Download Integrated Math 2 Unit 5 Right Triangle and Circular Trigonometry

Integrated Math 2
Unit 5
Right Triangle and Circular Trigonometry
Farmington Public Schools
Grade 9-11
Math
Hall, Lepi
Draft: 6/27/2003
Farmington Public Schools
Table of Contents
Unit Summary
………………….….…..page - 3
Stage One: Standards
Stage One identifies the desired results of the unit including the broad
understandings, the unit outcome statement and essential questions
that focus the unit, and the necessary knowledge and skills.
The Understanding by Design Handbook, 1999
…………………………….... page(s) 4-6
Stage Two: Assessment Package
Stage Two determines the acceptable evidence that students have acquired
the understandings, knowledge and skills identified in Stage One.
……………………………… page 7 - 8
Stage Three: Curriculum and Instruction
Stage Three helps teachers plan learning experiences and instruction that
aligns with Stage One and enables students to be successful in Stage two.
Planning and lesson options are given, however teachers are encouraged to
customize this stage to their own students, maintaining alignment with Stages
One and Two.
………………..……………… page 9 - 10
Appendices
….....………………………. page 11- 59
Hall, Lepi
Draft 6/27/2003
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2
Unit Summary
This unit on Right Triangle and Circular Trigonometry is the fifth unit of the year
following a unit on Power Models. It addresses the Operations, Estimation and
Approximation, Ratios Proportions and Percents, Measurement and Algebra and
Functions. The unit builds on previous knowledge of transformations, proportional
reasoning, characteristics of right triangles and basic knowledge of circles. It explores
trigonometric functions graphically, algebraically and numerically. Applications
include architecture, physics, and earth science. This is the first formal introduction of
Trigonometric topics. This unit will take between five and six weeks.
Hall, Lepi
Draft 6/27/2003
Farmington Public Schools
3
Stage One: Standards
Stage One identifies the desired results of the unit including the broad
understandings, the unit outcome statement and essential questions that focus
the unit, and the necessary knowledge and skills.
The Understanding by Design Handbook, 1999
Essential Understandings and Content Standards
Essential Math Understanding
2
Operations
Students will understand that people must
correctly select and apply appropriate number
operations in order to solve numerical problems
3
Estimation and Approximation
Students will understand that people must use
estimation and approximation in order to judge
the reasonableness of results and to guide their
mathematical thinking
4
Ratios, Proportions and Percents
Students will understand that people use ratios,
proportions and percents in order to represent
relationships between quantities and measures.
5
Measurement
Students will understand that people must
appropriately apply customary and metric
measurement units in order to approximate,
measure and compute length, area, volume, mass,
temperature, angle and time
9
Algebra and Functions
Students will understand that people must use
algebraic skills and concepts, including functions,
in order to describe real-world phenomena
symbolically and graphically, and to model
quantitative change.
Hall, Lepi
Draft 6/27/2003
Content Standard
The students will be able to:
c. Use appropriate methods for computing,
including mental math, estimation, paper-andpencil and calculator methods
a. Assess the reasonableness of answers to
problems arrived at using paper-and-pencil
techniques, mental math, formulas, calculators
and computers
a. Use ratios, proportions… to solve a wide
variety of situations, including real world
problems
d. Describe trigonometric ratios and apply them
to measuring triangles
d. Use techniques of algebra, geometry and
trigonometry to measure quantities indirectly
b. Model real-world phenomena using…
trigonometric functions
d. Translate among and use tabular, symbolic
and graphical representations of equations…and
functions
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Unit Outcome Statement
Consistently aligning all instruction with this statement will maintain focus in
this unit.
This unit will introduce students to trigonometric (a.k.a. periodic) functions and their
uses. Right triangle trigonometry will focus students’ attention on how to use
trigonometry to solve problems involving right triangles and the relationship between
angle of elevation, angle of depression, and line of sight in solving such problems.
Circular trigonometry will focus students’ attention on what periodic change is, how to
model periodic change graphically and symbolically, the relationship between area and
circumference of a circle and its parts (sectors, arcs) and how to measure angles in both
degrees and radians.
The students will:
Define sine, cosine and tangent for an acute angle of a right triangle.
Use the trigonometric ratios of sine, cosine and tangent to find missing sides and
angles in right triangles.
Graph the sine and cosine function to model periodic motion.
Recognize and perform transformations of the sine and cosine, both graphically and
symbolically using the form y = a sin (bx) +d or y = a cos (bx) +d.
Convert angle measurements between degrees and radians.
Find area, circumference, arc length and area of a sector for circles.
Essential Questions
These questions help to focus the unit and guide inquiry.
What is trigonometry?
How is trigonometry used to model and solve real world problems and situations?
How does trigonometry relate to the geometric models of triangles and circles?
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Draft 6/27/2003
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5
Knowledge and Skills
The Knowledge and Skills section includes the key facts, concepts, principles,
skills, and processes called for by the content standards and needed by students
to reach desired understandings.
The Understanding by Design Handbook, 1999
Knowledge
Students will:
Identify the parts of the sine and cosine models:
y= d + a sin(bx) and y= d + a cos (bx).
Define the sine, cosine and tangent of an acute angle of a right
triangle.
Be able to accurately draw a model using line of sight, angle of
depression and angle of elevation.
Know when to use formulas for area, circumference, arc length and
sectors of circles.
Be able to distinguish between the graphs of sine and cosine.
Skills/Processes
Students will:
Find the value for the sine, cosine and tangent of an acute angle.
Convert a radian measurement to degrees, and vice versa.
Use appropriate trigonometric functions to find the missing side
or angle of a right triangle.
Find the arc length, area, circumference, and area of a sector of a
circle.
Graph accurately the sine and cosine functions.
Thinking Skills
Students will:
Understand and apply how changes in a, b and d affect the graph
of the trigonometric functions (in isolation and combination).
Understand the connection between the transformations used in
the geometry unit taught earlier and trigonometric graphs.
Understand that all similar right triangles will have the same
values for their trigonometric ratios.
Understand how the sine and cosine functions model periodic
behavior of real world applications.
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Draft 6/27/2003
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6
Stage Two: Assessment Package
Stage Two determines the acceptable evidence that students have acquired the
understandings, knowledge and skills identified in Stage One.
Authentic Performance Task
Triangle Trig Project
Students will use the methods learned, similar triangle proportions and right
triangle trigonometry ratios to find the height of two “tall” structures on the high school
campus. They will describe the methods used, show all calculations, and give the pros
and cons of the methods used for each object that they are estimating the height of. This
will be turned in as a written report of their findings to preclude a job.
Tests, Quizzes, and Other Quick and Ongoing Checks for Understanding
Quiz 1
Trig Ratio Quiz
On this assessment students will use the trigonometric ratios for sine, cosine, and
tangent to identify the length of sides of triangles, both in fraction and decimal
form. Students will also find the measure of an angle using inverse trig ratios.
Quiz 2
Trigonometry Missing sides Quiz
On this assessment students will identify and use an appropriate method for
calculating the missing sides of right triangles.
Test 1
Right Triangle Trigonometry
On this test for right triangle trigonometry students will identify the parts of a
right triangle using a particular angle as reference, use the trig ratio or its inverse
appropriately to find lengths of sides or angles of a right triangle, translate word
problems into appropriate diagrams in order to solve for the missing parts.
Quiz 3
Trig graph quiz
On this non-calculator quiz for sine and cosine graphs, students will use their
knowledge of amplitude, period and vertical shift in order to match a graph with
its equation.
Test 2
Graphs of sine and cosine
On this test, students will be allowed to use a calculator to accurately graph
different sine and cosine functions, while paying attention to period, amplitude
and vertical shifts, as well as identify these components when given an equation
or graph, a word problem will be solved that shows the students have made the
connection between degrees and revolutions as well as how to graph the
information and come up with an appropriate equation for the situation.
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Draft 6/27/2003
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Test 3
Circle Test
On this test, students will use their knowledge or circles to find area,
circumference, arc length and area of a sector. They will convert between degree
and radian measurement, graph a trig function using radian measurement and
solve word problems involving circles.
Projects, Reports, Etc.
Circle Project
In this project each student will be assigned one of the satellites for one of the
planet in our solar system to investigate the concept of circumference and arc length as it
relates to satellite orbits.
Hall, Lepi
Draft 6/27/2003
Farmington Public Schools
8
Stage Three: Learning Experiences and Instruction
Stage Three helps teachers plan learning experiences and instruction that align
with Stage One and enables students to be successful in Stage Two.
Learning Experiences and Instruction
The learning experiences and instruction described in this section provide
teachers with one option for meeting the standards listed in Stage One. Teachers
are encouraged to design their own learning experiences and instruction, tailored
to the needs of their particular students.
High School
Lesson Topic
Right Triangle
Trigonometry
Applications
of right
triangle trig
Guiding Questions
Why does knowing the measure of
one acute angle of right triangle
determine the triangle shape?
What are the trig ratios?
What can you say about the ratio of
corresponding sides of similar right
triangles?
How do the trigonometric ratios aid
in solving for missing sides of right
triangles?
How do inverse trigonometric ratios
aid in solving for missing angles?
How are trigonometric ratios used to
solve real world problems?
Explain how are angle of depression,
angle of elevation and line of sight
related.
Suggested Sequence of Teaching and
Learning Activities
Investigation 2.2 Pg. 395-399
OYO Pg. 399
Sine, Cosine, and Tangent
worksheets
Pg. 408 O1, R2
Approx. 4 days
Investigation 2.3 Pg. 400- 405
OYO Pg. 405
Word Problem worksheet
Find missing length worksheet
Pg. 406 M1, M2, M3, O5
Authentic Assessment Trig
Problem Set trig problems
Approx. 6 days
Sine and
cosine graphs
Explain how trigonometry can be
used to model aspects of periodic
motion.
Describe the pattern of change for a
basic sine and cosine graph.
How are these trig graphs different
from other graph models we have
investigated this year?
Explain how the amplitude, period
and vertical shift of a trig function
relates to the graphs of sine and
cosine. How are each of these pieces
represented in the equation?
Paper Plate Activity
Investigation 3.3 Pg. 431-434
OYO pg. 435
Ferris Wheel Problem
Unit Circle and Sine/cosine
worksheet
OYO pg. 441
Graphing Sine/Cosine worksheet
Amplitude and Vertical Shift
worksheet
Trig graphs and matching equations
Pg. 442 M1, o3, o4, R1,
Pg. 452 #5 a
Approx. 8 days
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Circles
circumference
and area, arc
length and
area sectors
What is the relationship between arc
length and circumference? Area of
circle and area of sector?
How can arc length and area of a
sector be modeled in real world
situations?
How is central angle, arc length and
sector area related?
Circumference, Area and Sector
worksheet
Circumference and Arc Length
worksheet
Arc, Sectors etc. worksheet
Approx. 3 days
Radian and
degrees
What is a radian?
How are revolutions, radian
measurement and degrees related?
Explain how to change radian
measurement to degrees and vice
versa.
Investigation 3.2 Pg. 419-422 # 1-6
Investigation 3.4 pg. 436-44
Pg. 446 R2
Revolutions Worksheet
Radian/Degree conversion
worksheet
Approx. 5 days
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Appendices
Tests, quizzes, authentic assessment, and project attached.
Worksheets attached.
Hall, Lepi
Draft 6/27/2003
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11
Integrated Math 2
Trig Ratio Quiz
Name______________________________
Date _________
Use the diagram to the right to answer the following questions.
1. Measure of < C _________
2. Measure of < A _________
3. Length of side opposite < A ______
4.
Length of side adjacent to < B
5.
Length of side adjacent to < A ________
6.
Length of hypotenuse _________
11 ‘
5’
________
27
9.8’
Use ∆ DEF to find the following trigonometric ratios, then give the decimal value
for the trig ratio accurate to thousandths, (3 decimal places).
9
7. Sin < D ______ _______
8. Cos < D ______ _______
6m
9. Tan < D ______ _______
10.8 m
10. Sin < F ______ _______
11. Cos < F ______ ______
12. Tan < F _____ _______
Use the Pythagorean theorem to determine the missing side length (to the nearest tenth).
Then find the following trigonometric ratios, convert to decimal form to nearest
thousandths.
1.
13. Sin < R _____ ______
14. Cos < R______ ______
18 ft
15. Tan < R ______ ______
16. Sin < P ______ ______
M
17. Cos < P _____ ______
P
17 ft
18. Tan < P _____ ______
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For each of the trig ratios below fill in the correct angle name.
K
4
J
3
5
H
19.
3
= sin ∠ ______
5
Hall, Lepi
20.
4
= cos ∠ ______
5
Draft 6/27/2003
4
21. = tan ∠ ____
3
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Integrated Math 2
Trigonometry- Missing Sides Quiz
Name_________________________
Date __________________
Identify the trig function to be used to solve for x; include the set up and work!
1.
4
x
X = ____________
39º
2.
10
20º
X = ____________
x
3.
27
14º
X = ____________
x
Hall, Lepi
Draft 6/27/2003
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27º
4.
x
7
X = ____________
5.
12
x
X = ____________
57º
10
6. Given ∆ABC with right angle C, draw and label the triangle then find all the
missing parts.
∠A = 73° , c = 109 cm
Hall, Lepi
Draft 6/27/2003
Farmington Public Schools
15
Integrated Math 2
Triangle Trig Assessment
Name ______________________
Date _________
You have applied for a job with RJ Construction as a supervisor of construction. During
the interview, a discussion involving finding indirect measurements of tall structures
included both similar triangle measurement without knowing the angle measures and the
use of trigonometric ratios where knowing the angle is needed. The interviewer Mr. Hi
was interested in your ability to measure tall structures in more than one way in order to
ensure the accuracy of calculation. In order to get this job, you have agreed to complete
the following report for him showing your ability to use two different methods of finding
the height of a tall structure. The report is due within 5 days (school days) and the
directions for this report are as follows.
Directions:
1. Describe the two different methods of measuring the height of tall structures.
One method should use similar triangles, without knowing the angle
measures, and the other method should trigonometry, using the angle of
elevation. (One method using shadows on a sunny day, or mirrors.)
2. Use both methods to measure the height of two structures on the FHS campus.
(Ex. Flag pole, backstop on baseball field, height of building etc.)
3. In your report, identify and describe the structures whose height you measure.
For each structure and method of measurement, describe and give the step-bystep results of the measurements you made. Describe the mathematical
calculations you preformed to get the final results.
4. Compare the numerical results of the two measurement methods. Discuss the
possible sources of inaccuracy in each method, the relative difficulty of
applying each method, the conditions under which each method works best,
and the kind of structures that are measured accurately using each method.
Grading Rubric:
A. Format of the report follows directions
B. Clarity and accuracy of the descriptions of measurement methods.
C. Comparison of measurement results
D. Quality of discussion of pros and cons of measurement methods.
Hall, Lepi
Draft 6/27/2003
10%
35%
20%
35%
Farmington Public Schools
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Integrated Math 2
Test 1, Unit 6
Trigonometric Ratios
Name ______________________
Round all answers correctly to one decimal place!
1. Relative to angle P, label the 3 sides of the triangle shown to the right with the terms
“opposite”, “hypotenuse”, and “adjacent”.
P
2. Complete the following definition (circle the correct response): In a right triangle, the tangent
of an angle is defined as
a) the length of the side adjacent to the angle divided by the length of the side opposite the
angle.
b) the length of the hypotenuse divided by the length of the side adjacent to the angle.
c) the length of the side opposite the angle divided by the length of the hypotenuse.
d) the length of the side adjacent to the angle divided by the length of the hypotenuse.
e) the length of the side opposite the angle divided by the length of the side adjacent to the
angle.
Use the trigonometric table or your calculator to identify the following (round answers to
hundredths)
3.
Sin 27 = ____________
tan 53 = ___________ cos 45 = ___________
4.
the sine of the angle is .9965, the angle = ___________
the cosine of the angle is .0456, the angle = __________
the tangent of the angle 1.576, the angle = _________
In questions 5 – 9 round final answers to 2 decimal places and show all work! (no work = no
credit)
5. Find the length of x.
x = ________________
6. Find the length of x.
Hall, Lepi
x
5
25°
Draft 6/27/2003
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x = ________________
x
8.5
50°
7. Find the length of x.
x
10
x = ________________
40°
8. Find the measure of angle A.
A = _______________
A
25
12
9a. A right triangle uses the standard naming conventions with angles A, B, C and side lengths a,
b, c. If ∠B = 30° and c = 80 cm, find the measures of side BC two ways (one using
trigonometry and one using properties of “special” triangles). Show your work.
b. Did you get the same answer using each method? Why do you think this happened?
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Draft 6/27/2003
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10. A 15-meter cable is used to support a tree. The wire is staked to the ground and makes a 50°
angle with the ground. How high up the tree is the cable connected? Draw a sketch and show
all work.
11. A man climbs to the top of a 45-foot flagpole. He looks down at his friend on the ground at
an angle of depression of 35°. How far is his friend from the base of the flagpole? Draw a
sketch and show all work.
12. If a man with a height of 6 feet casts a 3.5-foot shadow on the ground, what angle are the
suns rays making with the ground? Draw a sketch and show all work.
Bonus: Karen is 1.6 meters tall looking at the top of a flagpole that is 10 meters away. The
angle of elevation is 42 ° . Find the height of the flagpole to the nearest tenth.
Integrated Math 2
Hall, Lepi
Name _________________
Draft 6/27/2003
Farmington Public Schools
19
Trig graph quiz
NO CALCULATOR
Date ________
Match each of the following graphs with one of the equations listed. All windows are 0 –
360 on the x axis and –5 to 5 on the y axis.
a. y = sin x
b. y = 3 sin x
c. y = 2 + cos x
e. y = cos 2 x
g. y = −1cos x
d . y = sin3 x
f . y = 3 + sin x
h. y = 2cos x
Hall, Lepi
_______
_________
_______
_______
________
_________
_______
_________
Draft 6/27/2003
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Integrated Math 2
Test, Graphs of Sine and Cosine (with Calculator)
Name ________________
1. Fill in the appropriate values for each function given below.
Trig Function
y = cos(3x)
y = 5sin(x) +1
y = 2cos(4x)
y = sin (2x) +3
y = cos(½x) -2
y = -3sin(5x)-1
Amplitude
Period (in degrees)
Vertical Shift
Problems 2-5. Accurately graph the following trig functions. Include several points.
2. Graph y = sin(x) + 2
3. Graph y = –3cos(x)
4. Graph y = cos(2x) +1
5. Graph y = 2sin(x) – 1
6. What is the amplitude of y = 3 + 2sin(4x)? ______________
7. What is the period of y = 3 + 2sin(4x)? _______________
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DRAFT: 06/24/2003
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21
8 – 11, write equations that fit each of the graphs shown.
8.
equation:
9.
equation:
10.
equation:
11.
equation:
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DRAFT: 06/24/2003
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22
12. A carnival ride, called The Wheel of Horror, is like a Ferris wheel that is inside a haunted house. You board
the ride from a platform at the height of the center the wheel. From your perspective, the Wheel of Horror
turns in a counterclockwise direction. When your seat is above the platform level, you are in the “belfry”
where you are bombarded with flying monsters. When your seat is below the level of the platform, you are in
the “dungeon” where equally scary creatures lurk.
a. The Wheel of Horror has a radius of 5 meters. Sketch The Wheel of Horror, being sure to mark the
locations of the platform, the belfry, and the dungeon.
b. You and a friend board the Wheel of Horror. Sketch a graph showing your distance, y, from the level of the
platform during 2 revolutions of the wheel (where x is in degrees). Mark the scale you use on the y axis.
Write an equation modeling this periodic motion.
EQUATION: y =_________________________________________
c. Indicate whether you are in the belfry or in the dungeon in each of the following intervals on the x-axis by marking
a 3in the appropriate place.
(0,180)
(180, 360)
(360, 540)
(540, 720)
Belfry
Dungeon
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DRAFT: 06/24/2003
Farmington Public Schools
23
Integrated Math 2
Circle Project
Name ____________________
Date __________
In the table below is the information for the radius, distance from the planet and the number of days for the orbital
period in days for the path of different satellites/planets in our solar system.
Your task is to determine:
the length in km. of the path a person would walk on the planet for a specific angle
the length of the orbit of the satellite for that same angle
how long it takes the satellite to travel that arc.
You are to do each of the above for:
a 90-degree rotation/movement
a 200-degree rotation/movement
a 315-degree rotation/movement.
In your solution be sure to discuss if these measurements are related and if so how?.
Satellite
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
Radius km Distance km Orbital period
day
Moon
1738
384000
27.3
Phobos
12
9370
.32
Deimos
8
23500
1.26
Almalthea 135
181000
.498
Io
1820
421600
1.77
Europa
1560
670900
3.55
Ganymede 2640
1070000
7.16
Calisto
2420
1880000
16.69
Himalia
85
11470000
250.6
Lysithia
12
11710000
260
Mimas
196
186000
.94
Euceladus 250
238000
1.37
Tethys
500
294700
1.888
Dione
560
377000
2.74
Rhea
765
527000
4.52
Titan
2575
1222000
15.95
Miranda
150
130000
1.41
Ariel
665
192000
2.52
Triton
1600
354000
5.88
Grading Rubric:
A. Format follows directions
B. Correct formulas used in determining responses
C. Clarity and accuracy of responses
D. Comparison of measurement results
Hall, Lepi
DRAFT: 06/24/2003
Eccentricity
.055
.015
.001
.003
0
0
.001
.01
.158
.13
.02
.004
0
.002
.001
.029
.017
.003
0
10%
35%
35%
20%
Farmington Public Schools
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Integrated Math 2
Circle Test
Name ___________________
Date ___________
1. If a circle has a diameter of 21 meters, then what is the radius of that circle? _______
2. What is the formula for the circumference of a circle? ___________
3. The distance from the center of the circle to a point on the circle is called the _______
4. In terms of π what is the circumference of a circle with radius 9 inches? ________
5. In terms of π what is the area of a garden with a diameter of 18 feet? ___________
6. What fraction of a circle does a central angel of 54 degrees cut? _________
7. How many degrees are in a complete circle? __________
8. How many radians are in a complete circle? __________
9. For the given circle find each of there values:
a. Radius
40 cm
b. Diameter
_______
c. Circumference
_______
d. Arc length
_______
e. Area of circle
_______
f. Area of sector
_______
135
10. For the given circle find each of these values:
a. Radius
_______
b. Diameter
12 inches
c. Circumference
_______
d. Arc length
_______
e. Area of circle
_______
f. Area of sector
_______
300
11. What simplified fraction of the circle is shown? _________
285
12. Convert each degree measurement to radian measurement in fraction form with correct units.
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DRAFT: 06/24/2003
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25
a. 1200
b. 1800
c. 300
d . 3300
e. 450
f . 2700
13. Convert each radian measurement to degrees.
π
2
π
3
b. 2π
c.
5
d. π
6
7
e. π
4
f.
a.
2
π
12
14. A large pizza has a diameter of 16 inches, and it is cut into 10 equal triangular slices. Half of the
pizza is topped with pepperoni and the other half has anchovies. Kate hates anchovies.
a. What is the area of one slice of Kate’s pizza? (show all formulas used and clearly state the
answer in both exact and approximate form.)
b. What is the length of the crust on her piece of pizza?
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15. The wheel of a bicycle has a radius of 16 inches. How many inches will the bicycle wheel travel in one
complete revolution? (give both exact and approximate answers)
Bonus:
Graph y = sin x using radian measurements on the x axis, give the period and amplitude of your graph..
Period ___________
Amplitude _____________
π
π
2π
2
2
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3π
DRAFT: 06/24/2003
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27
Integrated Math II
Radian and Degree Conversions
Name ____________________
A radian measures the length of an arc in relation to the radius of a circle.
If you have a circle with a radius of 1, what is the circumference? __________
The circumference of a circle is the arc length of one revolution of the circle. So one revolution of a circle with radius of 1,
measures 2π radians.
How many degrees are there in one revolution of a circle? __________
It is possible to convert degrees to radians and vice versa, using the fact that
2π radians = 360 degrees
so… π radians = _____ degrees
Example 1: Convert 135° to radian measure. Simplify answer and leave in terms of π.
135° 180°
To convert to radians, set up a proportion:
=
x
π
Solve for x:
x
135π
cross-multiply: 135π = 180x ⇒
=
180 180
135π
3π
isolate x by dividing both sides by 180: x =
⇒ x=
radians
180
4
Example 2: Convert
−3π
to degree measure.
5
⎛ −3π ⎞
⎜
⎟
π
5 ⎠
⎝
Set up a proportion:
=
x
180°
Solve for x:
⎛ −3π ⎞
cross-multiply, then simplify: π x = ⎜
⎟ 180 ⇒ π x = −108π
⎝ 5 ⎠
π x −108π
=
⇒ x = −108°
isolate x by dividing both sides by π:
π
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DRAFT: 06/24/2003
π
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Convert each degree measure to radian measure. Simplify answers and leave in terms of π.
1.300°
2. 36°
3. 6°
4.105°
5. -85°
6. 70°
7.75°
8. -860°
9. 1200°
Convert each radian measure to degree measure.
5π
11π
10.
radians
11.
radians
2
12
13.
16.
13π
radians
12
π
5
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radians
14. 4π radians
17.
5π
radians
4
DRAFT: 06/24/2003
12.
7π
radians
9
15.
−12π
radians
5
18. 3 radians
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Integrated Math II
Name ____________________
Revolutions
One revolution of a circle is equivalent to one rotation. Therefore, there are _____ degrees in one revolution and _____radians in
one revolution.
Example 1
How many degrees are swept through when a tire rotates
Step 1: Set up a proportion:
x
=
360°
1rev.
3
revolution?
4
3
rev.
4
⎛ 3⎞
Step 2: Solve for x: 1x = ⎜ ⎟ 360 ⇒ x = 270°
⎝4⎠
Example 2
How many revolutions are swept through when a wheel rotates 480°?
x
1rev.
=
480° 360°
360 x 480(1)
=
Step 2: Solve for x: 360 x = 480(1) ⇒
360
360
3
x=
revolution
4
Step 1: Set up a proportion:
Example 3
If the radius of a car tire measures 1 foot and the car travels 30 feet, how many revolutions did the tire make?
Round answer to the nearest hundredth.
Step 1: Find the circumference of the tire: C = 2π(1) ⇒ C = 2π feet
x
1rev.
=
30 ft 2π ft
2π ( x ) 1(30)
Step 3: Solve for x: 2π ( x ) = 1(30) ⇒
=
⇒ x = 4.77 revolutions
2π
2π
Step 2: Set up a proportion:
I. Practice
1. How many degrees are swept through when a tire rotates 1½ revolutions?
2.
How many radians are swept through when a tire rotates 3¾ revolutions?
3.
How many revolutions are swept through when a wheel rotates 216°?
4.
How many revolutions are swept through when a wheel rotations 5π radians?
5. If the radius of a tractor-trailer tire measures 1.5 feet and the tractor-trailer travels 150 feet, how many revolutions did the
tire make? Round answer to the nearest hundredth.
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Problem Set, Trig Problems
Name_________________________
Do problems: ___________________________________ and __________________
Directions: Include a diagram for each problem and SHOW ALL WORK! Round all answers to the nearest hundredth (2
decimal places) and include units.
1.
2.
Sam elevates his telescope 55° to spot the top of a building from a point on the ground 200 feet away.
a.
What trigonometric function of the 55° angle is equal to
b.
Use this function to find the height of the building.
A car traveled along a ramp raised 4° from the ground.
a. If the car traveled 750 meters along the ramp, how far did it rise during this distance?
b.
3.
h
?
200
If the car rises 10 meters, how far along the ramp did the car travel?
A 150-foot rope is tied from the top of a 100-foot-tall pole to a stake in the ground.
a. What is the angle between the rope and the ground?
b.
How far from the base of the pole is the stake?
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31
4.
The Aerial run in Snowbird, Utah, is 8,395 feet long. Its vertical drop is 2900 feet. If the slope were constant, estimate the
angle of elevation that the run makes with the horizontal.
5.
a.
b.
6.
A plane at a height of 5000 feet begins descending in a straight line toward a runway.
If the horizontal distance between the plane and the runway is 45,000 feet. What angle of depression must the plane use?
If the angle of depression from the plane to the runway is 5 degrees, what horizontal distance will the plane travel before
it hits the runway?
You lean a ladder 6.7 meters long against the wall.
a. If the ladder makes an angle of 63° with the ground, how high up is the top of the ladder?
b.
If the ladder makes an angle of elevation of 70° with the ground, how far is the base of the ladder from the base
of the wall?
7. You must order a new rope for the flagpole. To find out what length of rope is needed, you observe that the pole casts a
shadow 11.6 meters long on the ground. The angle of elevation of the sun is 36°. How tall is the pole?
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32
8.
Your cat is trapped on a tree branch 6.5 meters above the ground. Your ladder is only 6.7 meters long. If you place the
ladder’s tip on the branch, what angle will the ladder make with the ground?
9. The tallest freestanding structure in the world is the 553-meter tall CN Tower in Toronto, Ontario.
a. Suppose that at a certain time of day it casts a shadow 1100 meters long on the ground. What is the angle of elevation of
the sun at that time of day?
b. At around 11:00 AM, the angle of elevation of the sun is 75°. What is the length of the shadow cast by the tower?
10. A boat is anchored to the bottom of the sea with a 30-foot rope.
a. If a duck, 8 feet from the boat is swimming above the anchor, what is the angle of depression from the boat to the anchor?
b.
If the angle of elevation from the anchor to the boat is 25°. What is the horizontal distance from the boat to the anchor?
11. The top of a storage space is 4 meters above the main roof of a house. If the main roof is 9.1 meters above the ground and a
ladder that reaches the top of the storage space makes an able of 69° with the ground, how long is the ladder?
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12. Trevor is standing on top of a cliff 200 feet above a lake
a. If the measurement of the angle of depression to a boat on the lake is 21°. How far is the boat from the base of the cliff?
b.
If a person in the boat is 100 feet from the cliff, what is the angle of elevation to the person at the top of the cliff?
c.
If Trevor notices a hot air balloon that is 200 feet away horizontally, at an angle of elevation of 30°, how high above the
lake is the plane?
Extra Credit: John views the top of a water tower at an angle of elevation of 36°. He walks 120 meters in a straight line toward
the tower. Then he sights the top of the tower at an angle of elevation of 51°. How far is John from the base of the tower?
Hall, Lepi
DRAFT: 06/24/2003
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34
Integrated Math 2
Name __________________
Problem set Pg 406
M1) a)
Date ________________
A
C
63
|
|
|
|
|
|
|
123 m
b)
river
c)
B
distance
M2) - - - - - - - - - - - - - - - - - - - - - - - angle of descent
a)
altitude
Sketch this
for each of a, b and c with appropriate values.
b)
c)
d)
M3)
a)
b)
depth
of
crater
c)
angle of elevation
length of shadow
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DRAFT: 06/24/2003
d)
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35
Integrated Math 2
Pg. 408-10
Name _______________________
O1) a)
B
b) sin A =
sin A’ =
tan A =
tan A’ =
a
c
C
b
A
c)
R2)
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DRAFT: 06/24/2003
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36
Integrated Math II
Graphing Sine and Cosine
Name ____________________
Sketch each pair of functions on the same set of axes. Use 0° ≤ x ≤ 360° .
1.
y = cos x , y = 4 cos x
2.
4
1
y = sin x , y = − sin x
2
2
3
2
1
1
90
180
270
360
90
-1
-2
180
270
360
-1
-3
-4
-2
3. y = sin x , y = −3 + sin x
4.
y = cos x , y = cos 4 x
4
4
3
3
2
2
1
1
90
180
270
360
90
-1
-1
-2
-2
-3
-3
-4
-4
5. y = cos x , y = 1 + cos x
6.
2
1
1
180
270
360
90
-1
-1
-2
-2
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DRAFT: 06/24/2003
270
360
1
y = sin x , y = sin x
3
2
90
180
Farmington Public Schools
180
270
360
37
7. y = cos x , y = 2 cos x
y = sin x , y = 2 + sin x
8.
4
2
3
2
1
1
90
180
270
90
360
180
270
360
-1
-2
-1
-3
-4
-2
y = cos x , y = cos 2 x
9.
10. y = sin x , y = −1 + sin x
2
2
1
1
90
90
180
270
180
270
360
360
-1
-1
-2
-2
⎛1 ⎞
11. y = cos x , y = cos ⎜ x ⎟
⎝3 ⎠
12. y = sin x , y = 2 − sin x
4
4
3
3
2
2
1
1
90
180
270
360
90
-1
-1
-2
-2
-3
-3
-4
-4
Hall, Lepi
DRAFT: 06/24/2003
180
Farmington Public Schools
270
360
38
Integrated Math II
Practice with Circumference and Arc Lengths
Find the circumference (in terms of
1.
radius = 4 m
Name ________________________
) of the following circles.
2. r = 15 cm
3.
The arc of a circle is shown to the right:
6 in.
∩
A
4. Find these arc lengths (express in terms of
¼ of a circle with radius 8 meters =
Arc AB
)
B
½ of a circle with radius 6 inches =
¾ of a circle with radius 12 yards =
⅔ of a circle with radius 6 cm. =
⅛ of a circle with radius 24 feet =
⅜ of a circle with radius 4 meters =
⅓ of a circle with radius 1 foot =
A central angle is an angle with its vertex at the center of a circle.
5. What fraction (simplified) of the circle is each of these arcs that
are determined by the central angles given?
120
60
90
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30
Farmington Public Schools
39
6. What fraction (simplified) of the circle is each of these arcs that
are determined by the central angles given?
180º
270º
200º
36º
90
40º
45º
300º
7. Find the lengths of the arcs of these circles given these central angles and these radii. Express your answers in terms of
Central angle
Radius
60º
12 inches
90º
8 feet
120º
15 meters
20º
18 centimeters
300º
30 yards
45º
1
120º
1
180º
1
420º
1
540º
1
Hall, Lepi
DRAFT: 06/24/2003
.
Arc Length
Farmington Public Schools
40
Integrated Math II
Trig Ratio Practice
Name ________________________
B
A. A diagram of ∆ABC is given to the right.
Give the measures of the missing sides or angles.
13
5
1.
∠C
2.
∠A
3.
Side opposite ∠ A
4.
Side (leg) adjacent to ∠ B
5.
Side (leg) adjacent to ∠ A
6.
Hypotenuse
C
12
A
For ∆DEF as shown below, find the following trigonometric ratios…then divide and give the
II.
ratio to the nearest thousandth (3 decimal places).
24
F
E
Sin ∠ D = __________ =
7
Cos ∠ D = __________ =
25
Tan ∠ D = __________ =
D
Sin ∠ F = __________ =
Cos ∠ F = _________ =
Tan ∠F = _________ =
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41
III.Use the Pythagorean Theorem to determine the missing side length (to the nearest tenth).
Then find the following trigonometric ratios.
Then divide and give the ratio to the nearest thousandth (3 decimal places).
P
35
6
R
T
Sin ∠ R = __________ =
Cos ∠ R = __________ =
Tan ∠ R = __________ =
Sin ∠ P = __________ =
Cos ∠ P = _________ =
Tan ∠ P = _________ =
III.
For each of the trig ratios below fill in the correct angle:
M
4
3
N
5
P
3
= sin ∠ _____
5
Hall, Lepi
4
= cos ∠ _____
5
DRAFT: 06/24/2003
4
= tan ∠ _____
3
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42
Integrated Math 2
Areas and Sectors
name:
Find the area of each of these circles:
approximation.)
(Write the area in terms of π and then use your calculator to find an
1. r = 5 cm.
2. r = 8 in.
3. diameter = 12 inches
4. R = 1.5 m.
Complete this table. (Write the area in terms of π and then use your calculator to find an approximation.)
Use correct units>
Circle radius
Circle diameter
Area in terms of π
Approximate area
11 feet
20 cm.
25 mm.
7 m.
32 ft.
1 mile
Area of Sectors:
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11. r = 8 cm.
12.
r = 20 in.
80°
120°
Area of sector = __________
Area of sector = __________
13.
d = 18 m.
r = _________
Central Angle = 36°
Area of sector = __________
Complete this table. Use correct units.
Radius
Diameter
Central Angle
14 cm.
240°
5 feet
100°
30 mm.
Area of sector
90°
60°
8 m.
12 in.
30°
20. What is the area of a quarter circle with a radius of 15 mm.?
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44