = 2.5
Period = 5π
Displacement = 1
Displacement = −
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 107 of 163
π
3
)
5π
6
Columbus Public Schools 1/5/06
Trigonometry - E
Sine Curve Equation
Name
Match the sine curve equations on the right to their characteristics on the left. Place the letter of
the matching equation on the blank before its characteristics.
Amplitude
Period
Displacement
Sine Curve Equations
units left
a. sin .4(x +
units right
b. 1.5sin 4(x -
π
units right
c. 2sin 2(x -
π
)
units left
d. sin 3(x + π)
__________ 1.
1
2π
π
__________ 2.
2
π
π
__________ 3.
1.5
π
π
2
6
__________ 4.
1.5
2π
__________ 5.
1
2π
3
__________ 6.
2
4π
__________ 7.
1
3π
4
__________ 8.
1
5π
π
__________ 9.
1.5
π
π units right
i. 1.5sin (x +
__________ 10. 2
π
2π
units left
3
j. 2sin 2(x -
__________ 11. 1
3π
π
__________ 12. 2
π
π
__________ 13. 1.5
4π
2π units left
m. 2sin 4(x +
__________ 14. 1.5
10π
π units right
n. 2sin 2(x -
2
4
2
π
6
π
6
6
π
π units left
e. sin (x +
π
units right
f. 1.5sin .2(x – π)
units right
g. sin
units left
h. 1.5sin 2(x – π)
3
π
2
4
2
6
4
)
π
π
)
6
)
2
8
π
(x - )
2
3
units left
k. sin
units right
l. sin 3(x -
π
)
6
π
2π
)
3
4
)
1
π
(x - )
3
2
p. 1.5sin (x q. 1.5sin
Page 108 of 163
)
2
π
(x + )
2
3
o. 2sin
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
)
4
π
6
)
1
(x + 2π)
2
Columbus Public Schools 1/5/06
Trigonometry - E
Sine Curve Equation
Answer Key
Match the sine curve equations on the right to their characteristics on the left. Place the letter of
the matching equation on the blank before its characteristics.
Amplitude
Period
Displacement
Sine Curve Equations
units left
a. sin .4(x +
units right
b. 1.5sin 4(x -
π
units right
c. 2sin 2(x -
π
)
units left
d. sin 3(x + π)
____E____ 1.
1
2π
π
____J____ 2.
2
π
π
____B____ 3.
1.5
π
π
2
6
____I____ 4.
1.5
2π
____D____ 5.
1
2π
3
____O____ 6.
2
4π
____K____ 7.
1
3π
4
____A____ 8.
1
5π
π
____H____ 9.
1.5
π
π units right
i. 1.5sin (x +
____M____ 10. 2
π
2π
units left
3
j. 2sin 2(x -
____G____ 11. 1
3π
π
____C____ 12. 2
π
π
____Q____ 13. 1.5
4π
2π units left
m. 2sin 4(x +
____F____ 14. 1.5
10π
π units right
n. 2sin 2(x -
2
4
2
π
6
π
6
6
π
π units left
e. sin (x +
π
units right
f. 1.5sin .2(x – π)
units right
g. sin
units left
h. 1.5sin 2(x – π)
3
π
2
4
2
6
4
)
π
π
)
6
)
2
8
π
(x - )
2
3
units left
k. sin
units right
l. sin 3(x -
π
)
6
π
2π
)
3
4
)
1
π
(x - )
3
2
p. 1.5sin (x q. 1.5sin
Page 109 of 163
)
2
π
(x + )
2
3
o. 2sin
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
)
4
π
6
)
1
(x + 2π)
2
Columbus Public Schools 1/5/06
Finding the Inverse Without a Calculator
Trigonometry - F
Name
Use the unit circle to find the value of each of the following in both radians and degrees.
⎛
2⎞
1. sin-1 ⎜⎜ −
⎟⎟
⎝ 2 ⎠
2. cos-1
3. cot-1 1
4. csc-1 1
5. arcsec 2
6. tan-1 (-
7. arcsin
3
2
⎛
2⎞
9. arccos ⎜⎜ −
⎟⎟
⎝ 2 ⎠
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
2
2
3)
8. csc-1 2
⎛
3⎞
10. arctan ⎜⎜ −
⎟⎟
⎝ 3 ⎠
Page 110 of 163
Columbus Public Schools 1/5/06
Finding the Inverse Without a Calculator
Answer Key
Trigonometry - F
Use the unit circle to find the value of each of the following in both radians and degrees on the
interval [0, 2π].
⎛
2⎞
1. sin-1 ⎜⎜ −
⎟⎟
⎝ 2 ⎠
2. cos-1
315o, 225o
7π 5π
,
4 4
45o, 315o
π 7π
,
4 4
3. cot-1 1
4. csc-1 1
45o, 225o
π 5π
,
4 4
90o
5. arcsec 2
6. tan-1 (-
60o, 300o
π 5π
,
3 3
330o, 150o
11π 5π
,
6
6
7. arcsin
2
2
π
2
3
2
3)
8. csc-1 2
60o, 120o
π 2π
,
3 3
30o, 150o
π 5π
,
6 6
⎛
2⎞
9. arccos ⎜⎜ −
⎟⎟
⎝ 2 ⎠
135o, 225o
3π 5π
,
4 4
⎛
3⎞
10. arctan ⎜⎜ −
⎟⎟
⎝ 3 ⎠
330o, 150o
11π 5π
,
6
6
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 111 of 163
Columbus Public Schools 1/5/06
Trigonometry - F
Match the Inverse
Name
Match the graph of the right with the equation on the left.
_____ 1. y = sin-1x
A
_____ 2. y = cos-1x
-1
_____ 3. y = tan x
_____ 4. y = csc-1x
4.7124
4.7124
3.1416
3.1416
1.5708
1.5708
-6.28319 -3.14159
_____ 5. y = sec-1x
B
3.14159
6.28319 -6.28319 -3.14159
3.14159
-1.5708
-1.5708
-3.1416
-3.1416
-4.7124
-4.7124
6.28319
_____ 6. y = cot-1x
D
C
4.7124
4.7124
3.1416
3.1416
1.5708
1.5708
-6.28319 -3.14159
3.14159
6.28319
-1.5708
-3.1416
3.1416
-1.5708
-3.1416
-3.1416
-4.7124
-4.7124
E
F
4.7124
4.7124
3.1416
3.1416
1.5708
-6.28319
-3.14159
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
1.5708
3.14159
6.28319
-6.28319 -3.14159
-1.5708
-1.5708
-3.1416
-3.1416
-4.7124
-4.7124
Page 112 of 163
3.14159
6.28319
Columbus Public Schools 1/5/06
Trigonometry - F
Match the Inverse
Answer Key
Match the graph of the right with the equation on the left.
___E__ 1. y = sin-1x
A
___A__ 2. y = cos-1x
-1
___F__ 3. y = tan x
___D__ 4. y = csc-1x
4.7124
4.7124
3.1416
3.1416
1.5708
1.5708
-6.28319 -3.14159
___C__ 5. y = sec-1x
B
3.14159
6.28319 -6.28319 -3.14159
3.14159
-1.5708
-1.5708
-3.1416
-3.1416
-4.7124
-4.7124
6.28319
___B__ 6. y = cot-1x
D
C
4.7124
4.7124
3.1416
3.1416
1.5708
1.5708
-6.28319 -3.14159
3.14159
6.28319
-1.5708
-3.1416
3.1416
-1.5708
-3.1416
-3.1416
-4.7124
-4.7124
E
F
4.7124
4.7124
3.1416
3.1416
1.5708
-6.28319
-3.14159
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
1.5708
3.14159
6.28319
-6.28319 -3.14159
-1.5708
-1.5708
-3.1416
-3.1416
-4.7124
-4.7124
Page 113 of 163
3.14159
6.28319
Columbus Public Schools 1/5/06
Trigonometry - G
Daylight Hours
Name
The Astronomical Applications Department of the U. S. Naval Observatory gives the time of
sunrise and sunset for each day in the year. It is available at
http://aa.usno.navy.mil/data/docs/RS_OneYear.html. Your group will be assigned a city by your
teacher.
Each group will make a scatterplot of the day of the year vs. length of day for the assigned city.
You should enter the day of the year into L1, the time of sunrise into L2, the time of sunset into
L3, and L3-L2 into L4, and then graphing L1 vs. L4. Choose every tenth day. Be careful to
choose each tenth day. Do not use the days of the month. The entries in L1 should be 10, 20,
30, 40, …, 360.
If you wish to divide the work up among your group, all group members should follow the above
instructions for their portion of the calendar. Then each one should transfer L1 into L5 and L4
into L6. Then, one at a time, transfer L5 and L6 onto one of the calculators. After each transfer,
use Augment to add all of the days to one list and all of the daylight hours into one list.
Augment is accessed by LIST, OPS, 9: augment(
The commands are
1. Model the statplot using a sine function.
2. Estimate the amplitude of the function. On what days of the year, should the maximum and
minimum occur? (You can use the points from your graph or the exact days to find the
amplitude.)
3. Estimate the phase shift of the function by estimating the point of inflection of the graph. On
what days of the year should the points of inflection occur? (You can use the points from your
graph or the exact days to find the phase shift.)
4. What is the vertical shift of your graph? What does this represent in terms of hours of
daylight?
5. Write the equation for your model. Enter it into Y1 and check it for accuracy.
Sketch your scatterplot and the graph of your model.
6. Compare your graph to those of the other two cities. Why are they different?
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 114 of 163
Columbus Public Schools 1/5/06
Trigonometry - G
Daylight Hours
Answer Key
The Astronomical Applications Department of the U. S. Naval Observatory gives the time of
sunrise and sunset for each day in the year. It is available at
http://aa.usno.navy.mil/data/docs/RS_OneYear.html. Your group will be assigned a city by your
teacher.
Each group will make a scatterplot of the day of the year vs. length of day for the assigned city.
You should enter the day of the year into L1, the time of sunrise into L2, the time of sunset into
L3, and L3-L2 into L4, and then graphing L1 vs. L4. Choose every tenth day. Be careful to
choose each tenth day. Do not use the days of the month. The entries in L1 should be 10, 20,
30, 40, …, 360.
If you wish to divide the work up among your group, all group members should follow the above
instructions for their portion of the calendar. Then each one should transfer L1 into L5 and L4
into L6. Then, one at a time, transfer L5 and L6 onto one of the calculators. After each transfer,
use Augment to add all of the days to one list and all of the daylight hours into one list.
Augment is accessed by LIST, OPS, 9: augment(
The commands are
1. Model the statplot using a sine function. ANSWERS WILL VARY
2. Estimate the amplitude of the function. On what days of the year, should the maximum and
minimum occur? (You can use the points from your graph or the exact days to find the
amplitude.) The amplitudes vary according to the latitude of the city, the farther from the
equator, the larger the amplitude. The max should occur on the summer solstice, June 21.
The min should occur on the winter solstice, December 21 or 22, depending on the year.
3. Estimate the phase shift of the function by estimating the point of inflection of the graph. On
what days of the year should the points of inflection occur? (You can use the points from your
graph or the exact days to find the phase shift.) The points of inflection should occur on the
vernal equinox, March 21 or 22 and the autumnal equinox, September 22 or 23.
4. What is the vertical shift of your graph? What does this represent in terms of hours of
daylight?
This is number of hours of daylight at the equinox. It should be the same on all the graphs.
5. Write the equation for your model. Enter it into Y1 and check it for accuracy.
Sketch your scatterplot and the graph of your model.
Answers will vary.
6. Compare your graph to those of the other two cities. Why are they different? Different
latitudes.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 115 of 163
Columbus Public Schools 1/5/06
Tic-Tac Trig - 4 in a Row
Trigonometry - H
Name
Directions for Play:
Objective:
The object of the game is similar to that of tic-tac-toe; the winner is the first of two players to
place four tokens in a row, either vertically, horizontally, or diagonally.
Materials:
The materials necessary to play Trig tic-tac-times include a factor board and a game board (see
the next page) and forty translucent tokens of two different colors. One token of each color is
used as the factor marker, and the remainder are used as game tokens. The game board should
be laminated so that it can be saved from year to year. The tokens should be stored in a bag so
that they don't get lost.
Method of play:
Player 1 begins the game by placing a factor marker and one of player 2's factor markers on any
factors on the factor board. The product of these factors determines the placement of player 1's
game token. For example, player 1 could place a factor marker on sin(-x) and player 2's marker
on cot(x). Player 1 then would place a game token on -cos(x) because [sin(-x)][cot(x)] = -cos(x).
Note: factor markers can be placed on the same factor, resulting in squared factors.
Player 2 can move only player 2's factor marker (player l's marker remains in place) to another
factor on the factor board. In this example, player 2 could move her factor marker to csc(x). The
product of these new factors determines the placement of player 2's game token. In this example,
player 2 would place a game token on the product of sin(-x) (player l's marker) and csc(x)
(player 2's marker), or -1, on the game board.
Players must use a strategy of working backward to determine which products combined with the
available factors will win the game. These same problem-solving strategies become a part of the
defensive play of the game when a player wishes to block an opponent.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 116 of 163
Columbus Public Schools 1/5/06
Trigonometry - H
Tic-Tac Trig - 4 in a Row
Game Board
A
B
C
D
E
- cos x
-1
1
csc x cot x
- cot x csc x
sin x tan x
sin x
sin 2 x
cos x
- sin x tan x
cos x
1 – cos2x
sec2x - 1
1
sin (2x)
2
tan2x + 1
cos2x – 1
- csc x
sec2 x sin x
1 – sec2x
sec x sin2x
1
csc2 x – 1
cot x
cos x
csc2x – 1
cot2x + 1
1
sec2 x – 1
1 – sin2x
csc x tan x
1 – csc2x
− cos 2 x
sin x
- cos x
sec x cot x
-1
- sin x
- tan x
tan x
-1
cos2x csc x
sin2x
csc x sec x
- sec x tan x
- sin x cos x
1
- sec x
Factor Board
sin (-x)
tan(-x)
sin x
cos x
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
tan x
Page 117 of 163
cot x
sec x
csc x
cot (-x)
Columbus Public Schools 1/5/06
Tic-Tac Trig - 4 in a Row
Answer Key
Trigonometry - H
THIS IS FOR TEACHERS ONLY!!
Listed below are the possible solutions for each of the 45 squares in the game board.
A-1 sin(-x) cot(x)
sin(x) cot(-x)
tan(x) cot(-x)
sin(-x) csc(x)
B-1 tan(-x) cot(x)
C-1 cos(x) sec(x)
tan(-x) cot(-x)
tan(x) cot(x)
sin(x) csc(x)
D-1 cot(x) csc(x)
E-1 cot(-x) csc(x)
------------------------------------------------------------------A-2 sin(-x) tan(-x)
sin(x) tan(x)
B-2 cos(x) tan(x)
C-2 sin(-x) tan(x)
sin(x) tan(-x)
D-2 sin(-x) tan(x)
sin(x) tan(-x)
E-2 sin(-x) cot(-x)
sin(x) cot(x)
------------------------------------------------------------------A-3 sin(x) sin(x)
sin(-x) sin(-x)
B-3 tan(-x) tan(-x)
tan(x) tan(x)
1
C-3
sin(2x)
2
D-3 tan2x + 1
E-3 cos2x – 1
------------------------------------------------------------------A-4 cot(-x) sec(x)
B-4 tan(x) sec(x)
C-4 tan(x) tan(-x)
D-4 sin(-x) tan(-x)
sin(x) tan(x)
E-4 cos(x) sec(x)
tan(-x) cot(-x)
tan(x) cot(x)
sin(x) csc(x)
------------------------------------------------------------------A-5 cot(x) cot(x)
cot(-x) cot(-x)
B-5 cos(x) csc(x)
C-5 sin(-x) cot(-x)
sin(x) cot(x)
cot(-x) cot(-x)
D-5 cot(x) cot(x)
E-5 csc(x) csc(x)
------------------------------------------------------------------A-6 cos(x) sec(x)
tan(-x) cot(-x)
tan(x) cot(x)
sin(x) csc(x)
B-6 tan(-x) tan(-x)
tan(x) tan(x)
C-6 cos(x) cos(x)
D-6 tan(x) csc(x)
E-6 cot(x) cot(-x)
------------------------------------------------------------------A-7 cos(x) cot(-x)
B-7 sin(-x) cot(x)
sin(x) cot(-x)
C-7 cot(x) sec(x)
D-7 tan(-x) cot(x)
tan(x) cot(-x)
sin(-x) csc(x)
E-7 cos(x) tan(-x)
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 118 of 163
Columbus Public Schools 1/5/06
Trigonometry - H
------------------------------------------------------------------A-8 sin(-x) sec(x)
B-8 sin(x) sec(x)
C-8 tan(-x) cot(x)
tan(x) cot(-x)
sin(-x) csc(x)
D-8 cos(x) cot(x)
E-8 sin(x) sin(x)
sin(-x) sin(-x)
------------------------------------------------------------------A-9 sec(x) csc(x)
B-9 tan(-x) sec(x)
C-9 sin(-x) cos(x)
D-9 cos(x) sec(x)
tan(-x) cot(-x)
tan(x) cot(x)
sin(x) csc(x)
E-9 tan(-x) csc(x)
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 119 of 163
Columbus Public Schools 1/5/06
Law of Sines Spaghetti Investigation
Trigonometry - I
Name
1. Measure each of these line segments and break pieces of spaghetti equal in length to each.
a.
e.
c.
d.
2. Use combinations of three sides from a, e, c, and d to try to make triangles. Which
combinations of three of these can you use to make a triangle? Why do the others not work?
3. Use side a as the base and c as the hypotenuse to make a right triangle. Break a piece of
spaghetti to make the other leg of the triangle. Name this side b. Find the length. Compare the
answer with the length of the piece of spaghetti that you broke, side b. Draw the triangle and
label the sides. Use right triangle trigonometry to find the degree measure of the angle.
4. Use the pipe cleaner to form an angle congruent to the angle B. Make one side of the angle
the same length as c. Tape a piece of spaghetti to the other side. Place the piece of spaghetti for
side b to recreate the right triangle from #3. Label the vertices of the triangle ABC.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 120 of 163
Columbus Public Schools 1/5/06
-I
5. Break a piece of spaghetti slightly longer than b. Place one end of the spaghetti at vertexTrigonometry
A.
Align the spaghetti so that it touches the base of the triangle. How many ways can you do this?
Explain this in terms of the length of the piece of spaghetti and side b.
6. Draw the two possibilities. Use the Law of Sines to solve each of the triangles.
7. Set the last piece of spaghetti aside and break another piece slightly shorter than b. How
many triangles can you make now? Explain this in terms of the length of the piece of spaghetti
and side b.
8. Set the second piece of spaghetti aside and break a third piece a little longer than side c. How
many triangles can you make using this for the side from vertex A?
9. Draw the triangle and use the Law of Sines to solve the triangle.
PreCalculus Standard 4 and 5
Columbus Public Schools 1/5/06
Page 121 of 163
Trigonometry and Trigonometric Functions
You have been investigating what is known as The Ambiguous Case of the Law of Sines.
The ambiguous case occurs when you are given an acute angle B, the side b, and another side a
that is longer than b. To determine the number of solutions:
•
Check the value of
•
If
•
If
a sin B
b < asinB , there is no triangle.
asin B < b < a , there are two solutions.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 122 of 163
Columbus Public Schools 1/5/06
Law of Sines Spaghetti Investigation
Answer Key
Trigonometry - I
1. Measure each of these line segments and break pieces of spaghetti equal in length to each.
a.
4.7 cm
e.
7 cm
c.
d.
5.7 cm
9.5 cm
2. Use combinations of three sides from a, e, c, and d to try to make triangles. Which
combinations of three of these can you use to make a triangle? Why do the others not work?
All of the sides work to make triangles because they all work in the triangle inequality
3. Use side a as the base and c as the hypotenuse to make a right triangle. Break a piece of
spaghetti to make the other leg of the triangle. Name this side b. Find the length. Compare the
answer with the length of the piece of spaghetti that you broke, side b. Draw the triangle and
label the sides. Use right triangle trigonometry to find the degree measure of the angle.
Angle B measures 60.348º
b
9.5
B
4.7
4. Use the pipe cleaner to form an angle congruent to the angle B. Make one side of the angle
the same length as c. Tape a piece of spaghetti to the other side. Place the piece of spaghetti for
side b to recreate the right triangle from #3. Label the vertices of the triangle ABC.
A
spaghetti
C
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
pipe
cleaner
B
pipe cleaner
Page 123 of 163
Columbus Public Schools 1/5/06
-I
5. Break a piece of spaghetti slightly longer than b. Place one end of the spaghetti at vertexTrigonometry
A.
Align the spaghetti so that it touches the base of the triangle. How many ways can you do this?
Explain this in terms of the length of the piece of spaghetti and side b.
Two ways, one to the left and one to the right.
6. Draw the two possibilities. Use the Law of Sines to solve each of the triangles.
A
C1
C2
B
ANSWERS WILL VARY
7. Set the last piece of spaghetti aside and break another piece slightly shorter than b. How
many triangles can you make now? Explain this in terms of the length of the piece of spaghetti
and side b.
No triangles. Side b does not reach the other side.
8. Set the second piece of spaghetti aside and break a third piece a little longer than side c. How
many triangles can you make using this for the side from vertex A?
Just one.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 124 of 163
Columbus Public Schools 1/5/06
9. Draw the triangle and use the Law of Sines to solve the triangle.
A
C
B
Answers will vary.
You have been investigating what is known as The Ambiguous Case of the Law of Sines.
The ambiguous case occurs when you are given an acute angle B, the side b, and another side a
that is longer than b. To determine the number of solutions:
•
Check the value of
•
If
•
If
a sin B
b < asinB , there is no triangle.
asin B < b < a , there are two solutions.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 125 of 163
Columbus Public Schools 1/5/06
Law of Sines: Real-life Applications
Trigonometry - I
Name
1. A power company would like to run a high-voltage power line across a
canyon. It must extend from point A on the south side of the canyon to
point C on the north side. A surveyor has marked point B on the south side
of the canyon 100 feet from point A and has determined that the measure of
angle CAB is 42º, and the measure of angle ABC is 110º. What is the minimum
length of power line that will be needed?
2. A motorist is traveling on a straight and level highway at a constant
speed of 60 miles per hour. A mountain top with an angle of elevation of
10º is visible straight ahead. Five minutes later, the angle of elevation
to the top of the mountain is 25º. Find the height of the mountain top (in
feet) relative to the highway?
3. Two guy wires for a radio tower make angles with the horizontal of 58º
and 49º. If the ground anchors for each wire are 150 feet apart, find
a. the length of each wire, assuming the wires are taught
b. the height of the tower.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 126 of 163
Columbus Public Schools 1/5/06
4. An airplane is flying at a constant altitude of 3650 feet and a constant
speed of 660 feet per second (approximately 450 miles per hour) on a path
that will take it directly over the Eiffel Tower. At a certain point, the
angle of depression to the top of the tower is 22º. Four seconds later, the
angle of depression to the top of the tower is 34º. Estimate the height of
the Eiffel Tower.
Trigonometry - I
5. Two Coast Guard stations are located 10 miles apart on a coastline that
runs north-south. A distress signal is received from a ship with a bearing
of N 43º E from the southern station and a bearing of S 56º E from the
northern station. How far is the ship from each station? How far is the
ship from shore?
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 127 of 163
Columbus Public Schools 1/5/06
Law of Sines: Real-life Applications
Answer Key
Trigonometry - I
1. A power company would like to run a high-voltage power line across a
canyon. It must extend from point A on the south side of the canyon to
point C on the north side. A surveyor has marked point B on the south side
of the canyon 100 feet from point A and has determined that the measure of
∠ CAB is 42º, and the measure of ∠ ABC is 110º. What is the minimum
length of power line that will be needed?
sin 28 sin110
=
AC
100
C
A
AC = 200.16 ft
100 ft
B
2. A motorist is traveling on a straight and level highway at a constant
speed of 60 miles per hour. A mountain top with an angle of elevation of
10º is visible straight ahead. Five minutes later, the angle of elevation
to the top of the mountain is 25º. Find the height of the mountain top (in
feet) relative to the highway?
10o
A
26400 ft
sin15 sin10
=
26400 BM
155o
M
25o
B
BM = 17712.42 ft
H
sin 25 =
MH
17712.42
MH = 7485.59 ft
3. Two guy wires for a radio tower make angles with the horizontal of 58º
and 49º. If the ground anchors for each wire are 150 feet apart, find a. the length of each wire,
assuming the wires are taught and b. the height of the tower.
T
73o
sin 73 sin 58 sin 73 sin 49
h
=
=
TB
TA
150
150
58o z
49o
TB = 133.02 ft TA = 118.38 ft
A 150 ft B
z2 + h2 = 118.382 and (150 – z)2 + h2 = 133.022
Use these two equations to solve for z and h. z = 62.73 ft and h = 100.39 ft (h is the height of
the tower)
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 128 of 163
Columbus Public Schools 1/5/06
Trigonometry - I
4. An airplane is flying at a constant altitude of 3650 feet and a constant
speed of 660 feet per second (approximately 450 miles per hour) on a path
that will take it directly over the Eiffel Tower. At a certain point, the
angle of depression to the top of the tower is 22º. Four seconds later, the
angle of depression to the top of the tower is 34º. Estimate the height of
the Eiffel Tower.
2640 ft
A
B C
o
o
o
∠ BAT = 22 ; ∠ ABT = 146 ; ∠ BTA = 12
T
sin12 sin 22
TB = 4756.64 ft
=
TB
2640
TC
sin 34 =
TC = 2659.88 ft
4756.64
Height of Tower = 3650 – 2659.88 = 990.12 ft
5. Two Coast Guard stations are located 10 miles apart on a coastline that
runs north-south. A distress signal is received from a ship with a bearing
of N 43º E from the southern station and a bearing of S 56º E from the
northern station. How far is the ship from each station? How far is the
ship from shore?
N
56o
L
H
10 mi
43o
S
sin 81 sin 43
=
SH
10
NH = 6.90 mi
SH = 8.39 mi
LH
sin 43 =
8.39
LH = 5.72 mi
sin 81 sin 56
=
NH
10
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
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Sines Just Aren’t Enough
Trigonometry - I
Name
Note: Round all final answers to the nearest hundredth.
1. Robbie the Robot is placed on a level practice field facing due north.
He moves 50 feet in that direction. He then stops and rotates clockwise
through 148º (so that he is facing approximately southeast. Robbie then
moves 65 feet in that new direction and stops.
a. How far is Robbie from his starting point?
b. How many degrees clockwise should Robbie rotate in order to be facing the starting point?
2. Two ships start from the same point and sail in different directions, one
on a course of 40º clockwise from north at a rate of 6 miles per hour and
the other on a course of 150º clockwise from north at a rate of 8 miles per
hour. How far apart are the ships after 2.5 hours?
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 130 of 163
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3. In a major-league baseball diamond, the bases form a square with side
length 90 feet. The pitcher’s mound is 60.5 feet from home plate. Find the
distance from the pitcher’s mound to each of the other three bases.
Trigonometry - I
4. A 10-foot by 12-foot by 8-foot rectangular room is to be partitioned
using a triangular piece of canvas. Each of the vertices of the triangle is
to be anchored at three corners of the room, as shown in the figure below.
Find the measure of each vertex angle of the triangle.
10 ft
B
8 ft
C
A
12 ft
5. Margaret and Elizabeth live 1 mile apart. While talking to each other
on the phone during an electrical storm, they both notice a bolt of
lightning in the sky between their two homes. Margaret hears the thunder 5
seconds after the flash, while Elizabeth hears the thunder 4 seconds after
the flash. If the speed of sound is 1088 feet per second, describe the
location of the lightning bolt relative to Margaret’s position.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 131 of 163
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Sines Just Aren’t Enough
Answer Key
Trigonometry - I
Note: Round all final answers to the nearest hundredth.
1. Robbie the Robot is placed on a level practice field facing due north.
He moves 50 feet in that direction. He then stops and rotates clockwise
through 148º (so that he is facing approximately southeast. Robbie then
moves 65 feet in that new direction and stops.
a. How far is Robbie from his starting point?
32o
65 ft
50 ft
a2 = 502 + 652 – 2(50)(65)(cos 32)
a2 = 1212.687375
a = 34.82 ft
b. How many degrees clockwise should Robbie rotate in order to be facing the starting point?
65
57.28
=
sin B sin 32
65(sin 32) = 57.28(sin B)
65(sin 32)
sin B =
57.28
B = 36.97o
2. Two ships start from the same point and sail in different directions, one
on a course of 40º clockwise from north at a rate of 6 miles per hour and
the other on a course of 150º clockwise from north at a rate of 8 miles per
hour. How far apart are the ships after 2.5 hours?
15 mi
a2 = 152 + 202 – 2(15)(20)(cos 110)
a2 = 830.2121
a = 28.81 mi
110o
20 mi
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 132 of 163
Columbus Public Schools 1/5/06
Trigonometry - I
3. In a major-league baseball diamond, the bases form a square with side
length 90 feet. The pitcher’s mound is 60.5 feet from home plate. Find the
distance from the pitcher’s mound to each of the other three bases.
a2 = 902 + 60.52 – 2(90)(60.5)(cos 45)
a2 = 4059.857153
a = 63.72 ft. (pitcher’s mound to 1st or 3rd base)
90 ft
b
a __
90 ft 60.5 ft
902 + 902 = c2
16200 = c2
127.28 ft = c
b = 127.28 ft – 60.5 ft = 66.78 ft (pitcher’s mound to 2nd base)
4. A 10-foot by 12-foot by 8-foot rectangular room is to be partitioned
using a triangular piece of canvas. Each of the vertices of the triangle is
to be anchored at three corners of the room, as shown in the figure below.
Find the measure of each vertex angle of the triangle.
10 ft
B
8 ft
BC = 8 ft
C
A
AC =
122 + 102 =
AB =
82 +
(
244
244 = 2 61
)
2
= 308 = 2 77
∠ BCA = 90o
8
= .5121475197
tan A =
244
∠ BAC = 27.12o
∠ CBA = 90 – 27.12 = 62.88o
12 ft
5. Margaret and Elizabeth live 1 mile apart. While talking to each other
on the phone during an electrical storm, they both notice a bolt of
lightning in the sky between their two homes. Margaret hears the thunder 5
seconds after the flash, while Elizabeth hears the thunder 4 seconds after
the flash. If the speed of sound is 1088 feet per second, describe the
location of the lightning bolt relative to Margaret’s position.
43522 = 54402 + 52802 – 2(5440)(5280)(cos M)
cos M = .6707486631
M = 47.875o from the line of sight
L
5440 ft
4352 ft
M
E
5280 ft
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 133 of 163
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Trigonometry - J
A Variety of Problems
Name
Solve each of the following problems using trigonometric properties.
1. Joshua is flying a kite with 135 feet of string out. The angle of elevation of the kite varies
from 45 degrees to 60 degrees during the flying time. What is the range of heights the kite will
fly during this time?
2. A jet plane is descending from 6000 feet above ground into DFW airport. The angle of
depression between the pilot’s line of sight and the control tower is 5o. How many miles is the
plane from the control tower?
3. One of the tallest mountains in Alaska is 15,308 feet above sea level, just 14.25 miles from
the Pacific coast. What is the angle of elevation from the summit of the mountain to the
shoreline of the Pacific Coast?
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 134 of 163
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Trigonometry - J
4. What is the measure of the angle of depression from a tower that is 120 feet tall to a point that
is 525 feet away from the tower on level ground?
5. A golfer is standing at the tee, looking up to the green on a hill. If the tee is 36 yards lower
than the green and the angle of elevation from the tee to the hole is 12o. Find the distance from
the tee to the hole.
6. From the top of a roller coaster, 60 yards above the ground, a rider looks down and sees the
merry-go-round and Ferris wheel. If the angles of depression are 11o and 8o respectively, how
far apart are the merry-go-round and the Ferris wheel?
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 135 of 163
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Trigonometry - J
7. Kirk visits Yellowstone Park and Old Faithful on a perfect day. His eyes are 6 feet from the
ground and the geyser can reach heights ranging from 90 feet to 184 feet.
A) If Kirk stands 200 feet from the geyser and the eruption rises 175 feet in the air, what is the
angle of elevation to the top of the spray to the nearest tenth?
B) In the afternoon, Kirk returns and observes the geyser’s spray reach a height of 123 feet
when the angle of elevation is 37o. How far from the geyser is Kirk?
8. On July 20, 1969, Neil Armstrong became the first human to walk on the moon. During this
mission, the lunar lander Eagle traveled aboard Apollo II. Before sending Eagle to the surface of
the moon, Apollo 11 orbited the moon 3 miles above the surface. At one point, the onboard
guidance system measured the angle of depression to the far and near edges of a large crater.
The angle measured 16o and 29o, respectively. Find the distance across the crater.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 136 of 163
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Trigonometry - J
A Variety of Problems
Answer Key
Draw a diagram and solve each of the following problems using trigonometric properties.
1. Joshua is flying a kite with 135 feet of string out. The angle of elevation of the kite varies
from 45 degrees to 60 degrees during the flying time. What is the range of heights the kite will
fly during this time?
95.46 feet to 116.91 feet
2. A jet plane is descending from 6000 feet above ground into DFW airport. The angle of
depression between the pilot’s line of sight and the control tower is 5o. How many miles is the
plane from the control tower?
68580.31 feet = 12.99 miles
3. One of the tallest mountains in Alaska is 15,308 feet above sea level, just 14.25 miles from
the Pacific coast. What is the angle of elevation from the summit of the mountain to the
shoreline of the Pacific Coast?
11.5o
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 137 of 163
Columbus Public Schools 1/5/06
Trigonometry - J
4. What is the measure of the angle of depression from a tower that is 120 feet tall to a point that
is 525 feet away from the tower on level ground?
12.9o
5. A golfer is standing at the tee, looking up to the green on a hill. If the tee is 36 yards lower
than the green and the angle of elevation from the tee to the hole is 12o. Find the distance from
the tee to the hole.
173.2 yards
6. From the top of a roller coaster, 60 yards above the ground, a rider looks down and sees the
merry-go-round and Ferris wheel. If the angles of depression are 11o and 8o respectively, how
far apart are the merry-go-round and the Ferris wheel?
118.2 yards
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 138 of 163
Columbus Public Schools 1/5/06
Trigonometry - J
7. Kirk visits Yellowstone Park and Old Faithful on a perfect day. His eyes are 6 feet from the
ground and the geyser can reach heights ranging from 90 feet to 184 feet.
A) If Kirk stands 200 feet from the geyser and the eruption rises 175 feet in the air, what is the
angle of elevation to the top of the spray to the nearest tenth?
40.2o
B) In the afternoon, Kirk returns and observes the geyser’s spray reach a height of 123 feet
when the angle of elevation is 37o. How far from the geyser is Kirk?
155.3 feet
8. On July 20, 1969, Neil Armstrong became the first human to walk on the moon. During this
mission, the lunar lander Eagle traveled aboard Apollo II. Before sending Eagle to the surface of
the moon, Apollo 11 orbited the moon 3 miles above the surface. At one point, the onboard
guidance system measured the angle of depression to the far and near edges of a large crater.
The angle measured 16o and 29o, respectively. Find the distance across the crater.
5.1 miles
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 139 of 163
Columbus Public Schools 1/5/06
Trigonometry - J
The Big Wheel
Name
Megan's little brother Andrew has a bicycle with training wheels. The bicycle wheels have a 24
inch diameter and the training wheels have a 6 inch diameter. Andrew is riding at a steady pace
and the big wheels rotate once every 4 seconds. The training wheels have a 6 inch diameter.
The horizontal distance between the place where the front wheel touches the ground and where
the training wheel touches the ground is 12π inches.
As Andrew is riding down the street, he crosses a freshly painted stripe on the road. Answer the
following questions to find out when the paint on the front tire will be the same height as the
paint on the training wheel and what that height will be?
1. How much time elapses between when the front wheel crosses the stripe and the training
wheel crosses the stripe?
2. Complete the chart. Do not fill in the gray spaces.
Time
elapsed (in
sec)
Height (in
inches)of paint
on front wheel
Height (in
inches) of
paint
on training
wheel
0
.25
.5
.75
1
1.25
1.5
1.75
2
Time
elapsed (in
sec)
Height (in
inches)of
paint
on front
wheel
Height (in
inches) of
paint
on training
wheel
2.25
2.5
2.75
3
3.25
3.5
3.75
4
4.25
3. What is the period of rotation of the front wheel? Use t=0 as the time that the front wheel
crosses the stripe. When will the height of the paint first be one-half of its maximum
displacement?
4. What is the period of rotation of the training wheel? Use t=0 as the time that the front
wheel crosses the stripe. When is the first time that the height of the paint will be onehalf of its maximum displacement?
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
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Trigonometry - J
5. Use t=0 as the time that the front wheel crosses the stripe. Use the sine function to write
the height of the paint above the stripe as a function of time.
6. Remember that the training wheel reaches the stripe later than the front wheel. Write the
height of the paint on the training wheel as a function of time, still using t=0 as the time
that the front wheel crosses the stripe.
7. Graph the functions on your graphing calculator and find the first time that the paint is at
the same height on both wheels and what that height is.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 141 of 163
Columbus Public Schools 1/5/06
Trigonometry - J
The Big Wheel
Answer Key
Megan's little brother Andrew has a bicycle with training wheels. The bicycle wheels have a 24
inch diameter and the training wheels have a 6 inch diameter. Andrew is riding at a steady pace
and the big wheels rotate once every 4 seconds. The training wheels have a 6 inch diameter.
The horizontal distance between the place where the front wheel touches the ground and where
the training wheel touches the ground is 12π inches.
As Andrew is riding down the street, he crosses a freshly painted stripe on the road. Answer the
following questions to find out when the paint on the front tire will be the same height as the
paint on the training wheel and what that height will be?
1. How much time elapses between when the front wheel crosses the stripe and the training
wheel crosses the stripe?
2. Complete the chart. Do not fill in the gray spaces.
Time
elapsed (in
sec)
Height (in
inches)of paint
on front wheel
0
.25
.5
.75
1
1.25
1.5
1.75
2
0
12
24
Height (in
inches) of
paint
on training
wheel
0
3
6
3
0
3
6
Time
elapsed (in
sec)
2.25
2.5
2.75
3
3.25
3.5
3.75
4
4.25
Height (in
inches)of
paint
on front
wheel
12
0
Height (in
inches) of
paint
on training
wheel
3
0
3
6
3
0
3
6
3
3. What is the period of rotation of the front wheel? Use t=0 as the time that the front wheel
crosses the stripe. When will the height of the paint first be one-half of its maximum
displacement?
Period is 4. At 1 second
4. What is the period of rotation of the training wheel? Use t=0 as the time that the front
wheel crosses the stripe. When is the first time that the height of the paint will be onehalf of its maximum displacement?
Period is 1. at .75 seconds
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 142 of 163
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Trigonometry - J
5. Use t=0 as the time that the front wheel crosses the stripe. Use the sine function to write
the height of the paint above the stripe as a function of time.
y = 12sin( π2 ( x − 1)) + 12
6. Remember that the training wheel reaches the stripe later than the front wheel. Write the
height of the paint on the training wheel as a function of time, still using t=0 as the time
that the front wheel crosses the stripe.
y = 3sin(2π ( x − 1.75)) + 3
7. Graph the functions on your graphing calculator and find the first time that the paint is at
the same height on both wheels and the height at that time.
After 3.67 seconds, the height on each will be 1.57 inches.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 143 of 163
Columbus Public Schools 1/5/06
Find the Missing Side or Angle
Trigonometry - Reteach
Name
Instructions: Find the missing side or angle as indicated in each of the right triangles below.
1.
2.
x = ___________
10
x
28°
18
θ
3.
4.
9
x
θ = ___________
23
45°
c
30
x = ___________
c = ___________
70°
θ
5.
11
x
55°
6.
θ = ___________
x = __________
25
2
38°
17
7.
a
8.
8
65°
a = ___________
b
b = __________
9. Describe a situation when you would use sine. Use illustrations to support your answer.
10. Describe a situation when you would use cos-1. Use illustrations to support your answer.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 144 of 163
Columbus Public Schools 1/5/06
Find the Missing Side or Angle
Answer Key
Trigonometry - Reteach
Instructions:
Find the missing side or angle as indicated in each of the right triangles below.
1.
2.
10
x
x=
28°
18.81
23
51.5o
c=
21.21
x=
14.34
θ
3.
4.
9
x
18
θ=
x=
45°
c
30
8.46
70°
6.
6.
θ
θ=
x
55°
10.30
25
11
2
7.
8.
17
a
65°
8
7.93
a=
38°
b=
6.25
b
9. Describe a situation when you would use sine. Use illustrations to support your answer.
When you know the measure of an angle and the measure of either the opposite side or the
hypotenuse.
x
15
25°
10. Describe a situation when you would use cos-1. Use illustrations to support your answer.
When you know the measure of the adjacent side and the hypotenuse and want to find the
measure of the angle.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
10
x°
5
Page 145 of 163
Columbus Public Schools 1/5/06
Paper Plate
Trigonometry - Reteach
Name
The unit circle provides a simple way to remember and calculate the trigonometric ratios of the
special right triangles. In fact, you only need to remember the following:
•
the hypotenuse is always 1,
•
the legs of the 45°, 45°, 90° have a measure of
•
the legs of the 30°, 60°, 90° have measures of 1 and 3 ,
2
2
1
the side opposite the 30° angle = ,
2
and the side opposite the 60° angle = 3 .
2
•
•
2,
2
Instructions:
1. Fold the paper plate into fourths and then fold in half. Open the paper plate and mark the 45°
angles. Label each leg 2 . Whether the measure is positive or negative is determined by the
2
quadrant in which it lies.
2. Fold the paper plate into fourths again and then into thirds. Open the paper plate and mark the
30°, 60°, 90° angles and corresponding sides, remembering that the side across from the 30°
angle measures 1 and the side across from the 60° angle measures 3 . The quadrant
2
2
determines whether the measure is positive or negative.
When complete your plate will look like this:
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 146 of 163
Columbus Public Schools 1/5/06
Crop Circles
Trigonometry - Reteach
Name
Marti the Martian needs your help! He has a deadline to meet on his crop circle project and
doesn’t have time to complete the drawn plans to submit to his boss.
Instructions:
• Help Marti by drawing the circles or partial circles in the boxes provided. Knowing that you
may not speak Martian, Marti put the specifications in terms of the unit circle.
• Start at 0° and move in the direction indicated until you reach the place on the circle equal to
the given cosine and sine values. Shade the portion of the circle bounded by the arc you just
drew.
Example: Clockwise, cos θ = - 1 , sin θ = - 3
2
2
Step 1
Step 2
Start
here
0°
0°
Move
clockwise
⎛- 1 , - 3 ⎞
⎜ 2
2 ⎟⎠
⎝
Move
clockwise
Find the place on the circle for the
given cos and sin and shade back to 0°.
1. Counter-clockwise, cos θ = - 2 , sin θ = - 2
2
2
3. Counter-clockwise, cos θ =
3 , sin θ = - 1
2
2
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
2. Clockwise, cos θ =
2 , sin θ = 2
2
2
4. Clockwise, cos θ = 1 , sin θ = 3
2
2
Page 147 of 163
Columbus Public Schools 1/5/06
Trigonometry - Reteach
5. Clockwise, cos θ = - 3 , sin θ = 1
2
2
6. Clockwise, cos θ = 0 , sin θ = 1
7. Counter-clockwise, cos θ = -1, sin θ = 0
8. Clockwise, cos θ = 1, sin θ = 0
9. Counter-clockwise, cos θ = 0, sin θ = -1
10. Clockwise, cos θ =
3 , sin θ = 1
2
2
Thanks for
your help!
11. Give the exact value of the following (exact values are expressed as integers or fractions and
may include radicals; if there is no value, write undefined):
cos 30° =
sin 30° =
tan 30° =
cos 60° =
sin 60° =
tan 60° =
cos 45° =
sin 45° =
tan 45° =
cos 90° =
sin 90° =
tan 90° =
cos 0° =
sin 0° =
tan 0° =
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 148 of 163
Columbus Public Schools 1/5/06
Crop Circles
Answer Key
Trigonometry - Reteach
Marti the Martian needs your help! He has a deadline to meet on his crop circle project and
doesn’t have time to complete the drawn plans to submit to his boss.
Instructions:
• Help Marti by drawing the circles or partial circles in the boxes provided. Knowing that you
may not speak Martian, Marti put the specifications in terms of the unit circle.
• Start at 0° and move in the direction indicated until you reach the place on the circle equal to
the given cosine and sine values. Shade the portion of the circle bounded by the arc you just
drew.
Example: Clockwise, cos θ = - 1 , sin θ = - 3
2
2
Step 1
Step 2
Start
here
0°
0°
Move
clockwise
⎛- 1 , - 3 ⎞
⎜ 2
2 ⎟⎠
⎝
Move
clockwise
Find the place on the circle for the
given cos and sin and shade back to 0°.
1. Counter-clockwise, cos θ = - 2 , sin θ = - 2
2
2
3. Counter-clockwise, cos θ =
3 , sin θ = - 1
2
2
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
2. Clockwise, cos θ =
2 , sin θ = 2
2
2
4. Clockwise, cos θ = 1 , sin θ = 3
2
2
Page 149 of 163
Columbus Public Schools 1/5/06
Trigonometry - Reteach
5. Clockwise, cos θ = - 3 , sin θ = 1
2
2
6. Clockwise, cos θ = 0 , sin θ = 1
7. Counter-clockwise, cos θ = -1, sin θ = 0
8. Clockwise, cos θ = 1, sin θ = 0
9. Counter-clockwise, cos θ = 0, sin θ = -1
10. Clockwise, cos θ =
3 , sin θ = 1
2
2
Thanks for
your help!
12. Give the exact value of the following (exact values are expressed as integers or fractions and
may include radicals; if there is no value, write undefined):
cos 30° = 3
sin 30° = 1
tan 30° = 1 = 3
3
2
2
3
sin 60° = 3
tan 60° = 3
cos 60° = 1
2
2
sin 45° = 2
tan 45° = 1
cos 45° = 2
2
2
cos 90° = 0
sin 90° = 1
tan 90° = undefined
cos 0° = 1
sin 0° = 0
tan 0° = 0
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 150 of 163
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Memory Match – Up
Trigonometry - Reteach
Students can be put into groups of 3 – 4.
First place all cards face down and have each student take turns drawing two cards. If the two
cards drawn go together as a pair, then the student will keep it as a match. The student with the
most matches wins.
Note: There are 3 cards that say “1”. There are 2 cards that say “ 3 ”. There are 2 cards that
say “ 3 ”. There are 2 cards that say “ 1 ”. Make sure that students know that two cards with
2
2
the exact same expression on them are not considered a match. For example: A card with a “1”
on it does not match a card with a “1” on it. A card with a “1” on it is a match with a card that
has “tan 45º” on it.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
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Trigonometry - Reteach
Memory Match – Up Cards
Pythagorean
Theorem
2
45o
1
30o
?
45o
45o
?
?
60o
45o
?
1
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
3
Page 152 of 163
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Trigonometry - Reteach
3
1
2
2
2
1
tan 45º
sin 45º
sin 30º
cos 30º
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 153 of 163
Columbus Public Schools 1/5/06
Trigonometry - Reteach
30o
30o
?
60o
60o
?
It can be used to
solve for an acute
angle in a right
triangle.
3
2
sin
3
2
3
3
-1
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
1
2
Page 154 of 163
Columbus Public Schools 1/5/06
Trigonometry - Reteach
sin θ
cos θ
tan θ
opposite
hypotenuse
adjacent
hypotenuse
opposite
adjacent
leg2 + leg2 =
hypotenuse2
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
2
Page 155 of 163
Columbus Public Schools 1/5/06
Trigonometry - Reteach
tan 30º
sin 60º
cos 60º
tan 60º
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 156 of 163
Columbus Public Schools 1/5/06
Trigonometry - Reteach
Memory Match-Up
Answer Key
leg2 + leg2 = hypotenuse2
Pythagorean Theorem
sin 45º
opposite
hypotenuse
adjacent
hypotenuse
opposite
adjacent
1
2
3
2
3
3
2
2
tan 45º
1
sin 60º
cos 60º
3
2
1
2
tan 60º
3
sin θ
cos θ
tan θ
sin 30º
cos 30º
tan 30º
sin -1
?
It can be used to solve for an acute angle in
a right triangle.
1
45º
45º
?
2
?
45º
45º
2
?
30º
60º
1
30º
60º
?
3
? 30º
60º
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 157 of 163
Columbus Public Schools 1/5/06
Trigonometry - Extension
When Good Trig Goes Bad--In Search of Actual
Triangles
Name
Existence: answers the question “Can I find some object that has these
properties?”
Uniqueness: answers the question “Is there only one object with these
properties?”
1. SSS (Side-Side-Side):
A. The question: If we know three sides of a triangle, how many different
triangles does this describe? Consider our three demonstration examples:
5, 12, 13
4, 5, 11
4, 11, 15
B. In your own words, why do only some combinations of sides yield a triangle?
C. State the Triangle Inequality:
D. If the side-lengths check out by triangle-inequality, how many triangles can
we get with those exact sides? Here’s a step in the right direction: if
you know the three sides of a triangle, what else can we determine about the
triangle (and how do we do so)?
E. From this, we may conclude that given for three sides that obey triangle
inequality, exactly one triangle can be formed (i.e. there exists one unique
triangle).
2. AAA (Angle-Angle-Angle)
A. If we know just the three angle-measures, how many triangles are possible?
(Assumption: the given angles add up to
______________ ).
B. Let’s start with a familiar example--an equilateral triangle. What are the
angle-measures of an equilateral triangle?
C. How many triangles can we build with that exact combination of angle
measures?
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 158 of 163
Columbus Public Schools 1/5/06
Trigonometry - Extension
D. For another example, let’s consider the Pythagorean triple 3-4-5 (the two legs and
hypotenuse of a right triangle). What do you know about the triangle with sides 6-8-10? It
is ______________________ to 3-4-5, meaning that:
a. corresponding angles
b. corresponding sides
E. So, for a given angle-angle-angle combination, how many triangles are
possible?
3. AAS (Angle-Angle-Side) or ASA (Angle-Side-Angle):
A. To begin with, what can we find easily (without trig) in either case?
B. From here, what law can we easily apply?
C. Can you come up with any situation where we would not get a
unique triangle determined?
4. SSA (Side-Side-Angle)
A. Try to solve the following example: triangle ABC has a=5, b=8, and A=110º. Start by solving
for the measure of B. What problems do we run into?
B. Now consider this problem in more intuitive terms. Just like triangle-inequality, we have a
well-formedness condition relating sides and angles of a triangle. Specifically: the largest angle
must be opposite the largest side, and the smallest angle must be opposite the smallest side.
How can we apply this to our last example?
C. So, what conclusion can we reach here? Given 2 sides and a non-included angle, it is possible
that they not form a triangle (i.e. some SSA cases give 0 triangles possible).
Let’s examine the SSA cases more methodically:
1) A is obtuse: we can use this last observation: Can either of the other two angles be bigger than
an obtuse angle?
Thus, the bigger/smaller (circle one) of the two smaller sides must be opposite the obtuse angle
for us to have a triangle. Otherwise, we get zero possible triangles.
2) A is right. Again, can either of the remaining two angles be bigger than 90º?
So this case behaves just like our obtuse example. If the largest side is opposite the right-angle,
we get one triangle. Otherwise,we get 0 triangles possible.
PreCalculus Standard 4 and 5
Columbus Public Schools 1/5/06
Page 159 of 163
Trigonometry and Trigonometric Functions
3) A is acute. This is the most complicated scenario, so let’s pick it apart.
Trigonometry - Extension
To diagram this, always put the acute angle in the lower-left corner, and the 2 sides proceed
clockwise from here (ASS). Draw a baseline, the angle, and the first side. How could we now
find the height of this triangle?
Now consider the possible outcomes, depending on the length of the second
side (ASS):
a. If this side is smaller than the height, describe what result we get?
An example of this type: C=35º, c=3, b=8. Work through this example; how many triangles are
possible?
b. If this second side is equal to the height, then what do we know about the exact location of
that second side (where must it lie)? Do we get any triangles, and if so, how many?
We can illustrate this with: B=42º, b=6.69, a=10. Work through this example to verify your
claims.
c. If this second side is greater than the height, but smaller than the first side, where might this
side lie in relation to the height? Do we get any triangles, and if so, how many?
Try this example, and see how many triangles you can get with these properties: C=35º, b=8,
c=6.
d. If this second side is greater than the height, and also
greater than (or equal to) the first side, where must this side lie in
relation to the height? Do we get any triangles, and if so, how many?
Try this example, and see how many triangles you can get with
these properties: C=35º, b=8, c=11.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 160 of 163
Columbus Public Schools 1/5/06
Trigonometry - Extension
When Good Trig Goes Bad--In Search of Actual
Triangles
Answer Key
Existence: answers the question “Can I find some object that has these
properties?”
Uniqueness: answers the question “Is there only one object with these
properties?”
1. SSS (Side-Side-Side):
A. The question: If we know three sides of a triangle, how many different
triangles does this describe? Consider our three demonstration examples:
5, 12, 13
4, 5, 11
4, 11, 15
B. In your own words, why do only some combinations of sides yield a triangle?
Student answers will vary.
When the first 2 sides are connected, the third must be able to “make it back” to the first
vertex.
C. State the Triangle Inequality:
The sum of any 2 side-lengths must be greater than the third.
D. If the side-lengths check out by triangle-inequality, how many triangles can
we get with those exact sides? Here’s a step in the right direction: if
you know the three sides of a triangle, what else can we determine about the
triangle (and how do we do so)?
Use the Law of Cosines to get any 1 angle.
Then use Law of Sines or Law of Cosines to get a second angle.
E. From this, we may conclude that given for three sides that obey triangle
inequality, exactly one triangle can be formed (i.e. there exists one unique
triangle).
2. AAA (Angle-Angle-Angle)
A. If we know just the three angle-measures, how many triangles are possible?
Infinite number
(Assumption: the given angles add up to
180º ).
B. Let’s start with a familiar example--an equilateral triangle. What are the
angle-measures of an equilateral triangle?
60º-60º-60º
C. How many triangles can we build with that exact combination of angle
measures?
Infinite number
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 161 of 163
Columbus Public Schools 1/5/06
Trigonometry - Extension
D. For another example, let’s consider the Pythagorean triple 3-4-5 (the two legs and hypotenuse
of a right triangle). What do you know about the triangle with sides 6-8-10? It is
similar to 3-4-5, meaning that:
a. corresponding angles
b. corresponding sides
are congruent
are in proportion
E. So, for a given angle-angle-angle combination, how many triangles are
possible?
Infinite Number
3. AAS (Angle-Angle-Side) or ASA (Angle-Side-Angle):
A. To begin with, what can we find easily (without trig) in either case?
Third Angle
B. From here, what law can we easily apply?
Law of Cosines
C. Can you come up with any situation where we would not get a
unique triangle determined?
No
4. SSA (Side-Side-Angle)
A. Try to solve the following example: triangle ABC has a=5, b=8, and A=110º. Start by solving
for the measure of B. What problems do we run into?
sin110 sin B
Using Law of Sines
=
, so sinB = 1.50351---not possible
5
8
B. Now consider this problem in more intuitive terms. Just like triangle-inequality, we have a
well-formedness condition relating sides and angles of a triangle. Specifically: the largest angle
must be opposite the largest side, and the smallest angle must be opposite the smallest side.
How can we apply this to our last example?
B would have to be bigger than 110º, but that’s not possible since a triangle’s angles total
180º.
C. So, what conclusion can we reach here? Given 2 sides and a non-included angle, it is possible
that they not form a triangle (i.e. some SSA cases give 0 triangles possible).
Let’s examine the SSA cases more methodically:
1) A is obtuse: we can use this last observation: Can either of the other two angles be bigger than
an obtuse angle?
No; Bigger
Thus, the bigger/smaller (circle one) of the two smaller sides must be opposite the obtuse angle
for us to have a triangle. Otherwise, we get zero possible triangles.
2) A is right. Again, can either of the remaining two angles be bigger than 90º?
No
So this case behaves just like our obtuse example. If the largest side is opposite the right-angle,
we get one triangle. Otherwise,we get 0 triangles possible.
3) A is acute. This is the most complicated scenario, so let’s pick it apart.
Height = first-side*sine(angle)
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 162 of 163
Columbus Public Schools 1/5/06
Trigonometry - I
To diagram this, always put the acute angle in the lower-left corner, and the 2 sides proceed
clockwise from here (ASS). Draw a baseline, the angle, and the first side. How could we now
find the height of this triangle?
Now consider the possible outcomes, depending on the length of the second
side (ASS):
a. If this side is smaller than the height, describe what result we get?
No triangles because the “second side” can’t make it back down to the base to complete the
triangle.
An example of this type: C=35º, c=3, b=8. Work through this example; how many triangles are
possible?
b. If this second side is equal to the height, then what do we know about the exact location of
that second side (where must it lie)? Do we get any triangles, and if so, how many?
It must lie on the altitude (we have a right triangle). One triangle results.
We can illustrate this with: B=42º, b=6.69, a=10. Work through this example to verify your
claims.
c. If this second side is greater than the height, but smaller than the first side, where might this
side lie in relation to the height? Do we get any triangles, and if so, how many?
The second side could lie on either side of the altitude. Two possible triangles.
Try this example, and see how many triangles you can get with these properties: C=35º, b=8,
c=6.
d. If this second side is greater than the height, and also
greater than (or equal to) the first side, where must this side lie in
relation to the height? Do we get any triangles, and if so, how many?
To the right side of the altitude (or, outside the altitude). One triangle results.
Try this example, and see how many triangles you can get with
these properties: C=35º, b=8, c=11.
PreCalculus Standard 4 and 5
Trigonometry and Trigonometric Functions
Page 163 of 163
Columbus Public Schools 1/5/06
COLUMBUS PUBLIC SCHOOLS
MATHEMATICS CURRICULUM GUIDE
GRADE LEVEL STATE STANDARD 3, 4, and 5
Pre-Calculus
Geometry and Spatial Sense; Patterns,
Functions and Algebra; Data Analysis
and Probability
TIME RANGE
35 - 40 days
GRADING
PERIOD
4
MATHEMATICAL TOPIC 5
Noncartesian Representations
A)
B)
C)
D)
CPS LEARNING GOALS
Uses vectors to model and solve application problems.
Uses parametric equations to model and solve application problems.
Uses polar coordinates.
Expresses complex numbers in trigonometric form and computes sums, differences,
products, quotients, powers, and roots of complex numbers in trigonometric form.
COURSE LEVEL INDICATORS
Course Level (i.e., How does a student demonstrate mastery?):
9 Performs vector addition and vector multiplication by a scalar.
9 Determines if two or more vectors are parallel or perpendicular.
9 Given a vector, creates vectors which are parallel or perpendicular to it.
9 Uses trigonometric ratios to perform decomposition of vectors into its component horizontal
and vertical vectors.
9 Converts between Cartesian and parametric representations.
9 Uses a graphing calculator to model problems given in parametric format.
9 Converts back and forth between Cartesian and polar coordinates.
9 Graphs polar equations by hand and with a graphing calculator.
9 Identifies the form and properties of the graph of a polar equation.
9 Converts coordinates back and forth between trigonometric and Cartesian form.
9 Uses a calculator to verify the nth roots of a given complex number.
Previous Level:
9 Uses basic trigonometric identities (sine, cosine, and tangent).
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
Page 1 of 41
Columbus Public Schools 1/5/06
The description from the state, for the Number, Number Sense, and Operations says:
Students demonstrate number sense, including an understanding of number systems and
operations and how they relate to one another. Students compute fluently and make reasonable
estimates using paper and pencil, technology-supported and mental methods.
The grade-band benchmark from the state, for this topic in the grade band 11 – 12 is:
A. Demonstrate that vectors and matrices are systems having some of the properties as the real
number system.
B. Develop an understanding of properties o and representations for addition and multiplication
of vectors and matrices.
D. Demonstrate fluency in operations with real numbers, vectors, and matrices, using mental
computation or paper and pencil calculations for simple cases, and technology for more
complicated cases.
E. Represent and compute with complex numbers.
The description from the state, for the Geometry and Spatial Sense Standard says:
Students identify, classify, compare, and analyze characteristics, properties, and relationships of
one-, two- and three-dimensional geometric figures, and objects. Students use spatial reasoning,
properties of geometric objects, and transformations to analyze mathematical situations and
solve problems.
The grade-band benchmark from the state, for this topic in the grade band 11 – 12 is:
B. Represent transformations within a coordinate system using vectors and matrices.
The description from the state, for the Patterns, Functions, and Algebra Standard says:
Students use patterns, relations, and functions to model, represent, and analyze problem
situations that involve variable quantities. Students analyze, model and solve problems using
various representations such as tables, graphs, and equations.
The grade-band benchmarks from the state, for this topic in the grade band 11 – 12 are:
A. Analyze functions by investigating rates of change, intercepts, zeros, asymptotes and local
and global behavior. Analyze functions by investigating rates of change, intercepts, zeros,
asymptotes and local and global behavior.
D. Apply algebraic methods to represent and generalize problem situations involving vectors
and matrices.
The description from the state, for the Data Analysis and Probability Standard says:
Students pose questions and collect, organize represent, interpret, and analyze data to answer
those questions. Students develop and evaluate inferences, predictions, and arguments that are
abased on data.
The grade-band benchmark from the state, for this topic in the grade band 11 – 12 is:
D. Create and analyze tabular and graphical displays of data using appropriate tools, including
spreadsheets and graphing calculators.
The description from the state, for the Mathematical Processes Standard says:
Students use mathematical processes and knowledge to solve problems. Students apply
problem-solving and decision-making techniques, and communicate mathematical ideas.
The grade-band benchmark from the state, for this topic in the grade band 11 – 12 is:
A. Enter Text here
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
Page 2 of 41
Columbus Public Schools 1/5/06
PRACTICE ASSESSMENT ITEMS
Noncartesian Representations – A
If the forces F1 = 3, 4 , F2 = 2, −6 , and F3 = −1, −5 act together on point P, find the additional
force required to create equilibrium.
A)
−4, 7
B)
7, −4
C)
4, −7
D)
−7, 4
Which of the following vectors describes an 8 lb force acting in the direction of u = 3, −5 ?
A) 8 3, −5
B)
8
3, −5
34
C)
5
3, −5
34
D)
5
−5,3
34
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
Page 3 of 41
Columbus Public Schools 1/5/06
PRACTICE ASSESSMENT ITEMS
Noncartesian Representations – A
Answers/Rubrics
Low Complexity
If the forces F1 = 3, 4 , F2 = 2, −6 , and F3 = −1, −5 act together on point P, find the
additional force required to create equilibrium.
A)
−4, 7
B) 7, −4
C)
4, −7
D)
−7, 4
Answer: A
Moderate Complexity
Which of the following vectors describes an 8 lb force acting in the direction of u = 3, −5 ?
A) 8 3, −5
B)
8
3, −5
34
C)
5
3, −5
34
D)
5
−5,3
34
Answer: B
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
Page 4 of 41
Columbus Public Schools 1/5/06
PRACTICE ASSESSMENT ITEMS
Noncartesian Representations – A
Let u = <5, 8>. Find a vector v of form <x, y> such that u is perpendicular to v.
A 2200 pound car is parked on an inclined street, 14º from the horizontal. Find the magnitude of
the force that must be applied to keep the car from rolling downhill. Find the force perpendicular
to the street. Create a mathematical model (diagram) representing all information and show your
work.
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
Page 5 of 41
Columbus Public Schools 1/5/06
PRACTICE ASSESSMENT ITEMS
Noncartesian Representations – A
Answers/Rubrics
High Complexity
Let u = <5, 8>. Find a vector v of form <x, y> such that u is perpendicular to v.
Short Answer/Extended Response
A 2200 pound car is parked on an inclined street, 14º from the horizontal. Find the magnitude of
the force that must be applied to keep the car from rolling downhill. Find the force perpendicular
to the street. Create a mathematical model (diagram) representing all information and show your
work.
Solution: To keep the car from rolling
downhill, we need to find v. Using
v
trigonometry, sin 14o =
.
2200
v = 2200sin14o = 532.23 lb
2200 lb
v
14º
The force perpendicular to the street is: cos14o =
v = 2200cos14o = 2134.65 lb
v
.
2200
h
A 4-point response gives a proper diagram and uses appropriate ratios to get correct
answers.
A 3-point response gives an accurate diagram, but makes one mistake in setting-up or
calculating answers.
A 2-point response includes a diagram, but has two errors in labeling or calculations.
A 1-point response gives a properly labeled diagram but no set-up or calculations.
A 0-point response shows no mathematical understanding of the task.
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
Page 6 of 41
Columbus Public Schools 1/5/06
PRACTICE ASSESSMENT ITEMS
Noncartesian Representations – B
Which of the following describes the graph of the parametric equations x = 3 + t , y = 2 − 4t , t ≥ 0 ?
A. a line
B. a line segment
C. a ray
D. a parabola
E. a circle
6
Which of the following points corresponds to t = –2 in the parameterization x = −3t 2 , y = 8 + ?
t
A. (–3, 8)
B. (–12, 11)
C. (12, 5)
D. (–12, 5)
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
Page 7 of 41
Columbus Public Schools 1/5/06
PRACTICE ASSESSMENT ITEMS
Noncartesian Representations – B
Answers/Rubrics
Low Complexity
Which of the following describes the graph of the parametric equations x = 3 + t , y = 2 − 4t , t ≥ 0 ?
A. a line
B. a line segment
C. a ray
D. a parabola
E. a circle
Answer: C
Moderate Complexity
6
Which of the following points corresponds to t = –2 in the parameterization x = −3t 2 , y = 8 + ?
t
A. (–3, 8)
B. (–12, 11)
C. (12, 5)
D. (–12, 5)
Answer: D
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
Page 8 of 41
Columbus Public Schools 1/5/06
PRACTICE ASSESSMENT ITEMS
Noncartesian Representations – B
A ball is thrown straight up from level ground with its position above ground at any time t ≥ 0
given by x = 4, y = -16t2+80t+7. At what time will the rock be 88 ft. above the ground?
A. 3.590 sec and 1.410 sec
B. 3.50 sec
C. 1.410 sec
D. 3.50 and 7.1 sec
The parametric equations of flight on the moon are x = (v cos ϑ ) t and
y = (v sin ϑ )t − 2.66t 2 . Use your graphing calculator to determine the approximate horizontal
distance that a baseball tossed upward from the surface travels if it is thrown with an initial
velocity of 68 feet per second at an angle of 14 degrees relative to the surface and the amount of
time it remains above the surface. Show the equations entered, a sketch of the graph, and the
window used. Explain how you used the graph to find the answer.
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
Page 9 of 41
Columbus Public Schools 1/5/06
PRACTICE ASSESSMENT ITEMS
Noncartesian Representations – B
Answers/Rubrics
High Complexity
A ball is thrown straight up from level ground with its position above ground at any time t ≥ 0
given by x = 4, y = -16t2+80t+7. At what time will the rock be 88 ft. above the ground?
A. 3.590 sec and 1.410 sec
B. 3.50 sec
C. 1.410 sec
D. 3.50 and 7.1 sec
ANSWER: A
Short Answer/Extended Response
The parametric equations of flight on the moon are x = (v cos ϑ ) t and
y = (v sin ϑ )t − 2.66t 2 . Use your graphing calculator to determine the approximate horizontal
distance that a baseball tossed upward from the surface travels and the amount of time it
remains above the surface if it is thrown with an initial velocity of 68 feet per second at an angle
of 14 degrees relative to the surface. Show the equations entered, a sketch of the graph, the
window used. Explain how you used the calculator to find the answer.
Solution:
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PNG decompre
are needed to see thi
QuickTime™ and a
PNG decompressor
are needed to see this picture.
QuickTime™ and a
Photo - JPEG decompressor
are needed to see this picture.
QuickTime™ and a
Photo - JPEG decompressor
are needed to see this picture.
The object remains in the air for approximately 6.19 seconds and travels
approximately 408.4 ft. The student may trace or use the table to find the answer.
A 2-point response demonstrates appropriate use of the calculator and describes how the
proper x-coordinate is found.
A 1-point response describes the appropriate use of the calculator, but does not find the
correct answer.
A 0-point response shows no mathematical understanding of the task.
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
Page 10 of 41
Columbus Public Schools 1/5/06
PRACTICE ASSESSMENT ITEMS
⎛ 5π
Given a point in polar form ⎜ 8,
⎝ 6
to Cartesian coordinates?
⎛ 5π
A) x = 8sin ⎜
⎝ 6
Noncartesian Representations – C
⎞
⎟ , which of the following expressions can be used to convert
⎠
⎛ 5π ⎞
y = 8cos ⎜ ⎟
⎝ 6 ⎠
⎞
⎟,
⎠
2
⎛ 5π ⎞
B) x = 5 + ⎜ ⎟ ,
⎝ 6 ⎠
2
C)
2
x=8 ,
y=5
⎛ 5π ⎞
y=⎜
⎟
⎝ 6 ⎠
⎛ 5π ⎞
D) x = 8cos ⎜
⎟,
⎝ 6 ⎠
2
⎛ 5π ⎞
y = 8sin ⎜
⎟
⎝ 6 ⎠
Which one of the following polar coordinate pairs represents the same point as the point with
polar coordinates (−r , θ ) (assume that r does not equal 0)?
A) (r ,θ )
B) (r , θ + π )
C) (r , π − θ )
D) (r , −θ )
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
Page 11 of 41
Columbus Public Schools 1/5/06
PRACTICE ASSESSMENT ITEMS
Noncartesian Representations – C
Answers/Rubrics
Low Complexity
⎛ 5π ⎞
Given a point in polar form ⎜ 8, ⎟ , which of the following expressions can be used to convert
⎝ 6 ⎠
to Cartesian coordinates?
⎛ 5π
A) x = 8sin ⎜
⎝ 6
⎛ 5π ⎞
y = 8cos ⎜
⎟
⎝ 6 ⎠
⎞
⎟,
⎠
2
⎛ 5π ⎞
B) x = 52 + ⎜
⎟ ,
⎝ 6 ⎠
C)
x = 82 ,
y=5
⎛ 5π ⎞
y=⎜
⎟
⎝ 6 ⎠
⎛ 5π
D) x = 8cos ⎜
⎝ 6
⎞
⎟,
⎠
2
⎛ 5π ⎞
y = 8sin ⎜
⎟
⎝ 6 ⎠
Answer: D
Moderate Complexity
Which one of the following polar coordinate pairs represents the same point as the point with
polar coordinates (−r ,θ ) (assume that r does not equal 0)?
A) (r ,θ )
B) (r ,θ + π )
C) (r , π − θ )
D) (r , −θ )
Answer: B
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
Page 12 of 41
Columbus Public Schools 1/5/06
PRACTICE ASSESSMENT ITEMS
Noncartesian Representations– C
Which of the following gives a maximum r-value for the polar graph of r = 3 − 4sin θ ?
A) 9
B) 8
C) 7
D) 6
Convert the point (6, –3) from Cartesian to polar coordinates. Give four forms for your answer:
positive r with positive θ , positive r with negative θ , negative r with positive θ , and
negative r with negative θ .
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
Page 13 of 41
Columbus Public Schools 1/5/06
PRACTICE ASSESSMENT ITEMS
Noncartesian Representations – C
Answers/Rubrics
High Complexity
Which of the following gives a maximum r-value for the polar graph of r = 3 − 4sin θ ?
A) 9
B) 8
C) 7
D) 6
Answer: C
Short Answer/Extended Response
Convert the point (6, –3) from Cartesian to polar coordinates. Give four forms for your answer:
positive r with positive θ , positive r with negative θ , negative r with positive θ , and
negative r with negative θ .
Solution:
(
r = 62 + ( −3)2 = 45 = 3 5
θ = tan −1 ( −63 ) = −26.6°
)
four points: (3 5, −26.6° ),(3 5, 333.4° ),( −3 5,153.4° ),( −3 5, −206.6° )
A 2-point response correctly determines the radius, uses the tangent to determine the
angle, and properly determines the other three forms.
A 1-point response finds the correct radius and angle but makes at least one error in the
other forms.
A 0-point response shows no mathematical understanding of the task.
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
Page 14 of 41
Columbus Public Schools 1/5/06
PRACTICE ASSESSMENT ITEMS
Noncartesian Representations – D
Which of the following gives the number of distinct solutions of z 4 = 3 + 5i ?
A. 0
B. 1
C. 2
D. 3
E. 4
F. 5
3π
3π
⎛
+ i sin
Which Cartesian coordinate has a trigonometric form of 5 ⎜ cos
2
2
⎝
A. (5, 0)
⎞
⎟?
⎠
B. (0, 5)
C. (–5, 0)
D. (0, –5)
E. (0, –1)
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
Page 15 of 41
Columbus Public Schools 1/5/06
PRACTICE ASSESSMENT ITEMS
Noncartesian Representations – D
Answers/Rubrics
Low Complexity
Which of the following gives the number of distinct solutions of z 4 = 3 + 5i ?
A. 0
B. 1
C. 2
D. 3
E. 4
F. 5
Answer: E
Moderate Complexity
3π
3π
⎛
Which Cartesian coordinate has a trigonometric form of 5 ⎜ cos
+ i sin
2
2
⎝
A. (5, 0)
⎞
⎟?
⎠
B. (0, 5)
C. (–5, 0)
D. (0, –5)
E. (0, –1)
Answer: D
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
Page 16 of 41
Columbus Public Schools 1/5/06
PRACTICE ASSESSMENT ITEMS
Noncartesian Representations – D
What are the three cube roots of -1?
1
3
1
3
A. − −
i, − 1, and - +
i
2 2
2 2
B.
1
3
1
3
−
i, 1, and +
i
2 2
2 2
1
3
1
3
C. − −
i, 1, and - +
i
2 2
2 2
D.
1
3
1
3
−
i, − 1, and +
i
2 2
2 2
Using DeMoivre’s Theorem, determine the 3 cube-roots of 2 + 2i in standard form (a + bi).
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
Page 17 of 41
Columbus Public Schools 1/5/06
PRACTICE ASSESSMENT ITEMS
Noncartesian Representations– D
Answers/Rubrics
High Complexity
What are the three cube roots of -1?
1
3
1
3
A. − −
i, − 1, and - +
i
2 2
2 2
B.
1
3
1
3
−
i, 1, and +
i
2 2
2 2
1
3
1
3
C. − −
i, 1, and - +
i
2 2
2 2
D.
1
3
1
3
−
i, − 1, and +
i
2 2
2 2
ANSWER: D
Short Answer/Extended Response
Using DeMoivre’s Theorem, determine the 3 cube-roots of 2 + 2i in standard form (a + bi).
Solution: for 2 + 2i, r = 22 + 22 = 2 2
θ = tan −1 ( 22 ) = 45°
(
r2 = (
r3 = (
r1 =
) = ( 8 ,15) = 1.366 + .366i
8, 45 + 360 ) = ( 8 ,135 ) = −1 + i
8, 45 + 720 ) = ( 8 , 255 ) = −.366 − 1.366i
8, 45°
1
3
1
6
1
3
1
3
1
6
1
6
A 4-point response correctly applies DeMoivre’s Theorem to find the roots and expresses
them in standard form.
A 3-point response properly applies DeMoivre’s Theorem to find the roots and expresses
them in standard form with only one slight error
A 2-point response carries out the conversion into polar form properly and sets up the
DeMoivre’s Theorem appropriately but fails to put answers in standard form.
A 1-point response carries out the conversion into polar form properly
A 0-point response shows no mathematical understanding of the problem.
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
Page 18 of 41
Columbus Public Schools 1/5/06
Teacher Introduction
NonCartesian Representations
Although this topic includes many avenues of study, this course offers only an introduction into
vectors, parametrics, polar equations, and the trigonometric form of complex numbers. If time
allows, the text offers much greater depth into these areas of study.
Vectors are generally included in a high school physics course and would probably be studied in
depth there. They are included here to provide the mathematical background necessary to
understand them and to provide an introduction for students who do not study physics.
Parametrics offer the ability to graph relations which are not functions and to look at graphs
which are functions differently. The use of a parameter enables many real world applications
where there is a controlling factor, usually time, which is the basis for the creation of the graph.
Linear and quadratic parametric equations could easily be included in the earlier studies of lines
and parabolas.
The students should understand the basis of polar graphing, the pair consisting of a radius and an
angles, and should be able to convert back and forth between polar and Cartesian coordinates.
They should graph simple polar equations by hand and use a graphing calculator to graph
complicated polar equations. Students should recognize rose curves, limaçon curves, cardioids,
and lemniscate curves.
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
Page 19 of 41
Columbus Public Schools 1/5/06
TEACHING STRATEGIES/ACTIVITIES
Vocabulary:
vector, component, magnitude, scalar unit vector, dot product, projection, parameter, polar,
trigonometric form, DeMoivre's Theorem
Core:
Learning Goal A: Uses vectors to model and solve application problems.
1. Use the activity Current Swimming, included in this curriculum guide, to introduce
separating a vector into its components and adding vectors by adding components. Save this
activity for use later in the topic.
Learning Goal B: Uses parametric equations to model and solve application problems.
1. Use the activity Ships, included in this curriculum guide, to introduce parametric graphing.
2. Use the activity Parametric Swimming to emphasize the relationship between the component
form of vectors and parametrics.
Learning Goal C: Uses polar coordinates.
1. Have students do the activity Polar Flora, included in this curriculum guide to use the
graphing calculator to investigate polar graphing..
Learning Goal D: Expresses complex numbers in trigonometric form and computes sums,
differences, products, quotients, powers, and roots of complex numbers in trigonometric form.
1. Complete the exercise DeMoivre vs. Factoring to provide the connection between the
trigonometric form of roots and the more familiar quadratic formula.
RESOURCES
Learning Goal A:
Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 502-521
Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004):
Resource Manual pp. 117-120
Learning Goal B:
Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 522-533
Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004):
Resource Manual pp. 121-122
Learning Goal C:
Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 534-549
Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004):
Resource Manual pp. 123-126
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
Page 20 of 41
Columbus Public Schools 1/5/06
Learning Goal D:
Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 550-557
Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004):
Resource Manual pp. 127-128
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
Page 21 of 41
Columbus Public Schools 1/5/06
Vectors – A
Current Swimming
Name
Mary, Tamika, and Gabriella are in a contest to see who can swim across a river the fastest.
Each of them can swim at the rate of .2 m/sec in still water. The river is 50 meters wide and
flows from north to south at .1 m/sec. The girls are going to start on the western bank of the
river and swim to the eastern shore.
Each girl decides upon a strategy to cross the river, bearing in mind that the river is going to
affect the path. Mary decides to head straight across the river. Tamika decides to head 30º N of
E. Gabriella decides to head 30º S of E. Who will win the race?
Mary
1. If there were no current, Mary would head straight across the river. The current moves
her moves her.1 m due south. Complete the chart for her position, using (0,0) for her
initial position. Make a sketch illustrating her motion.
Time
(in seconds)
x-position
(in meters)
y-position
(in meters)
without current
effect of current on
y-position
y-position
(in meters)
with current
1
10
20
50
100
t
2. How many seconds does it take Tamika to cross the river?
3. Does she travel upstream, downstream or straight across the river?
4. How far upstream or downstream does she end up?
5. What is her displacement, the distance between her initial and final positions?
6. What are her speed and direction?
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
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Vectors – A
Tamika
7. If Tamika heads 30º N of E, draw a sketch of her path and use trigonometry to describe
her motion to the east and north.
8. Complete the chart for her position, using (0,0) for her initial position. Make a sketch
illustrating her motion.
Time
(in seconds)
x-position
(in meters)
y-position
(in meters)
without current
effect of
current on yposition
y-position
(in meters)
with current
1
10
20
50
100
t
9. How many seconds does it take Tamika to cross the river?
10. Does she travel upstream, downstream or straight across the river?
11. How far upstream or downstream does she end up?
12. What is her displacement, the distance between her initial and final positions?
13. What are her speed and direction?
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
Page 23 of 41
Columbus Public Schools 1/5/06
Vectors – A
Gabriella
14. If Gabriella heads 30º S of E, draw a sketch of her path and use trigonometry to describe
her motion to the east and north.
15. Complete the chart for her position, using (0,0) for her initial position. Make a sketch
illustrating her motion.
Time (in seconds)
x-position (in meters)
y-position (in meters)
without current
y-position (in meters)
with current
1
10
20
50
100
t
16. How many seconds does it take Gabriella to cross the river?
17. Does she travel upstream, downstream or straight across the river?
18. How far upstream or downstream does she end up?
19. What is her displacement, the distance between her initial and final positions?
20. What are her speed and direction?
21. All the girls would be swimming the same speed in still water. Who is actually moving
the fastest?, the slowest? Who reaches the eastern bank first?
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
Page 24 of 41
Columbus Public Schools 1/5/06
Current Swimming
Answer Key
Vectors – A
Name
Mary, Tamika, and Gabriella are in a contest to see who can swim across a river the fastest.
Each of them can swim at the rate of .2 m/sec in still water. The river is 50 meters wide and
flows from north to south at .1 m/sec. The girls are going to start on the western bank of the
river and swim to the eastern shore.
Each girl decides upon a strategy to cross the river, bearing in mind that the river is going to
affect the path. Mary decides to head straight across the river. Tamika decides to head 30º N of
E. Gabriella decides to head 30º S of E. Who will win the race?
Mary
1. If there were no current, Mary would head straight across the river. The current moves
her moves her.1 m due south. Complete the chart for her position, using (0,0) for her
initial position. Make a sketch illustrating her motion.
Time
(in seconds)
x-position
(in meters)
1
10
20
50
100
t
.2
2
4
10
20
.2t
y-position
(in meters)
without current
0
0
0
0
0
0
effect of water
on y-position
-.1
-1
-2
-5
-10
-.1t
y-position
(in meters)
with current
-.1
-1
-2
-5
-10
0--.1t
1. How many seconds does it take Mary to cross the river?
250 sec
2. Does she travel upstream, downstream or straight across the river?
Downstream
3. How far upstream or downstream does she end up?
25 m
4. What is her displacement, the distance between her initial and final positions?
55.902 m
5. What are her speed and direction? .223 m/s 26.565º Sof E
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
Page 25 of 41
Columbus Public Schools 1/5/06
Vectors – A
Tamika
6. If Tamika heads 30º N of E, draw a sketch of her path and use trigonometry to describe
her motion to the east and north.
.2
.2sin30
.2 cos 30
7. Complete the chart for her position, using (0,0) for her initial position. Make a sketch
illustrating her motion.
Time
(in seconds)
x-position
(in meters)
1
10
.2 cos 30º=.173
2 cos
30º=1.732
4 cos
30º=3.4641
10 cos
30º=8.660
20 cos
30º=17.321
.2t cos 30º
20
50
100
t
y-position
(in meters)
without current
.1
1
effect of current
on y-position
-.1
-1
y-position
(in meters)
with current
0
0
2
-2
0
5
-5
0
10
-10
0
.1t
-.1t
.1t+.1t=0
8. How many seconds does it take Tamika to cross the river?
288.675 sec
9. Does she travel upstream, downstream or straight across the river?
straight across
10. How far upstream or downstream does she end up?
0 meters
11. What is her displacement, the distance between her initial and final positions?
50 meters
12. What are her speed and direction?
.173 m/sec, 0º
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
Page 26 of 41
Columbus Public Schools 1/5/06
Vectors – A
Gabriella
13. If Gabriella heads 30º S of E, draw a sketch of her path and use trigonometry to describe
her motion to the east and north.
.2cos 30º
.2sin30
.2
14. Complete the chart for her position, using (0,0) for her initial position. Make a sketch
illustrating her motion.
Time
(in seconds)
x-position
(in meters)
1
10
20
50
100
t
.2 cos 30º=.173
2 cos 30º=1.732
4 cos 30º=3.4641
10 cos 30º=8.660
20 cos 30º=17.321
.2t cos 30º
y-position
(in meters)
without current
-.1
-1
-2
-5
-10
-.1t
Effect of
current on yposition
-.1
-1
-2
-5
-10
-.1t
y-position
(in meters)
with current
-.2
-2
-4
-10
-20
-.1t-.1t-.2t
15. How many seconds does it take Gabriella to cross the river?
288.675 sec
16. Does she travel upstream, downstream or straight across the river?
Downstream
17. How far upstream or downstream does she end up?
57.735 m
18. What is her displacement, the distance between her initial and final positions?
76.376 m
19. What are her speed and direction?
.265 m/s and 49.107º S of E
20. All the girls would be swimming the same speed in still water. Who is actually moving
the fastest?, the slowest? Who reaches the eastern bank first?
Fastest Gabriella
Slowest Tamika
Mary reaches the eastern bank first.
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
Page 27 of 41
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Parametric – B
Ships
Name
You are watching a radar screen that is a square with sides 1000 mm in length. Two ships, the
Minnow and the Pinafore, appear on your radar screen. As they come onto your screen, the
Minnow is at a point 900 mm from the bottom left corner of the screen along the lower edge.
The Pinafore is located at a point 100 mm above the lower left corner along the left edge. One
minute later the positions have changed. The Minnow has moved to a location on the screen that
is 3mm W and 2 mm N of its previous location. The Pinafore has moved 4mm E and 1 mm N.
They continue to move at constant speeds on their respective linear courses. WILL THE
MINNOW AND THE PINAFORE COLLIDE?
1. Use the table to compute the position of each ship on the radar screen at the times indicated.
In the final row, write an expression in terms of t that would provide a general description of the
position at any time t.
Minnow
Pinafore
Time (t)
x
y.
x
y
0
1
2
3
10
25
t
2. Write an equation for the path of the Minnow. What are the slope, x- and y-intercepts?
Write an equation for the path of the Pinafore. What are the slope, x- and y-intercepts?
How do the slope, and the x- and y-intercepts relate to the motion?
3. Set the graphing mode of your calculator to Simultaneous. Graph the equations on your
calculator using the window [0,1000] by [0,1000]. Where do the lines cross?
Does this
mean that the ships collide at that point?
How can you be sure?
4. Use the expressions you wrote for the x and y coordinates of each ship to determine the time
that each reaches the point of intersection of the paths.
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
Page 28 of 41
Columbus Public Schools 1/5/06
Minnow
Pinafore
Parametric – B
Do they arrive at the point of intersection at the same time?
5. Set the mode of your calculator to Parametric. Look at the y= menu. Notice that there are
now pairs of entries, one for x and one for y. Enter the expressions from the last row of the chart
for the Minnow in X1T and Y1T and the Pinafore in X2T and Y2T. You can enter the T by
using the key you use for x when in the function mode.
When you select window, the x and y settings should still be the same as when you graphed the
lines in #3. The t is called the parameter and is controlling x and y. Set TMIN at 0 and TMAX
10. Set TSTEP at 1. Hit graph. Change values for TMAX and TSTEP to see how they affect
the graph.
What do TMIN and TMAX do?
What does TSTEP do?
How does the parametric graph help you answer the question, "WILL THE SHIPS COLLIDE?
6. Two other ships, the Enterprise and Falcon, were spotted on a screen. The parametric
X 1T = 2t − 5
X 2T = t + 15
for the Enterprise and
for the
equations describing their paths were
Y 1T = 3t + 10
Y 2T = 4t − 10
Falcon. Use parametric equations to simulate their motion, adjusting you window as needed.
(They might not be visible on the radar screen at all times.) Do you think they collide? Show
the math that proves whether or not they collide.
What if these were space ships being sighted from the ground. Could you still be sure that they
collide? Explain.
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
Page 29 of 41
Columbus Public Schools 1/5/06
Parametric – B
7. What is the slope of the function that describes the path of the Enterprise?
Give a
point on the path of the Enterprise.. Write the function that describes the path of the
Enterprise.
What is the slope of the function that describes the path of the Falcon?
Give a point on the path of the Falcon..
Write the function that describes the path of the
Falcon.
2
8. The path of another ship is modeled by the equation y = − x + 10 . Write a pair of parametric
3
equations that describe this same line.
7. A rock is dropped from a tower that is 1000 ft. tall. At any time t measured in seconds, its
distance from the ground in feet is s = 1000 − 16t 2 . You cannot graph the path of the rock using
the function grapher on your calculator because it is a vertical line. Use the parametric mode to
graph the path of the rock. Give your equations here.
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
Page 30 of 41
Columbus Public Schools 1/5/06
Parametric – B
Ships
Answer Key
You are watching a radar screen that is a square with sides 1000 mm in length. Two ships, the
Minnow and the Pinafore, appear on your radar screen. As they come onto your screen, the
Minnow is at a point 900 mm from the bottom left corner of the screen along the lower edge.
The Pinafore is located at a point 100 mm above the lower left corner along the left edge. One
minute later the positions have changed. The Minnow has moved to a location on the screen that
is 3mm W and 2 mm N of its previous location. The Pinafore has moved 4mm E and 1 mm N.
They continue to move at constant speeds on their respective linear courses. WILL THE
MINNOW AND THE PINAFORE COLLIDE?
1. Use the table to compute the position of each ship on the radar screen at the times indicated.
In the final row, write an expression in terms of t that would provide a general description of the
position at any time t.
Minnow
Pinafore
Time (t)
x
y.
x
y
0
900
0
0
100
1
897
2
101
2
894
8
102
3
891
12
103
10
870
6
40
110
25
825
20
100
125
900-3t
50
4t
100+t
t
2. Write an equation for the path of the Minnow. What are the slope, x- and y-intercepts?
y=-2/3x +600
Write an equation for the path of the Pinafore. What are the slope, x- and y-intercepts?
y=-1/4x +100
How do the slope, and the x- and y-intercepts relate to the motion?
The slope indicates the direction of the motion. The y-intercept does not relate to motion, it
just indicates a particular position.
3. Set the graphing mode of your calculator to Simultaneous. Graph the equations on your
calculator using the window [0,1000] by [0,1000]. Where do the lines cross?
Does this
mean that the ships collide at that point?
How can you be sure?
(545.455, 236.364) Answers will vary
4. Use the expressions you wrote for the x and y coordinates of each ship to determine the time
that each reaches the point of intersection of the paths.
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
Page 31 of 41
Columbus Public Schools 1/5/06
Minnow
Pinafore
118.182 sec
136.364 sec
Parametric – B
Do they arrive at the point of intersection at the same time?
NO
5. Set the mode of your calculator to Parametric. Look at the y= menu. Notice that there are
now pairs of entries, one for x and one for y. Enter the expressions from the last row of the chart
for the Minnow in X1T and Y1T and the Pinafore in X2T and Y2T. You can enter the T by
using the key you use for x when in the function mode.
When you select window, the x and y settings should still be the same as when you graphed the
lines in #3. The t is called the parameter and is controlling x and y. Set TMIN at 0 and TMAX
10. Set TSTEP at 1. Hit graph. Change values for TMAX and TSTEP to see how they affect
the graph.
What do TMIN and TMAX do?
They give the starting and stopping time.
What does TSTEP do?
It gives how much the time increases from one point to the next.
How does the parametric graph help you answer the question, "WILL THE SHIPS COLLIDE?
You can see where they are at a given time.
6. Two other ships, the Enterprise and Falcon, were spotted on a screen. The parametric
X 1T = 2t − 5
X 2T = t + 15
for the Enterprise and
for the
equations describing their paths were
Y 1T = 3t + 10
Y 2T = 4t − 10
Falcon. Use parametric equations to simulate their motion, adjusting you window as needed.
(They might not be visible on the radar screen at all times.) Do you think they collide? Show
the math that proves whether or not they collide.
The window TMin 0, TMax 30, [0,100], [0,100] works well. The ships appear to collide.
Setting X1T=X2T gives t=20. Substituting 20 into Y1T and Y2T gives 70 in both cases.
This means that the ships were in the same place at this time and they do collide.
What if these were space ships being sighted from the ground. Could you still be sure that they
collide? Explain.
No, they could be in different planes.
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
Page 32 of 41
Columbus Public Schools 1/5/06
Parametric – B
7. What is the slope of the function that describes the path of the Enterprise?
Give a
point on the path of the Enterprise.. Write the function that describes the path of the
Enterprise.
What is the slope of the function that describes the path of the Falcon?
Give a point on the path of the Falcon..
Write the function that describes the path of the
Falcon.
Enterprise slope is 3/2. y=3/2x + 17.5
Falcon slope is 4. y=4x –70
2
8. The path of another ship is modeled by the equation y = − x + 10 . Write a pair of parametric
3
equations that describe this same line.
x=t, y=-2t + 10
7. A rock is dropped from a tower that is 1000 ft. tall. At any time t measured in seconds, its
distance from the ground in feet is s = 1000 − 16t 2 . You cannot graph the path of the rock using
the function grapher on your calculator because it is a vertical line. Use the parametric mode to
graph the path of the rock. Give your equations here.
x=5, y=1000 – 16t2. The number with x is arbitrary. It gives the equation of the vertical line
the rocket moves along.
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
Page 33 of 41
Columbus Public Schools 1/5/06
Parametric – B
Parametric Swimming
Name
Go back to the Activity Current Swimming. In this activity, you looked at the vectors for the
paths of Mary, Tamika, and Gabriella as they swam across a river and created tables that
described their motion and the effect of the river on their motion.
To illustrate the motion on your calculator, enter this window:
In Format, turn the axes off.
Tmin=
0
In Mode, choose Parametric and Simultaneous
Tmax= 250
For each swimmer, enter the formula for her x-position in X1T.
Tstep=
1
Enter the formula for her y-position with no current in Y1T.
Xmin= -10
Enter 0 in X2T.
Xmax= 50
Enter the formula for the current in Y2T
Xscl=
5
In X3T, enter X1T+X2T.
Ymin= -30
In Y3T, enter Y1T+Y2T
Ymax= 30
Yscl=
5
Sketch the diagram for each swimmer below. Identify the vector for the swimmer with no
current, the current, and the swimmer in the current.
Mary
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
Tamika
Page 34 of 41
Gabriella
Columbus Public Schools 1/5/06
Parametric – B
Parametric Swimming
Answer Key
Go back to the Activity Current Swimming. In this activity, you looked at the vectors for the
paths of Mary, Tamika, and Gabriella as they swam across a river and created tables that
described their motion and the effect of the river on their motion.
To illustrate the motion on your calculator, enter this window:
In Format, turn the axes off.
Tmin=
0
In Mode, choose Parametric and Simultaneous
Tmax= 290
For each swimmer, enter the formula for her x-position in X1T.
Tstep=
1
Enter the formula for her y-position with no current in Y1T.
Xmin= -10
Enter 0 in X2T.
Xmax= 50
Enter the formula for the current in Y2T
Xscl=
5
In X3T, enter X1T+X2T.
Ymin= -30
In Y3T, enter Y1T+Y2T
Ymax= 30
Yscl=
5
Sketch the diagram for each swimmer below. Identify the vector for the swimmer with no
current, the current, and the swimmer in the current.
Mary
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
Tamika
Page 35 of 41
Gabriella
Columbus Public Schools 1/5/06
Polar Flora
Polar – C
Name
Use your calculator to graph each of the polar equations. Sketch each graph and indicate the
window settings used to get a complete graph. Try to select the smallest interval possible
between θmin and θmax. Give the length of each petal of the rose graph. For each equation,
change sine to cosine (or cosine to sine) and see how the graph changes.
Sketch
How did the graph change?
y=2 sin 3θ
θ
y=3 sin 2θ
y=3 cos 2θ
y=5 cos 4θ
y=5 sin 4θ
y=cos 5θ
y=sin 5θ
Window settings
y=2 sin 3θ
y=2 cos 3
θ min
θ max
θ step
Xmin
Xmax
Ymin
Ymax
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
y=3 sin 2θ
y=3 cos 2θ
y =5 cos 4θ
y=5 sin 4θ
Page 36 of 41
y=cos 5θ
y=sin 5θ
Columbus Public Schools 1/5/06
Polar – C
1. What does the r-coordinate of each point represent?
2. What does the θ -coordinate of each point represent?
3. How do the choices of θ min, θ max, and θ step affect the graph?
4. How does the polar equation predict the length of each petal?
5. How does the polar equation predict the number of petals on the rose?
6. How did the graph change when you changed trig functions? Explain this in terms of the
relation of the graphs of the sine and cosine function.
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
Page 37 of 41
Columbus Public Schools 1/5/06
Polar – C
Polar Flora
Answer Key
Use your calculator to graph each of the polar equations. Sketch each graph and indicate the
window settings used to get a complete graph. Try to select the smallest interval possible
between θmin and θmax. Give the length of each petal of the rose graph. For each equation,
change sine to cosine (or cosine to sine) and see how the graph changes.
Sketch
How did the graph change?
y=2 sin 3θ
y=2cos 3θ
The petals are in different positions.
y=3 sin 2θ
y=3 cos 2θ
y=5 cos 4θ
y=5 sin 4θ
y=cos 5θ
y=sin 5θ
Window settingsAnswers are one possibility. Answers will vary.
y=2 sin 3θ
y=3 sin 2θ
y =5 cos 4θ
y=2 cos 3
y=3 cos 2θ
y=5 sin 4θ
θ min
0
0
0
θ max
2.10
3.1416
6.28
θ step
.1308996
.1308996
.1308996
Xmin
-6.1522856
-6.1522856
-8
Xmax
6.1522856
6.1522856
8
Ymin
-4
-4
-6
Ymax
4
4
6
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
Page 38 of 41
y=cos 5θ
y=sin 5θ
0
6.28
.1308996
-2
3
-1.5
1.5
Columbus Public Schools 1/5/06
Polar – C
1. What does the r-coordinate of each point represent? The distance from the pole.
2. What does the θ -coordinate of each point represent? The angle from the postive x-axis.
3. How do the choices of θ min, θ max, and θ step affect the graph? θ min and θ max
determine whether the graph is completed or repeated.
4. How does the polar equation predict the length of each petal? The coefficient of the trig
function determines the length of the petal.
5. How does the polar equation predict the number of petals on the rose? If the coefficient
of θ, n, is even, then there are 2n petals. If odd, there are n petals
6. How did the graph change when you changed trig functions? Explain this in terms of the
relation of the graphs of the sine and cosine function. The graphs are the same shape
and size but in different places. This corresponds to the graphs of sine and cosine
being out of phase.
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
Page 39 of 41
Columbus Public Schools 1/5/06
Complex in Trig. Form – D
DeMoivre vs. Factoring
Name
DeMoivre's Theorem is used to powers of complex numbers in trigonometric form and to find
real and nonreal roots of complex numbers.
If z=r(cos θ +i sin θ ), then DeMoivre's Theorem tells us that the n distinct complex numbers are
ϑ + 2π k
ϑ + 2π k
n
+ i sin
) , where k = 0, 1, 2, 3, 4, …, n – 1.
r (cos
n
n
1. Write the trigonometric form of -1.
2. Use DeMoivre's Theorem to find the cube roots of -1. Express your answer in both
trigonometric and a+bi form.
3. The roots of -1 are the solutions to the equation x3+1=0. What one the obvious solution
to the equation?
4. Use synthetic division to factor x3+1.
5. Use the quadratic formula to find the other two zeros.
6. Do the answers agree with the trigonometric formula?
7. Find the fourth root of a number using both of these methods.
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
Page 40 of 41
Columbus Public Schools 1/5/06
Complex in Trig. Form – D
DeMoivre vs. Factoring
Answer Key
DeMoivre's Theorem is used to powers of complex numbers in trigonometric form and to find
real and nonreal roots of complex numbers.
If z=r(cos θ +i sin θ ), then DeMoivre's Theorem tells us that the n distinct complex numbers are
ϑ + 2π k
ϑ + 2π k
n
+ i sin
) , where k = 0, 1, 2, 3, 4, …, n – 1.
r (cos
n
n
1. Write the trigonometric form of -1. z = -1 + 0i
2. Use DeMoivre's Theorem to find the cube roots of -1. Express your answer in both
trigonometric and a+bi form.
π
π 1
3
z1 = cos + i sin = +
i
3
3 2 2
π + 2π
π + 2π
z2 = cos
+ i sin
= −1 + 0i
3
3
π + 4π
π + 4π 1
3
z3 = cos
+ i sin
= −
i
3
3
2 2
3. The roots of -1 are the solutions to the equation x3+1=0. What one the obvious solution
to the equation? -1
4. Use synthetic division to factor x3+1.
−1 1 0 0 1
(x+1)(x2 +1)
−1 1 −1
1 −1 1 0
5. Use the quadratic formula to find the other two zeros.
1
3
z1 = +
i
2 2
1
3
z2 == −
i
2 2
6. Do the answers agree with the trigonometric formula?
YES
7. Find the fourth root of a number using both of these methods.
Answers will vary.
PreCalculus Standards 3, 4 and 5
Noncartesian Representations
Page 41 of 41
Columbus Public Schools 1/5/06