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TRIGONOMETRY UNIT NOTES
PART 1
1
Precalculus
Pretest- Trig
Name:____________________________________
Date:_____________________________________
1.
Given triangle ABC with hypotenuse AB. If AC = 5 and BC = 9, find:
a.
AB
b.
Angle A
2.
A man stands on the top of a 1000 foot building. He notices a car on the ground and an angle of
depression of 32 degrees. How far away from the base of the building is the car?
3.
A submarine uses SONAR to detect the passage of other vessels. A submarine detects an enemy
destroyer on the surface of the ocean at an angle of elevation of 61 degrees. In order to target the
destroyer, the sub needs to determine the range to the target (the straight line distance between the two
vessels). If the sub is traveling at a depth of 650 feet, what is the range of the target?
4.
Find cos 210 without using a calculator.
2
Precalculus
Lesson- Intro to Trigonometry
Name:____________________________________
Date:_____________________________________
Objective:
To review special right triangles and learn the basics about trig ratios.
DO NOW:
1)
Find the measure of the hypotenuse of a right triangle if the legs are 1 and 2.
2)
Given equilateral triangle ABC, find the length of the altitude to AB if BC=4.
3)
What is the length of the diagonal of square ABCD if AB=5?
__________________________________________________________________________________________
The Three Basic Trig Functions
ex: Find the length of the missing side of the triangle and the exact value of the three trigonometric functions of
the angle theta ( ) in the figures below:
1)
5
12
2)
7
11
3
3)
3
4)
2
5)
Find the values of the three trigonometric functions for angle
coordinates (-3, -5) lies on its terminal side.
7
1
in standard position if a point with the
4
Trigonometric Function Origin Timeline
Hindu text
Aryabhatiya of
Aryabhata is
written and refers
to a half-chord
known as ardhajya
Ca. 510 a.d.
ardha-jya or
half chord is
turned around
to jya-ardha
(“chordhalf”) which
in due time is
shortened to
jya or jiva (or
bowstring).
Edmund Gunter
coins the term:
cotangens,
which later
becomes
cotangent or
“complement of
tangent”
The first printed
table of secants
appeared in the
work Canon
doctrinae
triangulorum
By Leipzig
Ca. 1114-1187 a.d.
Aryabhatiya is translated
from Hindu to Arabic. Jiva
becomes Jiba which
becomes Jayb. When
translated into Latin from
Arabic, Jayb (bosom of a
garment) becomes Sinus
(meaning a bend or gulf, or
the bosom of a garment).
Anatomical meaning:
cavities or bays in the
facial bones
1551
1583
Tangent is
translated
from Latin
Tangere, to
touch.
1585-95
1620
Sinus is translated to
Sine in English which
means curve, fold,
pocket. Geometric
definition
established: a line
drawn from one
extreme of an arc of a
circle to the diameter
that passes through
its other extremity.
Sir Jonas Moore
introduces “Cos”
as the
abbreviation for
“cosinus”
1658
Edmund
Gunter coins
the term
cosinus
which means
“complement
of sinus”
1674
1700
The term
“Cosecant:”
complement of
secant is coined
5
Precalculus
Activity- Discovering trig functions
Objective:
To learn the origin of trig functions.
DO NOW:
What are the 3 trig ratios?
Name:__________________________________________
Date:___________________________________________
Follow the directions below and be sure to round each answer to the nearest ten-thousandth.
1.
Each person in your group should have a set of 3 triangles:
a.
45-45-90
b.
30-60-90
c.
53-37-90
2.
Measure the dimensions of each 45-45-90 right triangle (triangle 1) and write the results in the table provided.
3.
Create sine, cosine, and tangent ratios for your 45-45-90 right triangle (results should be rounded to the nearest ten-thousandth)
4.
Average the corresponding ratios with the other members in your group.
5.
Repeat the process using the 30-60-90 triangle using the 30 degree angle as reference (triangle 2)
6.
Then repeat the process using the 60 degree angle as a reference (also triangle 2 but from a different perspective)
7.
Finally, repeat the process using the 53-37-90 triangle (triangle 3)
6
Angle
45
1
30
2
60
Also
2
53
3
Length of Horizontal
Leg
Length of Vertical
Leg
Length of
Hypotenuse
Your Ratios (4 decimals)
Group Average Ratios
Class Average Ratios
Sin 45=
Sin 45=
Sin 45=
Cos 45=
Cos 45=
Cos 45=
Tan 45=
Tan 45=
Tan 45=
Sin 30=
Sin 30=
Sin 30=
Cos 30=
Cos 30=
Cos 30=
Tan 30=
Tan 30=
Tan 30=
Sin 60=
Sin 60=
Sin 60=
Cos 60=
Cos 60=
Cos 60=
Tan 60=
Tan 60=
Tan 60=
Sin 53=
Sin 53=
Sin 53=
Cos 53=
Cos 53=
Cos 53=
Tan 53=
Tan 53=
Tan 53=
7
Follow Up Questions
1.
What did we accomplish by averaging the corresponding ratios?
2.
Will this process work for the sine, cosine, tangent for an angle of any measure? Why/why not?
3.
Find the percent difference from the observed class averages and actual values of sine, cosine, and tangent for each of the angles?
Observed Actual
100
Formula:
Actual
Example:
Let’s say my class determined the sine of 45 degrees to be 0.7273. But, my calculator says it should be 0.7071.
I would take my answer and subtract from it the actual answer and then divide by the actual answer.
.7273 .7071
100 2.8567%
.7071
That tells me that my answer was off by 2.8567% for sine. So I would enter that in the appropriate box.
So…
Angle
45
% Difference for Sine
% Difference for Cosine
% Difference for Tangent
30
60
53
4.
Are there significant percent differences between the Observed and Actual trig ratios? What factors might cause such percent variations?
8
Precalculus
Homework- Discovering trig functions
Name:__________________________________________
Date:___________________________________________
Create a set of 4 similar triangles (triangles that have the same angles but are different sizes), on separate paper, with the same reference angle
(any angle except the ones used in this activity) and complete the table. You will need a protractor.
Triangle Measure of Horizontal Leg
Measure of Vertical Leg
Measure of Hypotenuse
Your Ratios (4 decimals)
Sin___=
Cos___=
Tan___=
Sin___=
Cos___=
Tan___=
Sin___=
Cos___=
Tan___=
Sin___=
Cos___=
Tan___=
Average Ratios:
Sin___=
Cos___=
Tan___=
9
Precalculus
Activity- What is a radian?
Name:____________________________________
Date:_____________________________________
Objective:
To discover what a radian is.
Follow the directions below and be sure to round each answer to the nearest ten-thousandth. Do all work
on separate paper.
1.
Use the paper provided to cut as many horizontal strips as you can. For convenience purposes, make
sure each is about ¾ inch wide.
2.
Use a piece of tape to tape the edges together to form a quasi-cylinder.
3.
Trace the circular edge on another sheet of paper and estimate the center.
4.
Measure the distance from the center to the circumference of the circle.
5.
Unfurl the paper and measure its length.
6.
Determine how many times the length of the radius goes into the distance found in step 5.
7.
Make a table comparing your width of paper, distance from center to edge, and quotient.
8.
Find the mean of all of your quotients. What does the mean represent?
9.
How many radians are in the circumference of a circle?
10.
1 radian is approximately equal to ______________ degrees.
11.
There are ________ radians in 180 degrees and there are ________ radians in 360 degrees.
12.
In your own words, a radian is…?
PRACTICE: As you do the practice problems, see if an equation for the conversion from radians to degrees
and degrees to radians becomes apparent.
Find the radian measure
for each degree measure:
Find the degree measure
for each radian measure:
1.
720
1.
2.
90
2.
3.
45
3.
4.
60
4.
2
2
6
2
3
10
Precalculus
Lesson: The building blocks of trig functions
Name:____________________________________
Date:_____________________________________
Building Blocks of the Unit Circle
Graph set up:
Quadrantal Angles:
Coterminal Angles:
Examples:
Find an angle that is COTERMINAL with each. *Note: it is often helpful to draw a diagram when solving
these types of problems.
1.
100º
2.
650º
3.
405º
4.
400º
11
Reference Angles:
Examples:
Find an angle that is the REFERENCE ANGLE of each. *Note: it is often helpful to draw a diagram when
solving these types of problems.
5.
100º
6.
650º
7.
405º
8.
400º
Function of a positive acute angle:
Examples:
Express the given function as a function of a positive acute angle.
1.
tan 225º
2.
cos 100º
3.
cos 405º
4.
6.
cos 400º
sin 650 º
5.
tan (-120º)
12
Precalculus
13
Precalculus
14
Find the exact value of each expression without using a calculator:
1.
2.
tan 135
sin 150
3.
4.
cos 210
cos 315
Practice:
Express the given function as a function of a positive acute angle and, if possible, find the exact function value.
5.
tan 225º
6.
cos 100º
7.
cos 405º
8.
sin 650º
9.
tan (-120º)
Find the exact value of the given expression.
11.
tan 135º + sin 330º
12.
sin 300º + sin (-240º)
10.
13.
cos 400º
(sin 60º)(cos 150º) – tan (-45º)
15
Precalculus
Review- Basic Trig Quiz
Name:____________________________________
Date:_____________________________________
Objective:
To review the following concepts in preparation for a test
I.
SOHCAHTOA
II. Coterminal, reference, and quadrantals
III. Finding exact trig values
IV. Unit Circle
MIXED PBLM SET
Answer each of the following neatly and completely and show all work. NO CALCULATORS.
1.
Find the exact value of each expression:
a.
cos 270º
b.
if cos
7
and sin
8
3.
Without finding , find the exact value of tan
4.
Find the values of the three trigonometric functions for angle
coordinates (-3, -5) lies on its terminal side.
5.
Given the following triangle find the measure of angle
3
sin 90º
0.
in standard position if a point with the
exactly
6
16
8
and tan
9
if sin
6.
Without finding , find the exact value of cos
7.
Find the values of the three trigonometric functions for angle
the coordinates (5, -4) lies on its terminal side.
8.
in standard position if a point with
Find the length of the missing side and the exact value of the three trigonometric functions of the angle
in each figure:
a)
b)
8
2
12
d)
9
7
7
e)
13
c)
3
5
9.
0.
11
Find the exact value of each expression without using a calculator:
a.
b.
sin 150
cos 210
c.
cos 315
17
Scrap Page
18
Deg 0
30
45
60
90
120
135
150
180
210
225
240
270
300
315
330
360
Rad
1
6
1
4
2
6
4
6
3
4
5
6
6
,
6
4
4
7
6
5
4
8
6
9
,
6
6
4
10
6
7
4
11
6
1
2
0
2
12
6
,
8
4
0
2
Sin
0
2
1
2
2
2
3
2
3
6
,
2
4
4
2
Cos
4
2
3
2
2
2
1
2
0
2
1
2
2
2
3
2
4
2
Tan
0
1
2
3
4
3
2
1
0
1
2
3
4
3
2
1
0
4
3
2
1
0
1
2
3
4
3
2
1
0
1
2
3
4
2
2
Csc
Sec
Cot
0
6
3
2
2
2
2
2
2
0
1
2
2
2
2
2
2
2
2
2
3
4
2
3
1
2
2
2
3
2
3
2
2
2
1
2
4
2
0
2
3
2
1
2
2
2
1
2
3
2
2
2
4
2
2
2
2
2
2
2
2
2
2
2
2
2
1
0
1
2
3
4
3
2
1
0
2
2
2
2
2
2
2
3
2
1
0
1
2
2
2
4
3
2
1
0
1
2
3
4
3
4
4
3
2
1
0
1
2
3
4
3
2
1
0
1
2
3
4
0
1
2
3
4
3
2
1
0
1
2
3
4
3
2
1
0
19
Precalculus
Lesson- Reciprocal Trig Functions
Name:____________________________________
Date:_____________________________________
Objective:
To learn about the reciprocal trig functions csc, sec, cot.
DO NOW:
Construct the unit circle in the space provided.
__________________________________________________________________________________________
Reciprocal Trig Functions
Examples:
Find the exact values of the following trig functions.
1.
sec 300
2.
cot 270
3.
4.
6.
5.
(se
csc (-210)
20
Precalculus
Lesson: The Wrapping Function
Name:____________________________________
Date:_____________________________________
Objective:
Use radians to solve trig functions
DO NOW:
Determine the exact value of sin (-45) without using the calculator.
Sketch a figure and find the coordinates for each circular point on the unit circle:
5
8
1.
2.
3
6
3.
7
6
Find the sine, cosine, and tangent of each radian measure:
3
5.
6.
2
4
4.
11
3
7.
2
21
Precalculus
Wkst: The Wrapping Function
Name:____________________________________
Date:_____________________________________
22
Precalculus
Lesson: Radians and Arc Length
Name:____________________________________
Date:_____________________________________
Objective:
Discuss the relationship among central angles, radii and arc lengths.
DO NOW:
Determine the exact value of tan (-45) without using the calculator.
__________________________________________________________________________________________
What is a central angle?
What is an arc length?
Relationship among central angle, radius and arc length:
Examples:
1.
Find the measure of a positive central angle that intercepts an arc of 14 cm on a circle of radius 5 cm.
2.
Find the length of the arc intercepted by a central angle of 3.5 radians on a circle of radius 6 m.
3.
A wheel of radius 18 cm is rotating at a rate of 90 revolutions per minute.
a.
How many radians per minute is this?
b.
How many radians per second is this?
c.
How far does a point on the rim of the wheel travel in one second?
d.
Find the speed of a point on the rim of the wheel in centimeters per second.
23
Precalculus
Lesson: Working with central angles, radii and
Arc lengths
Objective:
Name:_______________________________
Date:________________________________
find the length of an arc given the measure of the central angle
find the radian measure of a central angle given an arc and the radius
(1)
Find the measure of a central angle
(2)
Given a central angle of
opposite an arc of 24 cm in a circle with a radius of 4 cm.
2
, find the length of its intercepted arc in a circle of radius 14 cm, rounding to
3
the nearest tenth.
24
(3)
Given a central angle of 125 , find the length of its intercepted arc in a circle of radius 7 feet, rounding to
the nearest tenth.
(4)
An arc is 14.2 cm long and is intercepted by a central angle of 60 . Rounding to the nearest tenth, what
is the radius of the circle?
(5)
The diameter of a circle is 22 inches. If a central angle measures 78 , find the length of the intercepted
arc to the nearest tenth.
(6)
If the pendulum of a grandfather clock is 44 in long and swings through an arc of 6 , find the length, to
the nearest tenth of an inch, of the arc that the pendulum traces.
25
Precalculus
HW- Working with radian measure
Name:_____________________________
Date:______________________________
SHOW ALL WORK:
(1)
Find the exact radian measure of a central angle
(2)
Given a central angle of 128 , find the length of its intercepted arc in a circle of radius 5 centimeters.
(Round to the nearest tenth.)
(3)
An arc is 12 yards long and is intercepted by a central angle of
opposite an arc of 30 feet in a circle of radius 12 feet.
3
radians. Find the radius of the circle.
11
(Round to the nearest tenth.)
(4)
Find the exact radian measure of a central angle
meters.
opposite an arc of 27 meters in a circle of radius 18
26
SHOW ALL WORK:
(5)
Given a central angle of 147 , find the length of its intercepted arc in a circle of radius 10 meters.
(Round to the nearest tenth.)
(6)
An arc is 70.7 meters long and is intercepted by a central angle of
5
radians. Find the diameter of the
4
circle. (Round to the nearest tenth.)
(7)
The figure below shows a stretch of roadway where the curves are arcs of circles. Find the length of the
road from point A to point E. (Round to the nearest hundredth.)
0.70 mi
D
C
A
1.8 mi
80
84.5
1.46 mi
B
0.67 mi
E
27
Precalculus
Lesson/HW: Real world trig problems
Name:_______________________________
Date:________________________________
Objective:
solve real-world applications using trigonometric functions
(1)
Pizza is typically measured by its diameter. Steve orders a 14 inch pie and cuts it into six equal slices.
Find the length of the crust of each slice to the nearest tenth of an inch.
(2)
Use the diagram to the right to answer the following two questions:
(a) Find the angle measure in radians created by the two hands of this clock at 5:00.
(b) The minute hand of this clock is 12 centimeters long. Find how far the tip of this hand moves in
10 minutes. (Round to the nearest tenth.)
(3)
A belt connects a pulley of 2-inch radius with a pulley of 5-inch radius. If the larger pulley turns through
10 radians, through how many radians will the smaller pulley turn?
(4)
Through how many radians does a pulley of 10-centimeter diameter turn when 10 meters of rope is
pulled through it without slippage?
28
(5)
A wheel has a radius of 2 feet. As it turns, a cable connected to a
box winds onto the wheel.
(a)
(b)
225
How far (to the nearest tenth) does the box move if the
wheel turns 225 in a counterclockwise direction?
Find the number of degrees (to the nearest tenth) the wheel
must be rotated to move the box 5 feet.
BOX
2 ft
(6)
Through how many radians does a pulley of 6-inch diameter turn when 4 feet of rope is pulled through it
without slippage?
(7)
Two gears are interconnected. The smaller gear has a radius of 2 inches, and the larger gear has a radius
of 8 inches. The smaller gear rotates 330 . Through how many radians (to the nearest tenth) does the
larger gear rotate?
29
30
31
32
ANSWER THE FOLLOWING QUESTIONS SHOWING ALL WORK AND ROUND TO THE NEAREST TENTH:
(4)
The adjacent sides of a parallelogram measure 14 centimeters and 20 centimeters, and one angle
measures 57 . Find the area of the parallelogram.
(5)
The base of an isosceles triangle is 48.8 ft long and its vertex angle measures 38.6 . Find the length of
each leg.
(6)
A small rectangular park is crossed by two diagonal paths, each 280 m long, that intersect at a 34 angle.
Find the dimensions of the park.
33
Precalculus
Lesson- More geometric applications of trig
Name:_______________________________
Date:________________________________
Objective:
solve geometric applications using trigonometric functions
ANSWER THE FOLLOWING QUESTIONS SHOWING ALL WORK AND ROUND TO THE NEAREST TENTH:
(1)
If a regular pentagon is inscribed in a circle of radius 5.35 centimeters, find the length of one side of the
pentagon.
(2)
If a circle of radius 4 feet has a chord of length 3 feet, find the central angle that is opposite this chord to
the nearest degree.
34
ANSWER THE FOLLOWING QUESTIONS SHOWING ALL WORK AND ROUND TO THE NEAREST TENTH:
(3)
Find the perimeter of a square inscribed in a circle of radius 5 centimeters.
(4)
The sides of a parallelogram are 20 centimeters and 32 centimeters long. If the longer diagonal measures
40 centimeters, find the measures of the angles of the parallelogram.
(5)
Each base angle of an isosceles triangle measures 42 30 . The base is 14.6 meters long.
(a) Find the length of a leg of the triangle.
(b) Find the altitude of the triangle.
(c) Find the area of the triangle.
35
Trig Unit Part 1
Definitions, Laws & Formulas
The following trigonometric identities hold for all values of
expression is defined:
Reciprocal
Identities
where each
sin
1
csc
cos
1
sec
tan
1
cot
csc
1
sin
sec
1
cos
cot
1
tan
The following trigonometric identities hold for all values of
Quotient expression is defined:
Identities
sin
cos
tan
sin
cos
where each
cot
Two angles in standard position that have the same terminal side. If is the degree
Coterminal measure of an angle, then all angles measuring + 360k , where k is an integer,
Angles are coterminal with . Since angles differing in degree measure by multiples of
360 are equivalent, every angle has infinitely many coterminal angles.
A reference angle is the acute angle formed by the terminal side of the given angle
and the x-axis. So, for any angle , 0 < < 360 , its reference angle is defined
by:
(a)
, when the terminal side is in Quadrant I
(b) 180 – , when the terminal side is in Quadrant II
Reference Angles
(c)
– 180 , when the terminal side is in Quadrant III
(d) 360 – , when the terminal side is in Quadrant IV
If the measure of is greater than 360 or less than 0 , it can be associated with a
coterminal angle of positive measure between 0 and 360 .
This special relationship shows that cofunctions have equivalent values for
complementary angles:
sin = cos (90 – )
cos = sin (90 – )
Cofuntions
tan = cot (90 – )
cot = tan (90 – )
sec = csc (90 – )
csc = sec (90 – )
Degree & Radian 1 radian
Conversion
Formulas 1 deg ree
180
deg rees (or about 57.3 )
180
radians (or about 0.017 radian)
Coordinates of
for any points (x, y) on the unit circle, given angle , W( ) = (cos , sin )
Circular Points
36
For any angle in standard position with measure , a point P(x, y) on its terminal
Trigonometric
Functions of an
Angle in
Standard Position
side, and r
sin
csc
x2
y
r
r
y
y 2 , the trigonometric functions of are as follows:
cos
x
r
tan
sec
r
x
cot
y
x
x
y
Length of a The length of any circular arc s is equal to the product of the measure of the radius
Circular Arc of the circle r and the radian measure of the central angle that it subtends: s r
If is the measure of the central angle expressed in radians and r is the measure of
Area of a the radius of the circle, then the area of the sector, A, (region bounded by the
1 2
Circular Sector
r
central angle and the intercepted arc) is: A
2
If is the measure of the central angle expressed in radians and r is the measure of
Area of a the radius of the circle, then the area of the segment, A, (region bounded by the
1 2
Circular Segment
r
sin
intercepted arc and its chord)is: A
2
37
Precalculus
Review- Basic Trig Test 1 Part 1
Name:_______________________________
Date:________________________________
Find the exact value of each expression:
1.
sec
5.
sin 150
8.
Sketch a figure and find the coordinates for each circular point:
8
5
7
W
W
W
b.
c.
3
6
6
a.
3
2.
tan 135
3.
cot 270
6.
cos 210
7.
cos 315
4.
sec
d.
2
W
11
3
9.
Without finding , find the exact value of csc
if cos
7
and cot
8
0.
10.
Without finding , find the exact value of sec
if sin
8
and cot
9
0.
11.
Find the values of the six trigonometric functions for angle
the coordinates (5, -4) lies on its terminal side.
in standard position if a point with
12.
Find the values of the six trigonometric functions for angle
coordinates (-3, -5) lies on its terminal side.
in standard position if a point with the
13.
If csc
15.
Find the length of the missing side and the exact value of the six trigonometric functions of the angle
in each figure:
2.5 , find sin
a.
14.
If cot
0.75 , find tan
b.
7
11
8
7
38
Precalculus
Review- Basic Trig Test 1 Part 2
Name:_______________________________
Date:________________________________
(1)
(2)
Convert from degrees to radians in simplest fractional form:
a.
11.83
b.
47.5
c.
100
c.
7
rad
9
Convert from radians to degrees:
a.
10
rad
b.
1.3 rad
(3)
Find the measure of a central angle
(4)
Given a central angle of 18 , find the length of its intercepted arc in a circle of radius 5 feet. (Round to
the nearest tenth.)
(5)
An arc is 1.5 feet long and is intercepted by a central angle of
opposite an arc of 3 meters in a circle with a radius of 1 meter.
4
radians. What is the diameter of the
circle? (Round to the nearest tenth.)
(6)
A sector has an arc length of 6 feet and a central angle of 1.2 radians.
(a) Find the radius of the circle.
(b) Find the area of the sector.
(7)
A sector has an area of 15 square inches and a central angle of 0.2 radians.
(a) Find the radius of the circle to the nearest tenth.
(b) Find the arc length of the sector to the nearest tenth.
(8)
Given a central angle of 20 , find the length of the radius of the circle, to the nearest tenth, whose
intercepted arc has a length of 40 cm.
(9)
Steve rides his bike 3.5 kilometers. If the radius of the tire on his bike is 32 centimeters, determine the
number of radians that a spot on the tire will travel during the trip.
(10)
Two gears are interconnected. The smaller gear has a radius of 3 inches, and the larger gear has a radius
of 7 inches. The smaller gear rotates 250 . Through how many degrees, to the nearest tenth, does the
larger gear rotate?
(11)
Using the accompanying diagram, find the area of the shaded region, to the nearest tenth,
if a pentagon is inscribed in a circle that has a radius of 3.82 feet.
(12)
A regular octagon is inscribed in a circle with radius of 5 feet. Find the area of the octagon to the
nearest tenth.
39
TRIGONOMETRY UNIT NOTES
PART 2
40
Precalculus
Lesson- Inverse Trig Functions
Name:____________________________________
Date:_____________________________________
Objective:
To learn to use inverse trig functions to solve for an angle or angles
DO NOW:
Find the exact value of sec
5
4
__________________________________________________________________________________________
What is an inverse trig function? What is it used for?
Examples:
1.
Write in the form of an inverse function:
cos
2.
Write in the form of an inverse function:
cos 45
3.
Solve by finding the value of x to the nearest degree:
Sin 1 ( 1)
x
4.
Solve by finding the value of x to the nearest degree:
Arc cos
1
2
x
2
2
Find each value (put angles in radian measure). Round any decimals to the nearest hundredth.
5.
Arc tan
3
3
6.
cos 2 Sin
1
3
2
41
Precalculus
Lesson- Real world trig problems- SOHCAHTOA
Name:_______________________________
Date:________________________________
Objective:
solve real-world applications using right triangles and trigonometric functions
(1)
To measure the height of a cloud, you place a bright searchlight directly below the
cloud and shine the beam straight up. From a point 100 feet away from the
searchlight, the measure of the angle of elevation of the cloud is 83 12 . To the
nearest tenth of a foot, find the height of the cloud.
(2)
A ranger spots a fire from a 73 foot tower in Polynomial Park. The measure of the
angle of depression is 1 20 . To the nearest tenth of a foot, find how far the fire is
from the tower.
(3)
A lighthouse 25 meters tall stands at the top of a vertical cliff. A boatman directly offshore finds that, to
the nearest degree, the angles of elevation of the top and bottom of the lighthouse are 28 and 24 ,
respectively. To the nearest meter, find how far he is from the bottom of the cliff.
25 m
28
24
42
(4)
A large, helium-filled penguin is moored at the beginning of a parade route awaiting the start of the
parade. Two cables attached to the underside of the penguin makes angles of 48 and 40 with the
ground and are in the same plane as a perpendicular line from the penguin to the ground. If the cables
are attached to the ground 10 feet from each other, to the nearest tenth, how high above the ground is the
penguin?
10 ft
(5)
A person standing 100 feet from the bottom of a cliff notices a tower on top of the cliff. The angle of
elevation to the top of the cliff is 30 . The angle of elevation to the top of the tower is 58 . To the
nearest tenth, find the height of the tower.
58
30
100 ft
(6)
Romeo stands 200 feet away from Juliet’s castle. He looks up to her balcony at an
angle of elevation of 30 . He then sees a dove perched on the top of the castle,
shifting his gaze to an angle of elevation of 40 .
(a) If Romeo wants to climb up to Juliet’s balcony, how many feet, to the nearest
tenth, would he have to climb?
(b) To the nearest tenth, how many feet above the ground is the dove?
43
(7)
While hiking on a level path toward Colorado’s front range, Brian determines that the
angle of elevation to the top of Long’s Peak is 30 . Moving 1000 feet closer to the
mountain, Brian determines the angle of elevation to be 35 . To the nearest foot, how
much higher is the top of Long’s Peak than Brian’s elevation?
(8)
An exit ramp leading to a freeway overpass is 470 feet long and rises about 32 feet.
What is the average angle of inclination of the ramp to the nearest tenth of a degree?
(9)
Tanya places her surveyor’s telescope on the top of a tripod 5 feet above the ground. She measures an 8
elevation above the horizontal to the top of a tree that is 120 feet away. To the nearest hundredth, find
the height of the tree.
8
120 ft
5 ft
(10) A 75 foot long conveyor is used at Function Farm to put hay bales up for winter
storage. The conveyor is tilted to an angle of elevation of 22 .
(a) Rounding to the nearest tenth, to what height can the hay be moved?
(b) If Farmer Fred repositions the conveyor to an angle of 27 , how much higher can
hay be moved compared to its initial configuration, to the nearest tenth?
the
44
solve real-world applications using right triangles and trigonometric Junctions
0,„ '
[DO
1
71^
(4
Zf"
A lighthouse 25 meters tall stands at the top of a vertical cliff A boatman directly offshore finds
that, to the nearest degree, the angles of elevation of the top and bottom of the hghthouse are 28°
and 24°, respectively. To the nearest meter, find how far he is from the bottom of the cliff.
77^ 2^°
\
(3)
X
A ranger spots a fire from a 73 foot tower in Polynomial Park. The measure of the
angle of depression is 1°20'.- To the nearest tenth of a foot, fmd how far the fire is
^^thetower.
^,
.
^
(2)
•
Date:
Name:
To measure the height of a cloud, you place a bright searchlight directly below the
cloud and shine the beam straight up. From a point 100 feet away fi-om the
searchlight, the measure of the angle of elevation of the cloud is 83°12'. To the
nearest tenth of a foot, find the height of the cloud.
Objective:
Precalculus
Lesson- Real world trig problems
®
(6)
(5)
(4)
1(^0.
7X0 m " -
2 =
zoo
/W."?'
Jloo
6 _
(b) To the nearest tenth, how many feet above t
(a) If Romeo wants to climb up to Juliet's balco
tenth, would he have to climb?
Romeo stands 200 feet away from Juliet's castl
angle of elevation of 30°. He then sees a dove
shifting his gaze to an angle of elevation of 40°.
loo
A person standing 100 feet from the bottom of a
of elevation to the top of the cliff is 30°. The an
the nearest tenth, find the height of the tower.
Y-^lo
A large, helium-filled penguin is moored at the
parade. Two cables attached to the underside o
ground and are in the same plane as a perpendic
cables are attached to the ground 10 feet from e
ground is the penguin?
(7)
While hiking on a level path toward Colorado's front range, Brian determine
angle o f elevation to the top o f Long's Peak is 30° . Moving 1000 feet close
mountain. Brian determines the angle o f elevation to be 35° . To the nearest
much higher is the top o f Long's Peak than Brian's elevation?
3^
T l , 30-'~
X
/c,^
-2.0
4-
(8)
A n exit ramp leading to a freeway overpass is 470 feet long and rises abou
What is the average angle o f inclination o f the ramp to the nearest tenth o f
3Z
(9)
Tanya places her surveyor's telescope on the top o f a tripod 5 feet above
an 8° elevation above the horizontal to the top o f a tree that is 120 feet a
hundredth, fmd the height o f the tree.
(10)
A 75 foot long conveyor is used at Function Farm to put hay bales up for w
storage. The conveyor is tilted to an angle o f elevation o f 22° .
I
(a)
(b)
Sir.
Rounding to the nearest tenth, to what height can the hay be moved?
I f Farmer Fred repositions the conveyor to an angle o f 27° . how much hi
the hay be moved compared to its initial configuration, to the nearest te
?2
Q
I
-
3 ^ - 0
Precalculus
Lesson: Law of Cosines
Name:____________________________________
Date:_____________________________________
Objective:
solve triangles by using the Law of Cosines
Law of Cosines:
1.
Suppose a triangle ABC has side a = 4, side b = 7, and angle C = 54º. What is the measure of side C?
2.
Suppose a triangle XYZ has sides of x = 5, y = 6, and z = 7. What is the measure of the angle across
from the side of measure 6?
45
3.
Suppose a triangle ABC has side b = 2, side a = 5, and angle B = 27º. Find the measure of side c.
4.
Suppose a triangle ABC has side b = 4, side a = 5, and angle B = 27º. Find the measure of side c.
Exit Ticket: Complete on separate paper and hand in when finished.
1.
2.
In a triangle PQR we have p = 8 and r = 11. Angle Q is 47º. What is the length of side q?
A triangle XYZ has sides x = 1, y = 2, and z = 2.5. What is the measure of angle Y?
46
Precalculus
Lesson- Forces and the Law of Cosines
Name:____________________________________
Date:_____________________________________
Objective:
To determine the resultant force vector when given two force vectors and an included angle.
DO NOW:
If m A 30 , AC=5, and AB=7, solve the triangle. Find all sides to the nearest tenth and
angles to the nearest degree.
__________________________________________________________________________________________
Force- push or pull upon an object resulting from the object's interaction with another object.
Vector- a quantity of force having both magnitude and direction.
Examples:
1.
Two forces separated by 52 degrees acts on an object at rest. The magnitude of the two forces are 32
Newtons and 17 Newtons. Find the resultant force vector to the nearest Newton.
47
2.
A game of “Three Way Tug-O-War” is being played by a group of students. Two of the students are
trying to gang up on the other. They believe that it will be easier to win if they increase the angle they
create with the third person. Is that true? Justify your answer by providing examples.
3.
Two fisherman have hooked the same fish and they are trying to cooperatively reel it in. The angle the
fisherman make with the fish is 87 degrees. If the first fisherman’s line has a maximum tensile strength
223 Newtons and the second fisherman’s line has a maximum tensile strength of 401 Newtons and the
fishermans’ lines are at maximum strain, what is the resultant force applied to the fish?
4.
What is the angle separating two component force vectors whose magnitude are 15N and 17N
respectively if the resultant vector is 21N?
48
49
(2)
(3)
Given ABC where A = 13 , B = 65 20 , and a = 35:
(a)
Solve
(i)
(ii)
(iii)
(b)
Find the area of ABC, to the nearest tenth, using the formula K
ABC such that:
C is in DMS form
b is rounded to the nearest tenth
c is rounded to the nearest tenth
1
bc sin A
2
Given GHJ where g = 45.7, H = 111.1 , and J = 27.3 :
(a)
Solve GHJ, rounding answers to the nearest tenth
(b)
Find the area of
GHJ (to the nearest tenth)
50
Precalculus
Lesson: Real world trig problems
with Law of Cosines & Sines
Objective:
(1)
Name:_______________________________
Date:________________________________
solve real-world applications using Law of Sines and Law of Cosines
A derrick at the edge of a dock has an arm 25 meters long that makes a 122 angle with the floor of the
dock. The arm is to be braced with a cable 40 meters long from the end of the arm back to the dock. To
the nearest tenth of a meter, how far from the edge of the dock will the cable be fastened?
40 m
25 m
122
(2)
A triangular course for a 30 km yacht race has distances of 7 km, 9 km, and 14 km
long. Find the largest angle of the course to the nearest degree.
(3)
A lamppost tilts toward the sun at a 2 angle from the vertical and casts a 25 foot shadow. The angle
from the tip of the shadow to the top of the lamppost is 45 . Find the length of the lamppost to the
nearest tenth of a foot.
2
45
25 ft
51
(4)
Using the picture seen to the right, and rounding to the nearest tenth of a
meter, find the height of the tree.
110
23
120 m
(5)
Using the picture seen to the right, and rounding to the nearest hundredth,
find the area of the jib sail.
25
105
2.5 m
(6)
During an expedition, two hikers start at point A and head in a direction 30 west of
south to point B. They hike 6 miles from point A to point B. From point B, they
hike to point C and then from point C back to point A, which is 8 miles directly north
of point C. To the nearest tenth of a mile, how many miles did they hike from point
B
B to point C?
A
6 mi
30
8 mi
C
(7)
The Shaffers plan to fence a triangular parcel of their land. One side of the property is 75
feet in length. It forms a 38 angle with another side of the property, which has not yet
been measured. The remaining side is 95 feet in length.
(a) Help the Shaffers by finding, to the nearest tenth, the length of fence needed to
enclose this parcel of their land.
(b) Using your answer from part (a) and the given information, find the area of this parcel to the nearest
square foot.
52
(3)
s
\
(2)
(boo
^ (1)
C
C o
solve real-world applications using Law of Sines and Law of Cosines
Date:
Name:_
a''
+ lol^ -
So
<x Qos 1 2 2 °
(22°
or
S'O
Y-
13S'
-25.° !
A lamppost tilts toward the sun at a 2° angle from tlie vertical and casts a 25 foot shadow. The
angle from the tip of the shadow to the top of the lamppost is 45°. Find the length of the lamppost
to the nearest tenth of a foot.
C=
A triangular course for a 30 km yacht race has distances of 7 km, 9 km. and 14 km
long. Find the largest angle of the course to the neai-est degree.
=
C'-ZZ°
A derrick at the edge of a dock has an arm 25 meters long that makes a 122° angle with the floor of
the dock. The arm is to be braced with a cable 40 meters long from tlje end of the arm back to the
dock. To the nearest tenth of a meter, how far from the edge of the dock will the cable be
fastened?
Objective:
Precalculus
Lesson: Real world trig problems
with Law of Cosines & Sines
(7)
(6)
(5)
(4)
^
g3. .(3g
^COfO^-'i
(oo _
2 % U^-
+• 1-5"
(b) Using your answer from part (a) and tlie
nearest square foot.
(a) Help the Shaffers by finding, to the near
enclose this parcel of their land.
The Shaffers plan to fence a triangular parc
feet in length. It foiTns a 3,8° angle vrith ano
been measured. The remaining side is 95 f
C"^-
During an expedition, two hikers start at poi
west of south to point B. They hike 6 miles
point B, they hike to point C and then from
miles directly north of point C. To the near
did they hike from point B to point C?
Using the picture seen to the right, and rou
hundredth, fmd the area of the jib sail.
Using the picture seen to the right, and rou
of a meter, fmd the height of the tree.
Precalculus
Lesson:
Name:____________________________________
Determining the number of
Distinct triangles (ambiguous case)
Date:_____________________________________
Objective:
To determine the number of distinct triangles that can be formed given an angle and two
consecutive sides
DO NOW:
The sides of a triangle measure 6, 7, and 9. What is the largest angle in the triangle?
__________________________________________________________________________________________
Ambiguous Case: This is the case in the Law of Sines ( SSA) where there may be none, one, or two distinct
triangles for which you can solve.
There is a shortcut method to finding the number of distinct triangles that exist:
Assume: Given two sides and one opposite angle:
If a is acute:
a b sin a no solution
If a is obtuse:
a b sin a one solution
b a b sin a two solutions
a b
a b
a b
no solution
one solution
one solution
Examples:
How Many distinct triangles can be formed from the given information?
1. a
2 , b 3, m A 45
2. a 9, b 12, and m A 35
53
54
Trig Unit Part 2
Definitions, Laws & Formulas
The following trigonometric identities hold for all values of
expression is defined:
Reciprocal
Identities
where each
sin
1
csc
cos
1
sec
tan
1
cot
csc
1
sin
sec
1
cos
cot
1
tan
The following trigonometric identities hold for all values of
Quotient expression is defined:
Identities
sin
cos
tan
sin
cos
where each
cot
Two angles in standard position that have the same terminal side. If is the degree
Coterminal measure of an angle, then all angles measuring + 360k , where k is an integer,
Angles are coterminal with . Since angles differing in degree measure by multiples of
360 are equivalent, every angle has infinitely many coterminal angles.
A reference angle is the acute angle formed by the terminal side of the given angle
and the x-axis. So, for any angle , 0 < < 360 , its reference angle is defined
by:
(a) , when the terminal side is in Quadrant I
(b) 180 – , when the terminal side is in Quadrant II
Reference Angles
(c)
– 180 , when the terminal side is in Quadrant III
(d) 360 – , when the terminal side is in Quadrant IV
If the measure of is greater than 360 or less than 0 , it can be associated with a
coterminal angle of positive measure between 0 and 360 .
This special relationship shows that cofunctions have equivalent values for
complementary angles:
sin = cos (90 – )
cos = sin (90 – )
Cofuntions
tan = cot (90 – )
cot = tan (90 – )
sec = csc (90 – )
csc = sec (90 – )
Degree & Radian 1 radian
Conversion
Formulas 1 deg ree
180
deg rees (or about 57.3 )
180
radians (or about 0.017 radian)
Coordinates of
for any points (x, y) on the unit circle, given angle , W( ) = (cos , sin )
Circular Points
55
For any angle in standard position with measure , a point P(x, y) on its terminal
Trigonometric
Functions of an
Angle in
Standard Position
x2
side, and r
sin
csc
Inverses of the
Trigonometric
Functions
y 2 , the trigonometric functions of are as follows:
y
r
r
y
cos
x
r
tan
sec
r
x
cot
y
x
x
y
Trigonometric Function
y = sin x
Inverse Trigonometric Relation
x = sin-1 y or x = arcsin y
y = cos x
x = cos-1 y or x = arccos y
y = tan x
x = tan-1 y or x = arctan y
Length of a The length of any circular arc s is equal to the product of the measure of the radius
Circular
Arc of the circle r and the radian measure of the central angle that it subtends: s r
If is the measure of the central angle expressed in radians and r is the measure of
Area of a the radius of the circle, then the area of the sector, A, (region bounded by the
1 2
Circular Sector
r
central angle and the intercepted arc) is: A
2
If is the measure of the central angle expressed in radians and r is the measure of
Area of a the radius of the circle, then the area of the segment, A, (region bounded by the
1 2
Circular Segment
r
sin
intercepted arc and its chord)is: A
2
Let ABC be any triangle with a, b, and c, representing the measures of the sides
opposite the angles with measurements A, B, and C, respectively. Then the
Law of Sines
a
b
c
following is true:
sin A sin B sin C
Let ABC be any triangle with a, b, and c, representing the measures of the sides
opposite the angles with measurements A, B, and C, respectively. Then the
a2 = b2 + c2 – 2bc cos A
Law of Cosines following are true:
b2 = a2 + c2 – 2ac cos B
c2 = a2 + b2 – 2ab cos C
Let ABC be any triangle with a, b, and c, representing the measures of the sides
opposite the angles with measurements A, B, and C, respectively. Then the area K
can be determined using one of the following six formulas:
Area of Triangles
K
K
1
bc sin A
2
1 2 sin B sin C
a
sin A
2
K
K
1
ac sin B
2
1 2 sin A sin C
b
sin B
2
K
K
1
ab sin C
2
1 2 sin A sin B
c
sin C
2
56
Precalculus
Review- Trig Test 2
Name:_______________________________
Date:________________________________
ANSWER THE FOLLOWING QUESTIONS ON A SEPARATE SHEET OF PAPER AND SHOW ALL WORK!
(1)
A boatman is heading for the entrance of Hypotenuse Harbor in Trigonometry Town. At one point, he
measures the angle of elevation to the base of a lighthouse to be 10 . After traveling for 5 miles, the
angle of elevation measures 30 .
(a) Find, to the nearest tenth, how far away the boatman is from the entrance to the
harbor when he took the second angle reading.
(b) Using your answer from part (a), find, to the nearest tenth, how far away the
boatman is from the entrance to the harbor when he took the first angle
reading.
(c) Using your answer from part (a), find, to the nearest tenth,
how high the base of the lighthouse is above the entrance
to the harbor.
Hypotenuse
Harbor
5 mi
(2)
The rim of a basketball hoop is 10 feet above the ground. The free-throw line is 15 feet
from the basket rim. If the eyes of a basketball player are 6 feet above the ground, what is the angle of
elevation, to the nearest tenth of a degree, of the player’s line of sight when shooting a
free throw to the rim of the basket?
(3)
Looking out across Hypotenuse Harbor is a lighthouse that stands 175 feet tall. How far
from shore, to the nearest tenth, is a sailboat if the angle of depression from the top of the lighthouse is
13 15 ?
(4)
A soccer game is being played in Polynomial Park. Steve is standing 35 feet from one
post of the goal and 40 feet from the other post. Howie is standing 30 feet from one post
of
the same goal and 20 feet from the other post. If the goal is 24 feet wide, which player has a greater
angle to make a shot on goal? Show an algebraic solution and explain your answer in detail.
(5)
Two adjacent apartment buildings in Geometry Garden Estates share a triangular
courtyard. They plan to install a new gate to close the courtyard that forms an angle of
10 48 with one building and an angle of 48 20 with the second building, whose length is 527 feet.
(a) Find, to the nearest tenth, the area of the courtyard.
(b) Find, to the nearest tenth, the length of this new gate.
(6)
Anthony, Bill, and Chris all live in Trigonometry Town. Anthony lives on Angle
Avenue and is 6.4
miles away from Chris’ house. Bill lives on Binomial Boulevard
and is 3.8 miles
away from Chris’ house. Chris lives on the corner of Angle Avenue and Binomial
Boulevard, where the two streets intersect and form an angle that measures 67 40 . To the
nearest tenth of a mile, find the distance between Anthony’s house and Bill’s house.
57
ANSWER THE FOLLOWING QUESTIONS ON A SEPARATE SHEET OF PAPER AND SHOW ALL WORK!
(7)
Steve is competing in Trigonometry Town’s annual canoe race, which is to be run
over a triangular
course marked by buoys A, B, and C. The distance between A and B is 100 yards,
between B and C is 160 yards, and between C and A is 220 yards. In DMS form, and
rounded to the nearest minute, find the measure of the largest angle within this course.
(8)
A landscaper wants to plant begonias along the edges of a triangular plot of land in
Polynomial Park. Two angles of the triangle measure 95 and 40 . The side between
these two angles is 80 feet long.
(a) Find, to the nearest tenth, the missing sides of this triangular plot of land.
(b) Using your answers from part (a), find, to the nearest tenth, the perimeter of this triangular
plot of land.
(c) Using your answers from part (a), find, to the nearest tenth, the area of this triangular plot
of land.
(9)
Lafawnduh and Towanda are flying kites on a windy spring day. Lafawnduh has released
250 feet of string, and Towanda has released 225 feet of string. The angle that
Lafawnduh’s kite string makes with the horizontal is 35 , while the angle that Towanda’s
kite string makes with the horizontal is 42 .
(a) Draw a labeled diagram to model this situation.
(b) Which kite is higher and by how much (to the nearest tenth)?
(10) A landscaper is developing the grounds of Polynomial Park. He has sketched
plans for the area, as shown in the quadrilateral PQRS seen to the right.
(a) A walkway is to be constructed from point S to Q. Find, to the nearest
tenth of a foot, the length of this walkway.
(b) Using your answer from part (a), find, to the nearest tenth, the area of the
park.
(11) Find all solutions for each ABC given the following information, using the Laws of Sines and Cosines.
If no solutions exist, write none. Round each answer to the nearest tenth.
(a) a = 32.9 ft, b = 42.4 ft, c = 20.4 ft
(b) c = 15 in, A = 42 , B = 68
(c) a = 172 mi, c = 203 mi, A = 38.7
(d) A = 92.6 , B = 88.9 , a = 15.2 cm
58
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r,
3:1.•
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- a (I'z -^¥20. w jeer A
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