Download How long does it take to go around a Ferris wheel?

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TRIGONOMETRIC
GRAPHS, IDENTITIES,
AND EQUATIONS
How long does it take to go around
a Ferris wheel?
828
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CHAPTER
14
APPLICATION: Ferris Wheels
The Texas State Fair
is home
to America’s tallest Ferris wheel.
The Ferris wheel takes riders to
a height of 212 feet and has a
diameter of about 203 feet. The
maximum speed of the Ferris
wheel is 1.5 rotations per minute.
Think & Discuss
Height (feet)
The graph below shows a person’s height (in feet)
above the ground while riding the Ferris wheel at
maximum speed. Use the graph to answer the
questions below.
y
200
150
100
50
0
0
20
40
60
80 100 x
Time (seconds)
1. How many seconds does it take to reach the
maximum height from the minimum height?
2. How long does it take to complete one revolution?
3. How would the graph change if the Ferris wheel
rotated faster? if the Ferris wheel had a smaller
diameter?
Learn More About It
INT
In Example 5 on p. 842 you will use trigonometric
functions to model a person’s height above the
ground while riding a Ferris wheel.
NE
ER T
APPLICATION LINK Visit www.mcdougallittell.com
for more information about Ferris wheels.
829
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Study Guide
CHAPTER
14
PREVIEW
What’s the chapter about?
Chapter 14 is about trigonometry. In Chapter 14 you’ll learn
•
•
•
how to graph trigonometric functions and transformations of trigonometric graphs.
how to use trigonometric identities and solve trigonometric equations.
how to write and use trigonometric models.
K E Y VO C A B U L A RY
Review
• identity, p. 13
• domain, p. 67
• range, p. 67
• x-intercept, p. 84
• quadratic form, p. 346
PREPARE
• local maximum, p. 374
• local minimum, p. 374
• asymptote, p. 465
• sine, p. 769
• cosine, p. 769
• tangent, p. 769
New
• periodic function, p. 831
• cycle, p. 831
• period, p. 831
• amplitude, p. 831
• trigonometric identities, p. 848
Are you ready for the chapter?
SKILL REVIEW Do these exercises to review key skills that you’ll apply in this
chapter. See the given reference page if there is something you don’t understand.
Graph the function. (Review Example 1, p. 123; Example 2, p. 250; Example 2, p. 624)
STUDENT HELP
Study Tip
“Student Help” boxes
throughout the chapter
give you study tips and
tell you where to look for
extra help in this book
and on the Internet.
1. y = º|x + 2| º 4
2. y = 2(x + 2)2 + 3
3. (x + 4)2 + (y º 1)2 = 9
Solve the equation. (Review Example 5, p. 258; Example 1, p. 291)
4. x 2 + 7x º 8 = 0
5. 9x 2 º 25 = 0
6. 3x 2 º x º 5 = 0
Evaluate the function without using a calculator. (Review Example 4, p. 786)
π
8. tan 30°
9. cos 10. sin π
7. sin 60°
4
Evaluate the expression without using a calculator. Give your answer in both
radians and degrees. (Review Example 1, p. 793)
3
2
11. sinº1 12. cosº1 13. cosº1 0
14. tanº1 (º3
)
2
2
STUDY
STRATEGY
830
Chapter 14
Here’s a
study strategy!
Multiple Methods
There is often more than one
wa
Multiple methods can help in y to do an exercise.
three situations.
(1) If you get stuck using one
me
(2) Do an exercise more than thod, try another.
one
your understanding. (3) Check way to reinforce
your work by using a
different method.
Page 1 of 7
14.1
What you should learn
GOAL 1 Graph sine and
cosine functions, as applied
in Example 3.
GOAL 2
Graphing Sine, Cosine, and
Tangent Functions
GOAL 1
In this lesson you will learn to graph functions of the form y = a sin bx and
y = a cos bx where a and b are positive constants and x is in radian measure.
The graphs of all sine and cosine functions are related to the graphs of
Graph tangent
y = sin x
functions.
Why you should learn it
and
y = cos x
which are shown below.
y
M1
1
amplitude: 1
range:
1 ≤ y ≤ 1
FE
To model repeating reallife patterns, such as the
vibrations of a tuning fork
in Ex. 52.
AL LI
RE
GRAPHING SINE AND COSINE FUNCTIONS
π
3π
2
π
π
π
2
2
m 1
2π x
3π
2
period:
2π
Graph of y = sin x
y
M1
range:
1 ≤ y ≤ 1
amplitude: 1
2π
3π
2
π
m 1
π
2
π
2
1
π
3π
2
2π x
period:
2π
Graph of y = cos x
The functions y = sin x and y = cos x have the following characteristics.
1. The domain of each function is all real numbers.
2. The range of each function is º1 ≤ y ≤ 1.
3. Each function is periodic, which means that its graph has a repeating pattern
that continues indefinitely. The shortest repeating portion is called a cycle.
The horizontal length of each cycle is called the period. Each graph shown
above has a period of 2π.
π
4. The maximum value of y = sin x is M = 1 and occurs when x = + 2nπ
2
where n is any integer. The maximum value of y = cos x is also M = 1 and
occurs when x = 2nπ where n is any integer.
3π
5. The minimum value of y = sin x is m = º1 and occurs when x = + 2nπ
2
where n is any integer. The minimum value of y = cos x is also m = º1 and
occurs when x = (2n + 1)π where n is any integer.
1
6. The amplitude of each function’s graph is (M º m) = 1.
2
14.1 Graphing Sine, Cosine, and Tangent Functions
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CHARACTERISTICS OF Y = A SIN BX AND Y = A COS BX
The amplitude and period of the graphs of y = a sin bx and y = a cos bx,
where a and b are nonzero real numbers, are as follows:
amplitude = |a|
Examples
2π
period = and
|b|
2π
4
π
2
The graph of y = 2 sin 4x has amplitude 2 and period = .
1
3
1
3
2π
2π
The graph of y = cos 2πx has amplitude and period = 1.
For a > 0 and b > 0, the graphs of y = a sin bx and y = a cos bx each have five
2π
b
key x-values on the interval 0 ≤ x ≤ : the x-values at which the maximum and
minimum values occur and the x-intercepts.
y
(0, 0)
14 p 2πb , a
12 p 2πb , 0
EXAMPLE 1
y = a sin bx
y = a cos bx
y
(0, a)
2πb , 0
2πb , a
14 p 2πb , 0
x
34 p 2πb , a
12 p 2πb , a
34 p 2πb , 0
x
Graphing Sine and Cosine Functions
Graph the function.
a. y = 2 sin x
b. y = cos 2x
SOLUTION
2π
2π
a. The amplitude is a = 2 and the period is = = 2π. The five key points are:
b
1
Intercepts: (0, 0); (2π, 0);
14 • 2π, 2 = π2, 2
34 • 2π, º2 = 32π, º2
1
• 2π, 0 = (π, 0)
2
Maximum:
STUDENT HELP
Study Tip
In Example 1 notice
how changes in a and b
affect the graphs of
y = a sin bx and
y = a cos bx. When
the value of a increases,
the amplitude is greater.
When the value of b
increases, the period is
shorter.
832
Minimum:
y
1
π
2
3π
2
x
2π
2π
b. The amplitude is a = 1 and the period is = = π. The five key points are:
b
2
Intercepts:
14 • π, 0 = π4, 0;
34 • π, 0 = 34π, 0
Maximums: (0, 1); (π, 1)
Minimum:
12 • π, º1 = π2, º1
Chapter 14 Trigonometric Graphs, Identities, and Equations
y
2
π
x
Page 3 of 7
EXAMPLE 2
Graphing a Cosine Function
1
3
Graph y = cos πx.
SOLUTION
1
3
2π
b
2π
π
The amplitude is a = and the period is = = 2. The five key points are:
Intercepts:
Maximums:
Minimum:
14 • 2, 0 = 12, 0;
34 • 2, 0 = 32, 0
0, 13; 2, 13
12 • 2, º13 = 1, º13
y
2
3
2
x
..........
The periodic nature of trigonometric functions is useful for modeling oscillating
motions or repeating patterns that occur in real life. Some examples are sound waves,
the motion of a pendulum or a spring, and seasons of the year. In such applications,
the reciprocal of the period is called the frequency. The frequency gives the
number of cycles per unit of time.
EXAMPLE 3
Modeling with a Sine Function
MUSIC When you strike a tuning fork, the vibrations cause changes in the pressure
of the surrounding air. A middle-A tuning fork vibrates with frequency f = 440 hertz
(cycles per second). You strike a middle-A tuning fork with a force that produces a
maximum pressure of 5 pascals.
a. Write a sine model that gives the pressure P as a function of time t (in seconds).
b. Graph the model.
FOCUS ON
APPLICATIONS
SOLUTION
a. In the model P = a sin bt, the maximum pressure P is 5, so a = 5. You can use
the frequency to find the value of b.
1
period
b
2π
frequency = 440 = 880π = b
The pressure as a function of time is given by P = 5 sin 880πt.
1
1
b. The amplitude is a = 5 and the period is = .
ƒ
440
The five key points are:
RE
FE
L
AL I
4140 12 • 4140, 0 = 8180, 0
14 • 4140, 5 = 17160, 5
34 • 4140, º5 = 17360, º5
Intercepts: (0, 0); , 0 ;
OSCILLOSCOPE
The oscilloscope is a
laboratory device invented
by Karl Braun in 1897. This
electrical instrument
measures waveforms and
is the forerunner of today’s
television.
Maximum:
Minimum:
P
5
1
880
1
440
14.1 Graphing Sine, Cosine, and Tangent Functions
t
833
Page 4 of 7
GOAL 2
GRAPHING TANGENT FUNCTIONS
The graph of y = tan x has the following characteristics.
π
π
1. The domain is all real numbers except odd multiples of . At odd multiples of ,
2
2
the graph has vertical asymptotes.
2. The range is all real numbers.
3. The graph has a period of π.
C H A R A C T E R I S T I C S O F Y = A TA N B X
If a and b are nonzero real numbers, the graph of y = a tan bx has these
characteristics:
π
|b|
•
The period is .
•
There are vertical asymptotes at odd multiples of .
π
2|b|
π
3
The graph of y = 5 tan 3x has period and asymptotes at
Example
π
2(3)
π
6
nπ
3
x = (2n + 1) = + where n is any integer.
The graph at the right shows five key x-values
that can help you sketch the graph of
y = a tan bx for a > 0 and b > 0. These
are the x-intercept, the x-values where the
asymptotes occur, and the x-values halfway
between the x-intercept and the asymptotes.
At each halfway point, the function’s value is
either a or ºa.
y
a
π
2b
π
4b
π
2b
x
Graphing a Tangent Function
EXAMPLE 4
3
2
Graph y = tan 4x.
SOLUTION
π
b
π
4
The period is = .
y
Intercept:
(0, 0)
Asymptotes:
x = • , or x = ;
1 π
2 4
1 π
π
x = º • , or x = º
2 4
8
14 • π4, 32 = 1π6, 32;
º14 • π4, º32 = º1π6 , º32
Halfway points:
834
1
π
8
Chapter 14 Trigonometric Graphs, Identities, and Equations
π8
π
16
π
8
x
Page 5 of 7
GUIDED PRACTICE
✓
Concept Check ✓
Vocabulary Check
1. Define the terms cycle and period.
2. What are the domain and range of y = a sin bx, y = a cos bx, and y = a tan bx?
1
x
3. Consider the two functions y = 4 sin and y = sin 4x. Which function has the
3
3
greater amplitude? Which function has the longer period?
Skill Check
✓
Find the amplitude and period of the function.
4. y = 6 sin x
5. y = 3 cos πx
2
π
7. y = sin x
3
3
8. y = 5 sin 3πx
1
6. y = cos 3x
4
x
9. y = cos 2
Graph the function.
10. y = 3 sin x
11. y = cos 4x
12. y = tan 3x
1
13. y = sin πx
4
2
14. y = 5 cos x
3
15. y = 2 tan 4x
16.
PENDULUMS The motion of a certain pendulum
can be modeled by the function
d = 4 cos 8πt
where d is the pendulum’s horizontal displacement
(in inches) relative to its position at rest and t is the time
(in seconds). Graph the function. How far horizontally
does the pendulum travel from its original position?
d
PRACTICE AND APPLICATIONS
STUDENT HELP
MATCHING GRAPHS Match the function with its graph.
Extra Practice
to help you master
skills is on p. 959.
1
17. y = 2 sin x
2
1
18. y = 2 cos x
2
19. y = 2 sin 2x
1
20. y = 2 tan x
2
21. y = 2 cos 2x
22. y = 2 tan 2x
A.
y
B.
2
C.
y
1
x
y
1
x
π
π
π4
3π
π
4
x
π
x
STUDENT HELP
HOMEWORK HELP
Examples 1, 2: Exs. 17–49
Example 3: Exs. 51–55
Example 4: Exs. 17–22,
32–43
D.
E.
y
1
x
π
F.
y
y
1
1
π
2π
x
π
14.1 Graphing Sine, Cosine, and Tangent Functions
835
Page 6 of 7
INT
STUDENT HELP
NE
ER T
HOMEWORK HELP
Visit our Web site
www.mcdougallittell.com
for help with problem
solving in Exs. 44–49.
ANALYZING FUNCTIONS In Exercises 23–31, find the amplitude and period of
the graph of the function.
23.
24.
y
25.
y
y
0.5
1
π
1
26. y = cos πx
2
1
29. y = 5 cos x
2
x
1
2
x
27. y = sin 2x
1
30. y = 2 sin πx
2
2
π
2
π
1
28. y = 3 cos x
4
1
31. y = sin 4πx
3
GRAPHING Draw one cycle of the function’s graph.
1
32. y = sin x
4
1
33. y = cos x
5
1
34. y = tan πx
4
1
35. y = sin x
4
36. y = 4 cos x
37. y = 4 tan 2x
38. y = 3 cos 2x
39. y = 8 sin x
1
40. y = 2 tan x
3
1
1
41. y = sin πx
2
4
42. y = tan 4πx
43. y = 2 cos 6πx
WRITING EQUATIONS Write an equation of the form y = a sin bx, where a > 0
and b > 0, so that the graph has the given amplitude and period.
44. Amplitude: 1
Period: 5
1
47. Amplitude: 2
Period: 3π
FOCUS ON
CAREERS
RE
FE
L
AL I
INT
NE
ER T
CAREER LINK
www.mcdougallittell.com
836
46. Amplitude: 2
Period: 4
48. Amplitude: 4
π
Period: 6
Period: 2π
49. Amplitude: 3
1
Period: 2
50. LOGICAL REASONING Use the fact that the frequency of a periodic function’s
graph is the reciprocal of the period to show that an oscillating motion with
maximum displacement a and frequency ƒ can be modeled by y = a sin 2π ft
or y = a cos 2π ft.
51.
BOATING The displacement d (in feet) of a boat’s
water line above sea level as it moves over waves can
be modeled by the function
d = 2 sin 2πt
where t is the time (in seconds). Graph the height of
the boat over a three second time interval.
52.
MUSIC A tuning fork vibrates with a frequency of 220 hertz (cycles per
second). You strike the tuning fork with a force that produces a maximum
pressure of 3 pascals. Write a sine model that gives the pressure P as a function
of the time t (in seconds). What is the period of the sound wave?
53.
SPRING MOTION The motion of a simple
spring can be modeled by y = A cos kt where y is
the spring’s vertical displacement (in feet) relative
to its position at rest, A is the initial displacement
(in feet), k is a constant that measures the elasticity
of the spring, and t is the time (in seconds). Find
the amplitude and period of a spring for which
A = 0.5 foot and k = 6.
MUSICIAN
Musicians may
specialize in classical, rock,
jazz, or many other types
of music. This profession
is very competitive and
demands a high degree of
discipline and talent in order
to succeed.
45. Amplitude: 10
Chapter 14 Trigonometric Graphs, Identities, and Equations
A
A
x
Page 7 of 7
SIGHTSEEING In Exercises 54 and 55, use the following information.
Suppose you are standing 100 feet away from the base
Not drawn
of the Statue of Liberty with a video camera. As you
to scale
videotape the statue, you pan up the side of the statue
at 5° per second.
54. Write and graph an equation that gives the height
you
h of the part of the statue seen through the video
camera as a function of the time t.
h
†
100 ft
55. Find the change in height from t = 0 to t = 1, from t = 1 to t = 2, and from
t = 2 to t = 3. Briefly explain what happens to h as t increases.
Test
Preparation
56. MULTIPLE CHOICE Which function represents the graph shown?
A
¡
C
¡
E
¡
y = tan 5x
1
2
B
¡
y = 5 tan x
y = tan 5x
D
¡
y = 5 tan 2x
1
2
y
5
π2 , 5
π
π x
π
2
1
y = 5 tan x
8
57. MULTIPLE CHOICE Which of the following is an x-intercept of the graph of
1
π
y = sin x?
3
4
★ Challenge
A
¡
4
B
¡
C
¡
2
º6
D
¡
1
E
¡
4π
SKETCHING GRAPHS Sketch the graph of the function by plotting points.
Then state the function’s domain, range, and period.
58. y = csc x
59. y = sec x
60. y = cot x
MIXED REVIEW
GRAPHING Graph the quadratic function. Label the vertex and axis of
symmetry. (Review 5.1 for 14.2)
61. y = 2(x º 5)2 + 4
62. y = º(x º 3)2 º 7
63. y = 4(x + 2)2 º 1
64. y = º3(x + 1)2 + 6
3
65. y = (x º 1)2 º 2
4
66. y = 10(x + 4)2 + 3
CALCULATING PROBABILITY Find the probability of drawing the given
numbers if the integers 1 through 30 are placed in a hat and drawn randomly
without replacement. (Review 12.5)
67. an even number, then an odd number
68. the number 30, then an odd number
69. a multiple of 4, then an odd number
70. the number 19, then the number 20
FINDING REFERENCE ANGLES Sketch the angle. Then find its reference angle.
(Review 13.3)
71. 220°
72. º155°
73. 280°
74. º510°
35π
21π
17π
5π
75. 76. 77. º
78. º
3
4
6
8
79.
PERSONAL FINANCE You deposit $1000 in an account that pays 1.5%
annual interest compounded continuously. How long will it take for the balance
to double? (Review 8.6)
14.1 Graphing Sine, Cosine, and Tangent Functions
837
Page 1 of 1
ACTIVITY 14.1
Using Technology
Graphing Calculator Activity for use with Lesson 14.1
Graphing
Trigonometric Functions
Using the List feature and the trigonometric viewing window of a graphing
calculator, you can graph the trigonometric functions y = a sin bx, y = a cos bx,
and y = a tan bx and observe the effects of changing the values of a and b.
EXAMPLE
Graph y = a sin x for a = 1, 2, and 4.
SOLUTION
1 Use the graphing calculator’s List feature to
INT
STUDENT HELP
NE
ER T
KEYSTROKE
HELP
See keystrokes for
several models of
calculators at
www.mcdougallittell.com
enter the function as y1 = {1, 2, 4} sin x. This
represents the three functions y = sin x,
y = 2 sin x, and y = 4 sin x.
2 Select the trigonometric viewing window. In this
π
window, each tick mark on the x-axis represents 2
and each tick mark on the y-axis represents 1.
Y1={1,2,4}sin(X)
Y2 =
Y3 =
Y4 =
Y5 =
Y6 =
Y7 =
ZOOM MEMORY
1:ZBox
2:Zoom In
3:Zoom Out
4:ZDecimal
5:ZSquare
6:ZStandard
7 ZTrig
3 If the functions you graph have amplitudes much
greater than 1 or periods longer than 4π, you
may need to change the parameters of the
viewing window. Then you can use the
Maximum, Minimum, and Zero features to find
the amplitude and the period of each function.
EXERCISES
Use a graphing calculator’s List feature to graph the functions for the given
values of a in the same trigonometric viewing window. Find the amplitude
and the period of each function’s graph.
1
1. y = a sin x for a = , 3, 9
2. y = a sin x for a = 10, 20, 40
3
1
3. y = a cos x for a = , 1, 2
4. y = a cos x for a = 1, 2, 4
2
Use a graphing calculator’s List feature to graph the functions for the given
values of b in the same trigonometric viewing window. Find the amplitude
and the period of each function’s graph.
1
5. y = sin bx for b = 1, 2, 4
6. y = sin bx for b = , 1, 3
3
1
1
7. y = cos bx for b = , 1, 2
8. y = cos bx for b = , 1, 3
2
3
838
Chapter 14 Trigonometric Graphs, Identities, and Equations
Page 1 of 1
ACTIVITY 14.2
Developing Concepts
GROUP ACTIVITY
Group Activity for use with Lesson 14.2
Translating and
Reflecting Trigonometric Graphs
QUESTION
How can you graph reflections and horizontal and vertical
translations of graphs of trigonometric functions?
Work with a partner.
MATERIALS
graphing calculator
EXPLORING THE CONCEPT
1 Use a graphing calculator to graph each pair of functions in the same viewing
window. How are the graphs geometrically related?
a. y = cos x
3
c. y = sin 3x
5
3
y = º sin 3x
5
b. y = 2 sin x
y = ºcos x
y = º2 sin x
2 Use a graphing calculator to graph each pair of functions in the same viewing
window. How are the graphs geometrically related?
a. y = cos x
b. y = 2 sin x
y = cos x º π
2
c. y = sin 3x
y = 2 sin x + π
2
y = sin 3(x + π)
3 Use a graphing calculator to graph each pair of functions in the same viewing
window. How are the graphs geometrically related?
a. y = cos x
1
c. y = sin 3x
4
1
y = 4 sin 3x º 2
b. y = 2 sin x
y = cos x + 1
y = 2 sin x º 1
DRAWING CONCLUSIONS
1. Predict what the graph of each function looks like by making a sketch. Check
your prediction by graphing the function on a graphing calculator.
a. y = º3 cos x
3π
b. y = sin x + 4
c. y = sin x º 4
2. Predict what the graph of each function looks like by making a sketch. Check
your prediction by graphing the function on a graphing calculator.
1
a. y = º cos x º 2
2
b. y = º3 cos (x + π)
c. y = 2 sin (x º π) + 3
π
d. y = º4 sin x º º 6
4
3. CRITICAL THINKING Use the following phrases to describe how the graph of
each function in parts (a)–(d) is related to the graph of y = sin x.
•
•
•
shifted up 1 unit
shifted left 1 unit
•
•
shifted down 1 unit
shifted right 1 unit
reflected in a horizontal line
a. y = sin (x º 1) + 1
b. y = ºsin (x º 1) º 1
c. y = ºsin (x + 1) + 1
d. y = sin (x + 1) º 1
14.2 Concept Activity
839
Page 1 of 8
14.2
What you should learn
GOAL 1 Graph translations
and reflections of sine and
cosine graphs.
Translations and Reflections of
Trigonometric Graphs
GOAL 1
GRAPHING SINE AND COSINE FUNCTIONS
In previous chapters you learned that the graph of y = a • ƒ(x º h) + k is related to
the graph of y = |a| • ƒ(x) by horizontal and vertical translations and by a reflection
when a is negative. This also applies to sine, cosine, and tangent functions.
GOAL 2 Graph translations
and reflections of tangent
graphs, as applied in Ex. 61.
T R A N S F O R M AT I O N S O F S I N E A N D C O S I N E G R A P H S
Why you should learn it
To obtain the graph of
RE
FE
To model real-life
quantities, such as the height
above the ground of a person
rappelling down a cliff in
Example 7.
AL LI
y = a sin b(x º h) + k
or
y = a cos b(x º h) + k,
transform the graph of y = |a| sin bx or y = |a| cos bx as follows.
VERTICAL SHIFT Shift the graph k units vertically.
HORIZONTAL SHIFT Shift the graph h units horizontally.
REFLECTION If a < 0, reflect the graph in the line y = k after any vertical and
horizontal shifts have been performed.
Graphing a Vertical Translation
EXAMPLE 1
Graph y = º2 + 3 sin 4x.
SOLUTION
Because the graph is a transformation of the graph of y = 3 sin 4x, the amplitude is 3
2π
π
and the period is = . By comparing the given equation to the general equation
4
2
y = a sin b(x º h) + k, you can see that h = 0, k = º2, and a > 0. Therefore,
translate the graph of y = 3 sin 4x down 2 units.
The graph oscillates 3 units up and down from
its center line y = º2. Therefore, the maximum
value of the function is º2 + 3 = 1 and the
minimum value of the function is º2 º 3 = º5.
y
1
π
8
π
4
3π
8
π
2
The five key points are:
π4 π2 On y = k : (0, º2); , º2 ; , º2
π8 38π, º5
Maximum: , 1
Minimum:
✓CHECK
You can check your graph with a graphing calculator. Use the Maximum,
Minimum, and Intersect features to check the key points.
840
Chapter 14 Trigonometric Graphs, Identities, and Equations
x
Page 2 of 8
Graphing a Horizontal Translation
EXAMPLE 2
Graph y = 2 cos 2 x º π .
3
4
SOLUTION
Because the graph is a transformation of the graph of y = 2 cos 2x, the amplitude is 2
3
2π = 3π. By comparing the given equation to the general equation
and the period is 2
3
y = a cos b(x º h) + k, you can see that h = π, k = 0, and a > 0. Therefore,
4
translate the graph of y = 2 cos 2x right π unit. (Notice that the maximum occurs
3
π
unit to the right of the y-axis.)
4
4
The five key points are:
STUDENT HELP
Study Tip
When graphing
translations of functions,
you may find it helpful to
graph the basic function
first and then translate
the graph.
On y = k :
Maximums:
Minimum:
14 • 3π + π4, 0 = (π, 0);
34 • 3π + π4, 0 = 52π, 0
0 + π4, 2 = π4, 2;
3π + π4, 2 = 134π , 2
12 • 3π + π4, º2 = 74π, º2
EXAMPLE 3
y
3
π
2π
3π
x
Graphing a Reflection
Graph y = º3 sin x.
SOLUTION
Because the graph is a reflection of the graph of y = 3 sin x, the amplitude is 3
and the period is 2π. When you plot the five key points on the graph, note that the
intercepts are the same as they are for the graph of y = 3 sin x. However, when
the graph is reflected in the x-axis, the maximum becomes a minimum and the
minimum becomes a maximum.
The five key points are:
On y = k : (0, 0); (2π, 0);
Minimum:
Maximum:
12 • 2π, 0 = (π, 0)
14 • 2π, º3 = π2, º3
34 • 2π, 3 = 32π, 3
y
3
π
2
π
3π
2
2π
x
..........
The next example shows how to graph a function when multiple transformations are
involved.
14.2 Translations and Reflections of Trigonometric Graphs
841
Page 3 of 8
EXAMPLE 4
Combining a Translation and a Reflection
1
2
Graph y = º cos (2x + 3π) + 1.
SOLUTION
Begin by rewriting the function in the form y = a cos b(x º h) + k:
1
2
1
3π
+1
2
2
1
2π
3π
The amplitude is and the period is = π. Since h = º, k = 1, and a < 0, the
2
2
2
1
3π
graph of y = cos 2x is shifted left units and up 1 unit, and then reflected in the
2
2
y = º cos (2x + 3π) + 1 = º cos 2 x º º
line y = 1. The five key points are:
On y = k :
Minimums:
Maximum:
EXAMPLE 5
14 • π º 32π, 1 = º54π, 1;
34 • π º 32π, 1 = º34π, 1
0 º 32π, 1 º 12 = º32π, 12;
π º 32π, 1 º 12 = ºπ2, 12
12 • π º 32π, 1 + 12 = ºπ, 32
y
2
3π
2
π
π
2
x
Modeling Circular Motion
FERRIS WHEEL You are riding a Ferris wheel. Your height h (in feet) above the ground
at any time t (in seconds) can be modeled by the following equation:
π
15
h = 25 sin t º 7.5 + 30
The Ferris wheel turns for 135 seconds before it stops to let the first passengers off.
FOCUS ON
APPLICATIONS
a. Graph your height above the ground as a function of time.
b. What are your minimum and maximum heights above the ground?
SOLUTION
135
2π = 30. The wheel turns = 4.5 times
a. The amplitude is 25 and the period is π
30
15
in 135 seconds, so the graph shows 4.5 cycles.
The five key points are (7.5, 30), (15, 55), (22.5, 30), (30, 5), and (37.5, 30).
h
RE
FE
L
AL I
THE FERRIS
WHEEL was
invented in 1893 by George
Washington Gale Ferris for
the World’s Columbian
Exposition in Chicago. Each
car held 60 people.
NE
ER T
50
10
t
b. Since the amplitude is 25 and the graph is shifted up 30 units, the maximum
height is 30 + 25 = 55 feet and the minimum height is 30 º 25 = 5 feet.
INT
APPLICATION LINK
www.mcdougallittell.com
842
Chapter 14 Trigonometric Graphs, Identities, and Equations
Page 4 of 8
GOAL 2
GRAPHING TANGENT FUNCTIONS
Graphing tangent functions using translations and reflections is similar to graphing
sine and cosine functions.
T R A N S F O R M AT I O N S O F TA N G E N T G R A P H S
To obtain the graph of y = a tan b(x º h) + k, transform the graph of
y = |a| tan bx as follows.
•
•
Shift the graph k units vertically and h units horizontally.
Then, if a < 0, reflect the graph in the line y = k.
EXAMPLE 6
Combining a Translation and a Reflection
π
4
Graph y = º2 tan x + .
SOLUTION
The graph is a transformation of the graph of y = 2 tan x,
so the period is π. By comparing the given equation to
y
π
4
y = a tan b(x º h) + k, you can see that h = º, k = 0,
1
and a < 0. Therefore, translate the graph of y = 2 tan x
π
4
left unit and then reflect it in the x-axis.
4
3
4
π
3π
π
π
π
π
4
4
4
4
2•1
2•1
π
On y = k : (h, k) = º, 0
4
π
π
π
π
π
Halfway points: º º , 2 = º, 2 ; º , º2 = (0, º2)
4
2
4
4•1
4•1
x
Asymptotes: x = º º = º; x = º = Rappelling
EXAMPLE 7
Modeling with a Tangent Function
You are standing 200 feet from the base of a
180 foot cliff. Your friend is rappelling down
the cliff. Write and graph a model for your
friend’s distance d from the top as a function
of her angle of elevation †.
Not drawn
to scale
your
friend
d
180 d
you
†
200 ft
SOLUTION
Use the tangent function to write an equation
relating d and †.
op p
ad j
180 º d
2 00
tan † = = Definition of tangent
200 tan † = 180 º d
Multiply each side by 200.
d = º200 tan † + 180
Solve for d.
Distance from top (ft)
RE
FE
L
AL I
d
180
120
60
0
0
20
40
†
60
Angle (degrees)
14.2 Translations and Reflections of Trigonometric Graphs
843
Page 5 of 8
GUIDED PRACTICE
✓
Concept Check ✓
Vocabulary Check
? shifts a graph horizontally or vertically.
1. Complete this statement: A(n) 2. How is the graph of y = º2 cos 3x related to the graph of y = 2 cos 3x?
3. How is the graph of y = tan 2(x º π) related to the graph of y = tan 2x?
Skill Check
✓
State whether the graph of the function is a vertical shift, a horizontal shift,
and/or a reflection of the graph of y = 4 cos 2x.
4. y = 3 + 4 cos 2x
5. y = 4 cos (2x + 1)
6. y = º4 cos 2x
7. y = 4 cos 2(x + 1)
8. y = 4 cos (2x º 1) + 3
9. y = º3 º 4 cos 2x
Graph the function.
10. y = 3 sin (x + π)
11. y = º2 cos x + 1
13. y = ºsin π(x º 2) + 3
14. y = 4 cos 2(x º π) + 1 15. y = 5 º tan 2(x º π)
16.
12. y = 2 tan (x º π)
RAPPELLING Look back at Example 7. Suppose the cliff is 250 feet high
and you are 150 feet from the base. Write and graph an equation that gives your
friend’s distance from the top as a function of her angle of elevation.
PRACTICE AND APPLICATIONS
STUDENT HELP
Extra Practice
to help you master
skills is on p. 959.
TRANSFORMING GRAPHS Describe how the graph of y = sin x or y = cos x
can be transformed to produce the graph of the given function.
17. y = 2 + sin x
π
20. y = cos x + 2
π
23. y = 5 º cos x º 4
18. y = 5 º cos x
19. y = º2 + cos x
21. y = ºsin (x + π)
π
22. y = sin x º 2
24. y = º2 º sin (x º π)
3π
25. y = 3 + cos x + 4
MATCHING Match the function with its graph.
26. y = º2 + sin (2x + π)
27. y = ºsin (x + π)
π
29. y = cos x + 2
1
30. y = 1 + sin x
2
y
A.
28. y = º3 + cos x
3131..y = 1 + 2 cos
y
B.
C.
12x + π2
y
2
1
1
x
π
π
2
2
STUDENT HELP
HOMEWORK HELP
Examples 1–4: Exs. 17–55
Example 5: Exs. 57–60
Example 6: Exs. 32–55
Example 7: Exs. 61, 62
844
y
D.
y
E.
3
1
π
x
x
Chapter 14 Trigonometric Graphs, Identities, and Equations
π
x
π
2
x
y
F.
π x
3
Page 6 of 8
GRAPHING Graph the function.
INT
NE
ER T
HOMEWORK HELP
Visit our Web site
www.mcdougallittell.com
for help with Exs. 50–55.
π
33. y = cos x º 2
1
34. y = º4 sin x
4
35. y = 1 + cos (x + π)
36. y = º1 + cos (x º π)
37. y = 2 + 3 sin (4x + π)
38. y = º4 + sin 2x
39. y = 1 + 5 cos (x º π)
40. y = 2 º sin x
3π
41. y = sin x º º 2
2
π
42. y = cos 2x + º 2
2
π
43. y = 6 + 4 cos x º 2
1
44. y = tan x
2
π
45. y = 2 º tan x + 2
π
46. y = 2 tan x º 2
47. y = º1 + tan 2x
48. y = 3 + tan (x º π)
π
1
49. y = 1 + tan 2x º 4
2
STUDENT HELP
1
32. y = 2 + sin x
2
WRITING EQUATIONS In Exercises 50–55, write an equation of the graph
described.
50. The graph of y = sin 2πx translated down 5 units and right 2 units
51. The graph of y = 3 cos x translated up 3 units and left π units
π
52. The graph of y = 5 tan 4x translated left unit and then reflected in the x-axis
4
1
53. The graph of y = sin 6x translated down 1 unit and then reflected in the line
3
y = º1
1
3
54. The graph of y = cos πx translated down units and left 1 unit, and then
2
2
3
reflected in the line y = º
2
π
1
55. The graph of y = 4 tan x translated up 6 units and right unit, and then
2
2
reflected in the line y = 6
56. CRITICAL THINKING Explain how the graph of y = sin x can be translated to
become the graph of y = cos x.
57.
HEIGHT OF A SWING A swing’s height h (in feet) above the ground is
h = º8 cos † + 10
where the pivot is 10 feet above the ground, the rope is 8 feet long, and † is the
angle that the rope makes with the vertical. Graph the function. What is the
height of the swing when † is 45°?
8 ft
10 – h
8 ft †
10 ft
h
58.
BLOOD PRESSURE The pressure P (in millimeters of mercury) against the
walls of the blood vessels of a certain person is given by
8π
3
P = 100 º 20 cos t
where t is the time (in seconds). Graph the function. If one cycle is equivalent to
one heartbeat, what is the person’s pulse rate in heartbeats per minute?
14.2 Translations and Reflections of Trigonometric Graphs
845
Page 7 of 8
ANIMAL POPULATIONS In Exercises 59
and 60, use the following information.
Biologists use sine and cosine functions
to model oscillations in predator and prey
populations. The population R of rabbits
and the population C of coyotes in a
particular region can be modeled by
Coyote
population
decreases.
Rabbit
population
increases.
π
R = 25,000 + 15,000 cos t
12
π
C = 5000 + 2000 sin t
12
where t is the time in months.
Rabbit
population
decreases.
Coyote
population
increases.
59. Graph both functions in the same coordinate
plane and describe how the characteristics of
the graphs relate to the diagram shown.
60. LOGICAL REASONING Look at the diagram above.
Explain why each step leads to the next.
61.
62.
AMUSEMENT PARK At an amusement park you
watch your friend on a ride that simulates a free fall.
You are standing 250 feet from the base of the ride and
the ride is 100 feet tall. Write an equation that gives the
distance d (in feet) that your friend has fallen as a
function of the angle of elevation †. State the domain
of the function. Then graph the function.
WINDOW WASHERS You are standing
80 feet from a 300 foot building, watching as a
window washer lowers himself to the ground.
Write an equation that gives the window washer’s
distance d (in feet) from the top of the building as
a function of the angle of elevation †. State the
domain of the function. Then graph the function.
Not drawn to scale
The Tower
d
your friend
100 d
you
†
250 ft
Not drawn
to scale
window
d
washer
300 d
you
†
80 ft
Test
Preparation
63. MULTI-STEP PROBLEM You are at the top of a 120 foot building that straddles
a road. You are looking down at a car traveling straight toward the building.
a. Write an equation of the car’s distance from the base of the building as a
function of the angle of depression from you to the car.
b. Suppose the car is between you and a large road sign that you know is one
mile (5280 feet) from the building. Write an equation for the distance between
the road sign and the car as a function of the angle of depression from you to
the car.
c. Graph the functions you wrote in parts (a) and (b) in the same coordinate plane.
d.
Writing Describe how the graphs you drew in part (c) are geometrically
related.
★ Challenge
846
64.
FERRIS WHEEL Suppose a Ferris wheel has a radius of 20 feet and operates
at a speed of 3 revolutions per minute. The bottom car is 4 feet above the ground.
Write a model for the height of a person above the ground whose height when
t = 0 is h = 44.
Chapter 14 Trigonometric Graphs, Identities, and Equations
Page 8 of 8
MIXED REVIEW
CLASSIFYING CONICS Classify the conic section and write its equation in
standard form. (Review 10.6 for 14.3)
65. 36x2 + 25y2 º 900 = 0
66. 9x2 º 16y2 º 144 = 0
67. 10x2 + 10y2 º 250 = 0
68. 100x2 + 81y2 º 100 = 0
COMBINATIONS Find the number of combinations. (Review 12.2)
69. 8C7
70. 142C1
71. 10C3
72. 14C5
73. 5C3
74. 6C2
75. 7C6
76. 100C2
EVALUATING FUNCTIONS Evaluate the six trigonometric functions of the
angle †. (Review 13.1 for 14.3)
77.
78.
12
79.
10
26
†
†
†
10
3
20
80.
†
5
81.
8
†
40
82.
5
9
16
†
=
83.
BOX LUNCHES You have made eight different lunches for eight people.
How many different ways can you distribute the lunches? (Review 12.1)
QUIZ 1
Self-Test for Lessons 14.1 and 14.2
Find the amplitude and period of the function. (Lesson 14.1)
5
1. y = sin 7x
2
2. y = cos 2x
π
3. y = sin x
2
1
4. y = sin 2πx
4
5. y = 3 cos πx
3π
6. y = 4 cos x
2
7
7. y = cos 4x
3
1
8. y = sin x
3
1
9. y = 6 sin x
8
Graph the function. (Lessons 14.1, 14.2)
10. y = 2 sin πx
3
1
11. y = cos πx
2
2
12. y = ºsin 2x
1
13. y = 4 tan x
2
14. y = º4 + 2 sin 3x
15. y = 3 tan 2(x + π)
16. y = º3 cos (x + π)
1
π
17. y = º2 + cos (x º π) 18. y = 2 º 5 tan x + 2
3
19.
GLASS ELEVATOR You are standing 120 feet from the base of a 260 foot
building. You are looking at your friend who is going up the side of the building
in a glass elevator. Write and graph a function that gives your friend’s distance d
(in feet) above the ground as a function of her angle of elevation †. What is her
angle of elevation when she is 70 feet above the ground? (Lesson 14.2)
14.2 Translations and Reflections of Trigonometric Graphs
847
Page 1 of 7
14.3
What you should learn
GOAL 1 Use trigonometric
identities to simplify
trigonometric expressions
and to verify other identities.
Use trigonometric
identities to solve real-life
problems, such as comparing
the speeds at which people
pedal exercise machines in
Example 7.
GOAL 2
Why you should learn it
RE
FE
To simplify real-life
trigonometric expressions,
such as the parametric
equations that describe
a carousel’s motion
in Ex. 65.
AL LI
Verifying Trigonometric
Identities
GOAL 1
USING TRIGONOMETRIC IDENTITIES
In this lesson you will use trigonometric identities to evaluate trigonometric
functions, simplify trigonometric expressions, and verify other identities.
ACTIVITY
Developing
Concepts
Investigating Trigonometric Identities
Use a graphing calculator to graph each side of the equation in the same
viewing window. What do you notice about the graphs? Is the equation true
for (a) no x-values, (b) some x-values, or (c) all x-values? (Set your calculator
in radian mode and use º2π ≤ x ≤ 2π and º2 ≤ y ≤ 2.)
1. sin2 x + cos2 x = 1
2. sin (ºx) = ºsin x
3. sin x = ºcos x
4. cos x = 1.5
In the activity you may have discovered that some trigonometric equations are true
for all values of x (in their domain). Such equations are called trigonometric
identities. In Lesson 13.1 you used reciprocal identities to find the values of the
cosecant, secant, and cotangent functions. These and other fundamental identities
are listed below.
F U N DA M E N TA L T R I G O N O M E T R I C I D E N T I T I E S
RECIPROCAL IDENTITIES
1
sin †
csc † = 1
cos †
sec † = 1
tan †
cot † = TANGENT AND COTANGENT IDENTITIES
sin †
cos †
tan † = c os †
sin †
cot † = PYTHAGOREAN IDENTITIES
sin2 † + cos2 † = 1
1 + tan2 † = sec2 †
1 + cot2 † = csc2 †
COFUNCTION IDENTITIES
π2 sin º † = cos †
π2 cos º † = sin †
π2 tan º † = cot †
NEGATIVE ANGLE IDENTITIES
sin (º†) = ºsin †
848
cos (º†) = cos †
Chapter 14 Trigonometric Graphs, Identities, and Equations
tan (º†) = ºtan †
Page 2 of 7
Finding Trigonometric Values
EXAMPLE 1
3
π
Given that sin † = and < † < π, find the values of the other five trigonometric
5
2
functions of †.
SOLUTION
Begin by finding cos †.
sin2 † + cos2 † = 1
Write Pythagorean identity.
35 + cos † = 1
2
3
5
2
Substitute }} for sin †.
35 cos2 † = 1 º 53 from each side.
2
Subtract }}
16
25
cos2 † = 2
Simplify.
4
5
Take square roots of each side.
4
5
Because † is in Quadrant II, cos † is negative.
cos † = ±
cos † = º
Now, knowing sin † and cos †, you can find the values of the other four trigonometric
functions.
3
4
º
5
cos †
4
cot † = = 3 = º3
sin †
5
3
sin †
tan † = = = º 4
4
co s †
º
5
5
1
sin †
11 1
5
3
1
cos †
csc † = = = 3
EXAMPLE 2
5
1
5
4
sec † = = = º
4
º
5
Simplifying a Trigonometric Expression
Simplify the expression sec † tan2 † + sec †.
SOLUTION
sec † tan2 † + sec † = sec † (sec2 † º 1) + sec †
3
EXAMPLE 3
Pythagorean identity
= sec † º sec † + sec †
Distributive property
= sec3 †
Simplify.
Simplifying a Trigonometric Expression
π2 Simplify the expression cos º x cot x.
SOLUTION
π
cos º x cot x = sin x cot x
2
co s x
= sin x sin x
= cos x
Cofunction identity
Cotangent identity
Simplify.
14.3 Verifying Trigonometric Identities
849
Page 3 of 7
You can use the fundamental identities on page 848 to verify new trigonometric
identities. A verification of an identity is a chain of equivalent expressions showing
that one side of the identity is equal to the other side. When verifying an identity,
begin with the expression from one side and manipulate it algebraically until it is
identical to the other side.
Verifying a Trigonometric Identity
EXAMPLE 4
Verify the identity cot (º†) = ºcot †.
STUDENT HELP
Study Tip
Verifying an identity is
not the same as solving
an equation. When
verifying an identity you
should not use any
properties of equality,
such as adding the same
number or expression to
both sides.
SOLUTION
cos (º†)
sin (º†)
Cotangent identity
= cos †
ºsin †
Negative angle identities
= ºcot †
Cotangent identity
cot (º†) = Verifying a Trigonometric Identity
EXAMPLE 5
cot2 x
csc x
Verify the identity = csc x º sin x.
SOLUTION
cot2 x
csc2 x º 1
= csc x
csc x
c s c2 x
csc x
Pythagorean identity
1
csc x
Write as separate fractions.
= csc x º 1
csc x
Simplify.
= csc x º sin x
Reciprocal identity
= º EXAMPLE 6
Verifying a Trigonometric Identity
sin x
1 º cos x
1 + cos x
sin x
Verify the identity = .
SOLUTION
STUDENT HELP
Study Tip
In Example 6, notice
how multiplying by an
expression equal to 1
allows you to write an
expression in an
equivalent form.
850
sin x (1 + cos x)
sin x
= 1 º cos x
(1 º cos x)(1 + cos x)
1 + cos x
Multiply by }}
.
1 + cos x
sin x (1 + cos x)
1 º cos x
Simplify denominator.
sin x (1 + cos x)
sin x
Pythagorean identity
1 + cos x
sin x
Simplify.
= 2
= 2
= Chapter 14 Trigonometric Graphs, Identities, and Equations
Page 4 of 7
GOAL 2
USING TRIGONOMETRIC IDENTITIES IN REAL LIFE
In Lesson 13.7 you learned that parametric equations can be used to describe linear
and parabolic motion. They can be used to describe other types of motion as well.
Using Parametric Equations in Real Life
EXAMPLE 7
INT
STUDENT HELP
NE
ER T
HOMEWORK HELP
Visit our Web site
www.mcdougallittell.com
for extra examples.
PHYSICAL FITNESS You and Sara are riding exercise machines that involve pedaling.
The following parametric equations describe the motion of your feet and Sara’s feet:
YOU:
SARA:
x = 8 cos 4πt
x = 10 cos 2πt
y = 8 sin 4πt
y = 6 sin 2πt
In each case, x and y are measured in inches and t is measured in seconds.
a. Describe the paths followed by your feet and Sara’s feet.
b. Who is pedaling faster (in revolutions per second)?
SOLUTION
a. Use the Pythagorean identity sin2 † + cos2 † = 1 to eliminate the parameter t.
YOU:
FOCUS ON
APPLICATIONS
SARA:
x
= cos 4πt
8
x
= cos 2πt
10
y
= sin 4πt
8
y
= sin 2πt
6
cos2 4πt + sin2 4πt = 1
x8 + 8y = 1
2
Isolate the sine.
cos2 2πt + sin2 2πt = 1
1x0 + 6y = 1
2
2
Pythagorean identity
2
y2
x2
+ = 1
1 00
36
x2 + y2 = 64
Isolate the cosine.
Substitute.
Simplify.
Your feet follow a circle with a radius of 8 inches. Sara’s feet follow an ellipse
whose major axis is 20 inches long and whose minor axis is 12 inches long.
b. The number of revolutions per second for you
and Sara is the reciprocal of the common period
of the corresponding parametric functions.
YOU:
RE
FE
L
AL I
111
111
==2
ELLIPTICAL
TRAINERS provide
excellent aerobic exercise
by combining both lower and
upper body movements. The
smooth elliptical motion
produces less impact than
experienced on a treadmill.
2π
4π
1
2
SARA:
11 1
1
= = 1
2π
1
2π
In one second, your feet travel around 2 times and Sara’s feet travel around
1 time. So, you are pedaling faster.
✓CHECK
To check your results, set a graphing calculator to parametric, radian, and
simultaneous modes. Enter both sets of parametric equations with 0 ≤ t ≤ 1 and a
t-step of 0.01. As the paths are graphed, you can see that your path is traced faster.
14.3 Verifying Trigonometric Identities
851
Page 5 of 7
GUIDED PRACTICE
✓
Concept Check ✓
Vocabulary Check
1. What is a trigonometric identity?
2. Is sec (º†) equal to sec † or ºsec †? How do you know?
3. Verify the identity 1 º sin2 x cot2 x = sin2 x. Is there more than one way to
verify the identity? If so, tell which way you think is easier and why.
4. ERROR ANALYSIS Describe what is wrong with the simplification shown.
cos x º cos x sin2 x = cos x º cos x (1 + cos2 x)
= cos x º cos x º cos3 x
= ºcos3 x
Skill Check
✓
Find the values of the other five trigonometric functions of †.
3 π
5. cos † = º, < † < π
5 2
2
π
6. tan † = , 0 < † < 3
2
4 3π
7. sec † = , < † < 2π
3 2
1
3π
8. sin † = º, π < † < 2
2
Simplify the expression.
(sec x + 1)(sec x º 1)
9. tan x
π
10. sin º x sec x
2
π
11. cos2 º x + cos2 (ºx)
2
Verify the identity.
1
12. = ºcsc x
sin (ºx)
15.
13. cot x tan (ºx) = º1 14. csc x tan x = sec x
PHYSICAL FITNESS Look back at Example 7 on page 851. Suppose your
friend Pete starts riding another machine that involves pedaling. The motion of
5π
5π
his feet is described by the equations x = 10 cos t and y = 6 sin t. What
2
2
type of path are his feet following?
PRACTICE AND APPLICATIONS
STUDENT HELP
Extra Practice
to help you master
skills is on p. 959.
852
FINDING VALUES Find the values of the other five trigonometric functions of †.
π
1
16. cos † = , 0 < † < 2
5
5
π
18. sin † = , 0 < † < 6
2
3
π
17. tan † = , 0 < † < 8
2
9 3π
20. cot † = º, < † < 2π
4 2
11 π
21. cos † = º, < † < π
12 2
7
π
22. csc † = , 0 < † < 5
2
10
3π
23. sec † = º, π < † < 3
2
1 π
24. tan † = º, < † < π
6 2
3π
25. sec † = 2, < † < 2π
2
5
3π
26. csc † = º, π < † < 3
2
3π
27. cot † = º3
, < † < 2π
2
3 π
19. sin † = , < † < π
5 2
Chapter 14 Trigonometric Graphs, Identities, and Equations
Page 6 of 7
SIMPLIFYING EXPRESSIONS Simplify the expression.
STUDENT HELP
HOMEWORK HELP
Example 1: Exs. 16–27
Examples 2, 3: Exs. 28–43
Examples 4–6: Exs. 44–53
Example 7: Exs. 61–64
28. cot x sec x
cos (ºx)
29. sin (ºx)
31. sin x (1 + cot2 x)
π
32. 1 º sin2 º x
2
tan º x
2
33. csc x
sin (ºx)
35. + cos2 (ºx)
csc x
cos2 x tan2 (ºx) º 1
36. cos2 x
π
34. cos º x csc x
2
2
π π π cos º x º 1
2
39. 1 + sin (ºx)
tan º x sec x
2
38. 1 º csc2 x
2
37. sec x º tan x
π sec x sin x + cos º x
2
41. 1 + sec x
cot x cos x
40. π
tan (ºx) sin 2 º x
30. sec x cos (ºx) º sin2 x
π
43. tan º x cot x º csc2 x
2
42. cot2 x + sin2 x + cos2 (ºx)
VERIFYING IDENTITIES Verify the identity.
44. cos x sec x = 1
45. tan x csc x cos x = 1
π
46. cos º x cot x = cos x
2
47. 2 º sec2 x = 1 º tan2 x
48. sin x + cos x cot x = csc x
cos2 x + sin2 x
49. = cos2 x
1 + tan2 x
sin2 (ºx)
50. = cos2 x
tan2 x
sin º x º 1
2
51.
= º1
1 º cos (ºx)
1 + sin x
cos x
52. + = 2 sec x
cos x
1 + sin x
cos (ºx)
53. = sec x + tan x
1 + sin (ºx)
π IDENTIFYING CONICS Use a graphing calculator set in parametric mode to
graph the parametric equations. Use a trigonometric identity to determine
whether the graph is a circle, an ellipse, or a hyperbola. (Use a square viewing
window.)
54. x = 6 cos t, y = 6 sin t
55. x = 5 sec t, y = tan t
56. x = 2 cos t, y = 3 sin t
57. x = 8 cos πt, y = 8 sin πt
58. x = 2 cot 2t, y = 3 csc 2t
t
t
59. x = cos , y = 4 sin 2
2
60. CRITICAL THINKING A function ƒ is odd if ƒ(ºx) = ºƒ(x). A function ƒ is
even if ƒ(ºx) = ƒ(x). Which of the six trigonometric functions are odd? Which
of them are even?
61.
SHADOW OF A SUNDIAL The length s of a shadow cast by a vertical
gnomon (column or shaft on a sundial) of height h when the angle of the sun
above the horizon is † can be modeled by this equation:
h sin (90° º †)
sin †
s = This equation was developed by Abu Abdullah al-Battani (circa A.D. 920).
Show that the equation is equivalent to s = h cot †. Source: Trigonometric Delights
14.3 Verifying Trigonometric Identities
853
Page 7 of 7
GLEASTON WATER MILL In Exercises 62–64, use the
following information.
Suppose you have constructed a working scale model of
the water wheel at the Gleaston Water Mill. The parametric
equations that describe the motion of one of the paddles on
each of the water wheels are as follows.
Actual waterwheel:
Scale model:
x = 9 cos 8πt
x = 0.5 cos 15πt
y = 9 sin 8πt
y = 0.5 sin 15πt
In each case, x and y are measured in feet and t is measured in minutes.
62. Use a graphing calculator to graph both sets of parametric equations.
63. How is your scale model different from the actual waterwheel?
64. How many revolutions does each wheel make in 5 minutes?
Test
Preparation
65. MULTI-STEP PROBLEM The Dentzel Carousel in Glen Echo Park near
Washington, D.C., is one of about 135 functioning antique carousels in the
United States. The platform of the carousel is about 48 feet in diameter and
makes about 5 revolutions per minute. Source: National Park Service
a. Find parametric equations that describe the ride’s motion.
b. Suppose the platform of the carousel had a diameter of 38 feet and made
about 4.5 revolutions per minute. Find parametric equations that would
describe the ride’s motion.
2
c.
Writing How are the parametric equations affected when the speed is
changed? How are the equations affected when the diameter is changed?
★ Challenge
66. Use the definitions of sine and cosine from Lesson 13.3 to derive the
Pythagorean identity sin2 † + cos2 † = 1.
67. Use the Pythagorean identity sin2 † + cos2 † = 1 to derive the other Pythagorean
identities, 1 + tan2 † = sec2 † and 1 + cot2 † = csc2 †.
MIXED REVIEW
QUADRATIC EQUATIONS Solve the equation by factoring. (Review 5.2 for 14.4)
68. x2 º 5x º 14 = 0
69. x2 + 5x º 36 = 0
70. x2 º 19x + 88 = 0
71. 2x2 º 7x º 15 = 0
72. 36x2 º 16 = 0
73. 9x2 º 1 = 0
EVALUATING EXPRESSIONS Evaluate the expression without using a
calculator. Give your answer in both radians and degrees. (Review 13.4 for 14.4)
º12
2
74. cosº1 2
75. tanº1 3
3
76. sinº1 º
2
1
77. sinº1 2
78. tanº1 (º1)
79. cosº1
GRAPHING Draw one cycle of the function’s graph. (Review 14.1)
854
80. y = 4 sin x
81. y = 2 cos x
82. y = tan 4πx
83. y = 3 tan πx
84. y = 5 cos 2x
85. y = 10 sin 4x
Chapter 14 Trigonometric Graphs, Identities, and Equations
Page 1 of 7
14.4
What you should learn
GOAL 1 Solve a
trigonometric equation.
Solving Trigonometric
Equations
GOAL 1
SOLVING A TRIGONOMETRIC EQUATION
In Lesson 14.3 you verified trigonometric identities. In this lesson you will solve
trigonometric equations. To see the difference, consider the following equations:
GOAL 2 Solve real-life
trigonometric equations,
such as an equation for the
number of hours of daylight
in Prescott, Arizona, in
Example 6.
sin2 x + cos2 x = 1
Equation 1
sin x = 1
Equation 2
Equation 1 is an identity because it is true for all real values of x. Equation 2,
however, is true only for some values of x. When you find these values, you are
solving the equation.
Why you should learn it
RE
FE
To solve many types
of real-life problems, such
as finding the position of
the sun at sunrise in
Ex. 58.
AL LI
Solving a Trigonometric Equation
EXAMPLE 1
Solve 2 sin x º 1 = 0.
SOLUTION
First isolate sin x on one side of the equation.
2 sin x º 1 = 0
Write original equation.
2 sin x = 1
Add 1 to each side.
1
2
sin x = Divide each side by 2.
1
2
1
2
π
6
One solution of sin x = in the interval 0 ≤ x < 2π is x = sin–1 = . Another
π
6
5π
6
such solution is x = π º = .
Moreover, because y = sin x is a periodic function, there are infinitely many other
solutions. You can write the general solution as
π
6
x = + 2nπ
or
5π
6
x = + 2nπ
where n is any integer.
✓CHECK
1
You can check your answer graphically. Graph y = sin x and y = in the
2
same coordinate plane and find the points where the graphs intersect.
You can see that there are infinitely many such points.
y
y 12
1
STUDENT HELP
x
Look Back
For help with inverse
trigonometric functions,
see p. 792.
π
2
y sin x
14.4 Solving Trigonometric Equations
855
Page 2 of 7
Solving a Trigonometric Equation in an Interval
EXAMPLE 2
Solve 4 tan2 x º 1 = 0 in the interval 0 ≤ x < 2π.
SOLUTION
4 tan2 x º 1 = 0
Write original equation.
4 tan2 x = 1
Add 1 to each side.
1
4
tan2 x = Divide each side by 4.
1
2
tan x = ±
Take square roots of each side.
Use a calculator to find values of x for
tan-1 (.5)
.463647609
tan-1 (-.5)
-.463647609
1
2
which tan x = ±, as shown at the right.
The general solution of the equation is
STUDENT HELP
Study Tip
Note that to find the
general solution of a
trigonometric equation,
you must add multiples
of the period to the
solutions in one cycle.
x ≈ 0.464 + nπ
or
x ≈ º0.464 + nπ
where n is any integer. The solutions that are in the interval 0 ≤ x < 2π are:
x ≈ 0.464
x ≈ 0.464 + π ≈ 3.61
x ≈ º0.464 + π ≈ 2.628
x ≈ º0.464 + 2π ≈ 5.82
✓CHECK
Check these solutions by substituting them back into the original
equation.
EXAMPLE 3
Factoring to Solve a Trigonometric Equation
Solve sin2 x cos x = 4 cos x.
SOLUTION
sin2 x cos x = 4 cos x
sin2 x cos x º 4 cos x = 0
Write original equation.
Subtract 4 cos x from each side.
2
cos x (sin x º 4) = 0
Factor out cos x.
cos x (sin x + 2)(sin x º 2) = 0
Factor difference of squares.
Set each factor equal to 0 and solve for x, if possible.
cos x = 0
STUDENT HELP
Study Tip
Remember not to divide
both sides of an equation
by a variable expression,
such as cos x.
856
sin x + 2 = 0
3π
π
x = or x = 2
2
sin x º 2 = 0
sin x = º2
sin x = 2
Because neither sin x = º2 nor sin x = 2 has a solution, the only solutions in the
3π
2
π
2
interval 0 ≤ x < 2π are x = and x = .
π
2
3π
2
The general solution is x = + 2nπ or x = + 2nπ where n is any integer.
Chapter 14 Trigonometric Graphs, Identities, and Equations
Page 3 of 7
Using the Quadratic Formula
EXAMPLE 4
Solve cos2 x º 4 cos x + 1 = 0 in the interval 0 ≤ x ≤ π.
SOLUTION
STUDENT HELP
Look Back
For help with the
quadratic formula,
see p. 291.
Since the equation is in the form au2 + bu + c = 0, you can use the quadratic
formula to solve for u = cos x.
cos2 x º 4 cos x + 1 = 0
Write original equation.
4 ± (º
4
)2
º4(1
)(
1)
2(1)
Quadratic formula
= = 2 ± 3
4 ± 12
2
Simplify.
≈ 3.73 or 0.268
Use a calculator.
cos x = x ≈ cosº1 3.73 or x ≈ cosº1 0.268
Use inverse cosine.
No solution
Use a calculator if possible.
≈ 1.30
In the interval 0 ≤ x ≤ π, the only solution is x ≈ 1.30. Check this in the original
equation.
..........
When solving a trigonometric equation, it is possible to obtain extraneous solutions.
Therefore, you should always check your solutions in the original equation.
An Equation with Extraneous Solutions
EXAMPLE 5
INT
STUDENT HELP
NE
ER T
HOMEWORK HELP
Visit our Web site
www.mcdougallittell.com
for extra examples.
Solve 1 º cos x = 3 sin x in the interval 0 ≤ x < 2π.
SOLUTION
1 º cos x = 3 sin x
Write original equation.
(1 º cos x) 2 = (3 sin x)2
1 º 2 cos x + cos2 x = 3 sin2 x
Multiply.
1 º 2 cos x + cos2 x = 3(1 º cos2 x)
Pythagorean identity
4 cos2 x º 2 cos x º 2 = 0
Quadratic form
2 cos2 x º cos x º 1 = 0
Divide each side by 2.
(2 cos x + 1)(cos x º 1) = 0
Factor.
2 cos x + 1 = 0 or cos x º 1 = 0
2π
3
cos x = º
1
2
cos x = 1
4π
3
x=0
x = or x = Square both sides.
4π
3
Zero product property
Solve for cos x.
Solve for x.
The apparent solution x = does not check in the original equation. The only
2π
3
solutions in the interval 0 ≤ x < 2π are x = 0 and x = .
14.4 Solving Trigonometric Equations
857
Page 4 of 7
FOCUS ON
CAREERS
GOAL 2
SOLVING TRIGONOMETRIC EQUATIONS IN REAL LIFE
Solving a Real-Life Trigonometric Equation
EXAMPLE 6
METEOROLOGY The number h of hours of sunlight per day in Prescott, Arizona,
can be modeled by
π
6
h = 2.325 sin (t º 2.667) + 12.155
RE
FE
L
AL I
METEOROLOGIST
INT
Operational
meteorologists, the largest
group of specialists in the
field, use complex computer
models to forecast the
weather. Other types of
meteorologists include
physical meteorologists
and climatologists.
where t is measured in months and t = 0 represents January 1. On which days of the
year are there 13 hours of sunlight in Prescott? Source: Gale Research Company
SOLUTION
Method 1
Substitute 13 for h in the model and solve for t.
π
6
2.325 sin (t º 2.667) + 12.155 = 13
π
6
2.325 sin (t º 2.667) = 0.845
NE
ER T
π
6
CAREER LINK
sin (t º 2.667) ≈ 0.363
www.mcdougallittell.com
π
π
(t º 2.667) ≈ sinº1 (0.363) or (t º 2.667) ≈ π º sinº1 (0.363)
6
6
π
(t º 2.667) ≈ 0.371
6
π
(t º 2.667) ≈ π º 0.371 ≈ 2.771
6
t º 2.667 ≈ 0.709
t º 2.667 ≈ 5.292
t ≈ 3.38
t ≈ 7.96
The time t = 3.38 represents 3 full months plus (0.38)(30) ≈ 11 days, or April 11.
Likewise, the time t = 7.96 represents 7 full months plus (0.96)(31) ≈ 30 days, or
August 30. (Notice that these two days occur about 70 days before and after June
21, which is the date of the summer solstice, the longest day of the year.)
Method 2
Use a graphing calculator. Graph the equations
π
6
y = 2.325 sin (x º 2.667) + 12.155
y = 13
in the same viewing window. Then use the Intersect feature to find the points of
intersection.
Intersection
X=3.3773887 Y=13
858
Intersection
X=7.9566113 Y=13
From the screens above, you can see that t ≈ 3.38 or t ≈ 7.96. The time t = 3.38
is about April 11, and the time t = 7.96 is about August 30.
Chapter 14 Trigonometric Graphs, Identities, and Equations
Page 5 of 7
GUIDED PRACTICE
Vocabulary Check
Concept Check
✓
✓
1. What is the difference between a trigonometric equation and a trigonometric
identity?
2. Name several techniques for solving trigonometric equations.
3. ERROR ANALYSIS Describe the error(s) in the calculations shown.
1
2
1
cos x = 2
cos2 x = cos x
π
3
5π
3
Solution for 0 ≤ x < 2π
5π
3
General solution
x = or x = π
3
x = + 2π or x = + 2π
Skill Check
✓
Solve the equation in the interval 0 ≤ x < 2π.
4. 2 cos x + 4 = 5
5. 3 sec2 x º 4 = 0
6. tan2 x = cos x tan2 x
7. 5 cos x º 3
= 3 cos x
Find the general solution of the equation.
9. 4 sin x = 3
8. 3 csc x + 5 = 0
2
2
10. 1 + tan x = 6 º 2 sec x
12.
11. 2 º 2 cos2 x = 5 sin x + 3
METEOROLOGY Look back at the model in Example 6 on page 858.
On which days of the year are there 10 hours of sunlight in Prescott, Arizona?
PRACTICE AND APPLICATIONS
STUDENT HELP
Extra Practice
to help you master
skills is on p. 960.
CHECKING SOLUTIONS Verify that the given x-value is a solution of the
equation.
π
15. 4 cos2 x º 3 = 0, x = 6
5π
14. csc x º 2 = 0, x = 6
π
3
16. 3 tan x º 3 = 0, x = 4
5π
17. 2 sin4 x º sin2 x = 0, x = 4
13π
18. 2 cot4 x º cot2 x º 15 = 0, x = 6
13. 5 + 4 cos x º 1 = 0, x = π
SOLVING Find the general solution of the equation.
19. 2 cos x º 1 = 0
20. 3 tan x º 3
=0
21. sin x = sin (ºx) + 1
22. 4 cos x = 2 cos x + 1
STUDENT HELP
23. 4 sin2 x º 2 = 0
24. 9 tan2 x º 3 = 0
HOMEWORK HELP
25. sin x cos x º 2 cos x = 0
26. 2
cos x sin x º cos x = 0
Examples 1, 3: Exs. 13–32
Examples 2, 4, 5:
Exs. 13–18, 33–56
Example 6: Exs. 57–59
π
3
5π
3
2
27. 2 sin x º sin x = 1
28. 0 = cos2 x º 5 cos x + 1
29. 1 º sin x = 3
cos x
30. si
n
x = 2 sin x º 1
31. cos x º 1 = ºcos x
32. 6 sin x = sin x + 3
31. }} + 2nπ, }} + 2nπ
14.4 Solving Trigonometric Equations
859
Page 6 of 7
SOLVING Solve the equation in the interval 0 ≤ x < 2π. Check your solutions.
33. 5 cos x º 3 = 0
34. 3 sin x = sin x º 1
35. tan2 x º 3 = 0
36. 10 tan x º 5 = 0
2
37. 2 cos x º sin x º 1 = 0
38. cos3 x = cos x
39. sec2 x º 2 = 0
40. tan2 x = sin x sec x
41. 2 cos x = sec x
42. cos x csc2 x + 3 cos x = 7 cos x
APPROXIMATING SOLUTIONS Use a graphing calculator to approximate
the solutions of the equation in the interval 0 ≤ x < 2π.
43. 3 tan x + 1 = 13
44. 8 cos x + 3 = 4
45. 4 sin x = º2 sin x º 5
46. 3 sin x + 5 cos x = 4
FINDING INTERCEPTS Find the x-intercepts of the graph of the given function
in the interval 0 ≤ x < 2π.
47. y = 2 sin x + 1
48. y = 2 tan2 x º 6
49. y = sec2 x º 1
50. y = º3 cos x + sin x
FINDING INTERSECTION POINTS Find the points of intersection of the graphs
of the given functions in the interval 0 ≤ x < 2π.
tan2 x
51. y = 3
y = 3 º 2 tan x
54. y = sin2 x
y = 2 sin x º 1
57.
52. y = 9 cos2 x
53. y = tan x sin x
y = cos2 x + 8 cos x º 2
y = cos x
56. y = 4 cos2 x
55. y = 2 º sin x tan x
y = cos x
y = 4 cos x º 1
OCEAN TIDES The tide, or depth of the ocean near the shore, changes
throughout the day. The depth of the Bay of Fundy can be modeled by
π
6.2
d = 35 º 28 cos t
where d is the water depth in feet and t is the time in hours. Consider a day in
which t = 0 represents 12:00 A.M. For that day, when do the high and low tides
FOCUS ON
APPLICATIONS
1
2
occur? At what time(s) is the water depth 3 feet?
58.
POSITION OF THE SUN Cheyenne, Wyoming, has a latitude of 41°N.
At this latitude, the position of the sun at sunrise can be modeled by
326π5
D = 31 sin t º 1.4
where t is the time in days and t = 1 represents January 1. In this model, D
represents the number of degrees north or south of due east that the sun rises.
Use a graphing calculator to determine the days that the sun is more than 20°
north of due east at sunrise.
RE
FE
L
AL I
THE BAY OF
FUNDY is the
portion of the Atlantic Ocean
that runs along the southern
coast of New Brunswick in
Canada. It is the site of the
highest tides in the world.
INT
NE
ER T
APPLICATION LINK
www.mcdougallittell.com
860
59.
METEOROLOGY A model for the average daily temperature T (in degrees
Fahrenheit) in Kansas City, Missouri, is given by
21π2
T = 54 + 25.2 sin t + 4.3
where t is measured in months and t = 0 represents January 1. What months
have average daily temperatures higher than 70°F? Do any months have average
daily temperatures below 20°F? Source: National Climatic Data Center
Chapter 14 Trigonometric Graphs, Identities, and Equations
Page 7 of 7
Test
Preparation
60. MULTIPLE CHOICE What is the general solution of the equation
cos x + 2 = ºcos x? Assume n is an integer.
A
¡
x = + 2nπ
π
4
B
¡
x = + 2nπ or x = + 2nπ
C
¡
x = + 2nπ
3π
4
D
¡
x = + 2nπ or x = + 2nπ
E
¡
x = + nπ or x = + nπ
3π
4
π
4
7π
4
3π
4
5π
4
5π
4
61. MULTIPLE CHOICE Find the points of intersection of the graphs of y = 2 + sin x
and y = 3 º sin x in the interval 0 ≤ x < 2π.
π6, 52, 56π, 52
¡ π3, 12, 43π, 12
★ Challenge
π6, 12, 56π, 12
11π 5
, ¡ π6, 52, 6 2
A
¡
B
¡
D
E
C
¡
π3, 52, 43π, 52
MATRICES In Exercises 62 and 63, use the following information.
Matrix multiplication can be used to rotate a point (x, y) counter clockwise about
the origin through an angle †. The coordinates of the resulting point (x§, y§) are
determined by the following matrix equation:
cos † ºsin †
sin † cos †
x
x§
=
y
y§
y
62. The point (4, 1) is rotated counter clockwise about
π
the origin through an angle of . What are the
3
†
(x ´, y ´)
coordinates of the resulting point?
EXTRA CHALLENGE
(x, y)
63. Through what angle † must the point (2, 4) be
x
rotated to produce (x§, y§) = (º2 + 3, 1 + 23 )?
www.mcdougallittell.com
MIXED REVIEW
USING COMPLEMENTS Two six-sided dice are rolled. Find the probability of
the given event. (Review 12.4)
64. The sum is less than or equal to 10.
65. The sum is not 4.
66. The sum is not 2 or 12.
67. The sum is greater than 3.
GRAPHING Graph the function. (Review 14.1, 14.2 for 14.5)
68. y = sin 3x
69. y = 2 cos 4x
70. y = 10 sin 2x
1
71. y = 3 cos x
2
1
72. y = tan x
4
1
73. y = 3 tan x º 2
2
1
74. y = cos x + π
2
3π
75. y = 3 + tan x º 2
77.
76. y = ºsin 3π(x + 4) + 1
43 ft
SURVEYING Suppose you are trying to
determine the width w of a small pond. You stand
at a point 43 feet from one end of the pond and
50 feet from the other end. The angle formed
by your lines of sight to each end of the pond
measures 45°. How wide is the pond? (Review 13.6)
45
you
w
50 ft
14.4 Solving Trigonometric Equations
861
Page 1 of 7
E X P L O R I N G DATA
A N D S TAT I S T I C S
14.5
What you should learn
Model data with a
sine or cosine function.
Modeling with
Trigonometric Functions
GOAL 1
Graphs of sine and cosine functions are called sinusoids. When you write a sine or
cosine function for a sinusoid, you need to find the values of a, b > 0, h, and k for
y = a sin b(x º h) + k
GOAL 1
GOAL 2 Use technology to
write a trigonometric model,
as applied in Example 4.
Why you should learn it
RE
or
y = a cos b(x º h) + k
2π
where |a| is the amplitude, is the period, h is the horizontal shift, and k is the
b
vertical shift.
EXAMPLE 1
Writing Trigonometric Functions
Write a function for the sinusoid.
π3 , 4
y
a.
b.
(0.5, 5)
5
2
x
x
0.5
π
3
FE
To model many types
of real-life quantities,
such as the temperature
inside and outside an igloo
in Ex. 30.
AL LI
WRITING A TRIGONOMETRIC MODEL
(1.5, 15)
(0, 4)
SOLUTION
a. Since the maximum and minimum values of the function occur at points
equidistant from the x-axis, the curve has no vertical shift. And because the
minimum occurs on the y-axis, the graph is a reflection of a cosine curve with
no horizontal shift. Therefore, the function has the form y = a cos bx.
2π
b
2π
3
The period is = , so b = 3.
The amplitude is |a| = 4. Since the graph is a reflection, a = º4.
The function is y = º4 cos 3x.
b. Since the maximum and minimum values of the function do not occur at points
equidistant from the x-axis, the curve has a vertical shift. To find the value of k,
add the maximum and minimum values and divide by 2:
M+m
2
5 + (º15)
2
º10
2
k = = = = º5
Because the graph crosses the y-axis at y = k, the graph is a sine curve with no
horizontal shift. Therefore, the function has the form y = a sin bx º 5.
2π
b
The period is = 2, so b = π.
INT
STUDENT HELP
NE
ER T
HOMEWORK HELP
Visit our Web site
www.mcdougallittell.com
for extra examples.
862
Mºm
2
5 º (º15)
2
20
2
The amplitude is |a| = = = = 10.
Since the graph is not a reflection, a = 10 > 0.
The function is y = 10 sin πx º 5.
Chapter 14 Trigonometric Graphs, Identities, and Equations
Page 2 of 7
RE
FE
L
AL I
Meteorology
EXAMPLE 2
Modeling a Sinusoid
Write a trigonometric model for the average daily temperature in Birmingham,
Alabama. Source: National Climatic Data Center
Temperature (F)
Daily Temperature in Birmingham
T
80
60
40
20
0
0
2
t
6
8
10
4
Months since January 1
SOLUTION
Notice that the graph crosses the T-axis at the minimum point. So if you model the
temperature curve with a cosine function, there is a reflection but no horizontal shift.
The mean of the maximum and minimum values is 60, so there is a vertical shift of
k = 60.
2π
b
π
6
The period is = 12, so b = .
The amplitude is |a| = 20, and because the graph is a reflection it follows that
a = º20.
RE
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Ferris Wheel
π
The model is T = º20 cos t + 60 where t is measured in months and t = 0
6
represents January 1.
EXAMPLE 3
Modeling Circular Motion
A Ferris wheel with a radius of 25 feet is rotating at a rate of 3 revolutions per
minute. When t = 0, a chair starts at the lowest point on the wheel, which is 5 feet
above the ground. Write a model for the height h (in feet) of the chair as a function
of the time t (in seconds).
SOLUTION
When the chair is at the bottom of the Ferris wheel, it is 5 feet above the ground, so
m = 5. When it is at the top, it is 5 + 2(25) = 55 feet above the ground, so M = 55.
M+m
2
55 + 5
2
60
2
The vertical shift for the model is k = = = = 30.
When t = 0, the height is at its minimum, so the model is a cosine function with
a < 0 and no horizontal shift.
Mºm
55 º 5
50
The amplitude is |a| = = = = 25. Because a < 0 it follows that
2
2
2
a = º25.
Since the Ferris wheel is rotating at 3 revolutions per minute, it completes one
2π
b
π
10
revolution in 20 seconds. The period is = 20, so b = .
π
10
A model for the height of the chair as a function of time is h = º25 cos t + 30.
14.5 Modeling with Trigonometric Functions
863
Page 3 of 7
GOAL 2
SINUSOIDAL MODELING USING TECHNOLOGY
There are two ways you can model a set of data points whose scatter plot appears
sinusoidal. One way is to estimate the minimum and maximum values and use the
technique shown in Example 2. Another way is to use a graphing calculator that has
a sinusoidal regression feature. The advantage of the second method is that it uses
all of the data points to find the model.
RE
FE
L
AL I
Meteorology
EXAMPLE 4
Using Sinusoidal Regression
The average daily temperature T (in degrees Fahrenheit) in Fairbanks, Alaska, is
given in the table. Time t is measured in months, with t = 0 representing January 1.
Write a trigonometric model that gives T as a function of t.
0.5
2.5
3.5
4.5
5.5
6.5
7.5
8.5
9.5
10.5
11.5
º10.1 º3.6 11.0
30.7
48.6
59.8
62.5
56.8
45.5
25.1
2.7
º6.5
t
T
1.5
Source: National Climatic Data Center
SOLUTION
Begin by entering the data in a graphing calculator and drawing a scatter plot.
L1
L2
.5
-10.1
1.5
-3.6
2.5
11
3.5
30.7
4.5
48.6
L1(1)=.5
1
L3
Enter the data.
2
Draw a scatter plot.
Because the scatter plot appears sinusoidal, you can fit the data with a sine function
to get the following model:
T = 37.4 sin (0.518t º 1.72) + 26.5
SinReg
y=a*sin(bx+c)+d
a=37.40756055
b=.5184527958
c=-1.717020267
d=26.50468918
3
Perform a sinusoidal
regression.
4
Graph the model and the data in
the same viewing window.
After obtaining the model, you should graph it in the same viewing window as the
scatter plot to see how well the model fits the data. In this case, the model is a good fit.
✓CHECK
As a check on the reasonableness of the model, notice that the period
2π
is ≈ 12, which is the number of months in a year.
0.518
864
Chapter 14 Trigonometric Graphs, Identities, and Equations
Page 4 of 7
GUIDED PRACTICE
✓
Concept Check ✓
Vocabulary Check
?.
1. Complete this statement: Graphs of sine and cosine functions are called 2. Which two points are most useful when writing a sinusoidal model for a given
graph or set of data? Explain.
3. Describe a characteristic of a sinusoidal graph that you would model with a
cosine function rather than a sine function.
Skill Check
✓
Write a function for the sinusoid.
y
4.
y
5.
π
,1
2
2
(0, 1)
x
π
3
π
, 1
6
x
1
(1, 3)
4
Write a function for the sinusoid with maximum at A and minimum at B.
6. A(0, 8), B(π, º2)
9.
7. A(π, 10), B(3π, 4)
8. A(6, 5), B(2, 1)
FERRIS WHEEL Look back at Example 3. Suppose the Ferris wheel rotates
at a rate of 4 revolutions per minute and has a radius of 20 feet. Write a model
for the height h (in feet) of the chair as a function of the time t (in seconds).
PRACTICE AND APPLICATIONS
STUDENT HELP
WRITING FUNCTIONS Write a function for the sinusoid.
Extra Practice
to help you master
skills is on p. 960.
10.
y
3
π4 , 5
11.
(0, 3)
5
x
x
π
2
π
4
3π4 , 5
π4 , 3
12.
y
y
13.
π
,2
2
3
(1, 4)
4
x
π
3
(0, 4)
π6 , 2
STUDENT HELP
14.
Example 1:
Example 2:
Example 3:
Example 4:
Exs. 10–27
Exs. 30, 32
Exs. 31, 33
Exs. 34, 35
2
14 , 2
3
,0
4
y
15.
y
HOMEWORK HELP
x
1
2
(0, 2)
3
x
x
1
(2, 6)
14.5 Modeling with Trigonometric Functions
865
Page 5 of 7
WRITING TRIGONOMETRIC FUNCTIONS Write a function for the sinusoid with
maximum at A and minimum at B.
16. A(0, 3), B(2π, º3)
17. A(π, 8), B(3π, º8)
18. A(9π, 1), B(3π, º5)
π
19. A , 4 , B(0, 2)
3
20. A(2, 6), B(6, 0)
21. A(3, 7), B(1, º3)
2π
22. A, 9, B(2π, 5)
3
25. A(0, 0), B(3π, º8)
π
23. A , 11 , B(0, º1)
6
24. A(0, 5), B(4, º13)
26. A(6, 2), B(0, º4)
π
π
27. A , º2 , B , º6
4
12
28. CRITICAL THINKING Any sinusoid can be modeled by either a sine function or
a cosine function. Using the graph from part (a) of Example 1, find the values of
a, b, h, and k for the model y = a sin b(x º h) + k. Use identities to show that
the model you found is equivalent to the model from part (a) of Example 1.
29.
Writing Since any sinusoid can be modeled by either a sine function or a
cosine function, you can choose the type of function that is more convenient.
For a sinusoid whose y-intercept occurs halfway between the maximum and
minimum values of the function, tell which type of function you would use to
model the graph. Explain your answer.
CLIMATE CONTROL
Eskimos use igloos as temporary
shelter from harsh winter weather.
The graph shows the temperatures
inside and outside an igloo
throughout a typical winter day.
Write a sinusoidal model for the
outside temperature T (in degrees
Fahrenheit) as a function of the
time of day t (in hours since
midnight). Then write sinusoidal
models for the floor-level
temperature and for the sleepingplatform temperature.
Igloo Temperatures
T
40
Temperature (F)
30.
20
0
4
8
12
16
20
20
Hours since 8 A.M.
Source: Scientific American
FOCUS ON
APPLICATIONS
RE
FE
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AL I
THE S.S. BEAVER
was the first steamship on the Pacific Coast. It
was built in the 1830s and
sailed out of British
Columbia, Canada, for over
50 years.
866
31.
STEAMSHIPS The paddle wheel of the S.S. Beaver was 13 feet in diameter
and revolved 30 times per minute when moving at top speed. Using this speed
and starting from a point at the very top of the wheel, write a model for the
height h (in feet) of the end of a paddle relative to the water’s surface as a
function of the time t (in minutes). (Assume the paddle is 2 feet below the
water’s surface at its lowest point.) Source: S.S. Beaver: The Ship That Saved the West
32.
OCEAN TIDES The height of the water in a bay varies sinusoidally over
time. On a certain day off the coast of Maine, a high tide of 10 feet occurred at
5:00 A.M. and a low tide of 2 feet occurred at 1:00 P.M. Write a model for the
height h (in feet) of the water as a function of time t (in hours since midnight).
33.
SEWING MACHINES In front of the Antique Sewing Machine Museum in
Arlington, Texas, is the largest sewing machine in the world. The flywheel, which
turns as the machine sews, is 5 feet in diameter. Write a model for the height h
(in feet) of the handle on the flywheel as a function of the time t (in seconds),
assuming that the wheel makes a complete revolution every 2 seconds and that
the handle starts at its minimum height of 4 feet above the ground.
Chapter 14 Trigonometric Graphs, Identities, and Equations
t
Page 6 of 7
In Exercises 34 and 35, use a graphing calculator to
find a sinusoidal model that fits the data.
STATISTICS
CONNECTION
METEOROLOGY The average daily temperature T (in degrees Fahrenheit) in
34.
Moline, Illinois, is given in the table. Time t is measured in months, with t = 0
representing January 1. Find a model for the data. Source: National Climatic Data Center
t
0.5
1.5
2.5
3.5
4.5
5.5
6.5
7.5
8.5
9.5
10.5
11.5
T
19.9
24.8
37.4
50.4
61.4
71.1
75.2
72.7
64.7
53.0
39.6
25.4
HEATING DEGREE-DAYS For any given day, the number of degrees that
35.
the average temperature is below 65°F is called the degree-days for that day. This
figure is used to calculate how much is spent on heating. The table below gives
the total number T of degree-days for each month t in Dubuque, Iowa, with t = 1
representing January. Find a model for the data. Source: Workshop Math
1
t
T
Test
Preparation
2
3
1420 1204 1026
4
5
6
7
8
9
10
11
12
546
260
78
12
31
156
450
906
1287
36. MULTIPLE CHOICE During one cycle, a sinusoid has a minimum at (18, 44)
and a maximum at (30, 68). What is the amplitude of this sinusoid?
A
¡
6
B
¡
C
¡
12
22
D
¡
24
E
¡
48
37. MULTIPLE CHOICE During one cycle, a sinusoid has a maximum at (4, 12) and
a minimum at (12, º2). What is the period of this sinusoid?
★ Challenge
A
¡
8
B
¡
C
¡
8π
16
D
¡
16π
E
¡
32
38. FINDING A FUNCTION Write a sinusoidal function whose graph has a
π2 minimum at (π, 4) and a maximum at , 7 .
MIXED REVIEW
FINDING PROBABILITY Two six-sided dice are rolled. Find the probability of
the given event. (Review 12.4)
39. The sum is 3.
40. The sum is 6.
41. The sum is 12.
42. The sum is odd.
43. The sum is 5 or 6.
44. The sum is less than 7.
EVALUATING FUNCTIONS Evaluate the function without using a calculator.
(Review 13.3 for 14.6)
45. cos (º225°)
46. cos 240°
47. sin (º30°)
48. tan 330°
7π
49. tan º
6
5π
50. sin 4
FINDING AREA Find the area of ¤ABC having the given side lengths.
(Review 13.6)
51. a = 3, b = 8, c = 10
52. a = 12, b = 20, c = 25
53. a = 5, b = 9, c = 11
54. a = 4, b = 6, c = 7
55. a = 6, b = 11, c = 14
56. a = 8, b = 13, c = 20
14.5 Modeling with Trigonometric Functions
867
Page 7 of 7
QUIZ 2
Self-Test for Lessons 14.3–14.5
Simplify the expression. (Lesson 14.3)
π
1. tan º x sec x
2
co s x
2. 1 º sin2 x + sec x
π
3. cos (ºx) cos º x
2
Find the general solution of the equation. (Lesson 14.4)
4. 4 cos2 x º 3 = 0
6. 3
tan2 x + 4 tan x = º3
5. 3 sin2 x º 8 sin x = 3
Write a function for the sinusoid with maximum at A and minimum at B.
(Lesson 14.5)
3π
π
7. A , 5 , B , º5
4
4
10.
8. A(0, 3), B(3π, 1)
9. A(π, 6), B(0, 2)
METEOROLOGY The average daily temperature T (in degrees Fahrenheit) in
Detroit, Michigan, is given in the table. Time t is measured in months, with
t = 0 representing January 1. Find a model for the data. (Lesson 14.5)
0.5
1.5
2.5
3.5
4.5
5.5
6.5
7.5
8.5
9.5
10.5
11.5
T
22.9
25.4
35.7
47.3
58.4
67.6
72.3
70.5
63.2
51.2
40.2
28.3
INT
t
NE
ER T
Music and Math
THEN
NOW
APPLICATION LINK
www.mcdougallittell.com
MUSIC AND MATH have been studied together for many centuries. For
instance, 2500 years ago Pythagoras discovered that when the ratio of
the lengths of two strings is a whole number, plucking the strings
produces harmonious tones.
TODAY we know that musical notes can be modeled by sine functions.
The function y = sin 2πƒx models a note with frequency ƒ (in hertz)
where x represents the length of time (in seconds) that the note is played.
By adding the functions modeling two different notes, you can analyze
the sound that occurs when the notes are played together.
1. Write functions that model notes with frequencies 10 hertz, 11 hertz, 12 hertz,
13 hertz, 14 hertz, and 15 hertz.
Use a graphing calculator to graph the sum of the models for 10 hertz and
15 hertz. How many cycles of the function occur in 1 second?
2.
3. How many cycles per second are there in the function that describes what you hear
when notes with frequencies of 10 and 14 hertz are played together? 10 and 13 hertz?
10 and 12 hertz?
Harps first
appeared.
c. 3000 B . C .
First piano built.
1964
c. A . D . 700
1709
The lyre is introduced.
868
Chapter 14 Trigonometric Graphs, Identities, and Equations
Synthesizer invented.
Page 1 of 6
14.6
What you should learn
GOAL 1 Evaluate
trigonometric functions of
the sum or difference of
two angles.
Use sum and
difference formulas to solve
real-life problems, such as
determining when pistons in
a car engine are at the same
height in Example 6.
GOAL 2
Why you should learn it
RE
GOAL 1
SUM AND DIFFERENCE FORMULAS
In this lesson you will study formulas that allow you to evaluate trigonometric
functions of the sum or difference of two angles.
SUM AND DIFFERENCE FORMULAS
SUM FORMULAS
DIFFERENCE FORMULAS
sin (u + v) = sin u cos v + cos u sin v
sin (u º v) = sin u cos v º cos u sin v
cos (u + v) = cos u cos v º sin u sin v
cos (u º v) = cos u cos v + sin u sin v
tan u + tan v
tan (u + v) = 1 º tan u tan v
tan u º tan v
tan (u º v) = 1 + tan u tan v
In general, sin (u + v) ≠ sin u + sin v. Similar statements can be made for the other
trigonometric functions of sums and differences.
FE
To model real-life
quantities, such as the
size of an object in an
aerial photograph in
Ex. 58.
AL LI
Using Sum and
Difference Formulas
Evaluating a Trigonometric Expression
EXAMPLE 1
Find the exact value of (a) cos 75° and (b) tan π.
12
SOLUTION
a. cos 75° = cos (45° + 30°)
Substitute 45° + 30° for 75°.
= cos 45° cos 30° º sin 45° sin 30°
Sum formula for cosine
2 3
2 1
º =
2
2
2 2
Evaluate.
6 º 2
= 4
Simplify.
π
π
π
b. tan = tan º 3
4
12
π
π
3
π
12
π
tan º tan 3
4
= π
π
1 + tan tan 3
3 º 1
Difference formula for tangent
4
= Evaluate.
= 2 º 3
Simplify.
1 + (3 )(1)
✓CHECK
π
4
Substitute }} º }} for }}.
Try checking these results with a calculator. For instance, evaluate
6 º 2
cos 75° and to see that both have the same value.
4
14.6 Using Sum and Difference Formulas
869
Page 2 of 6
Using a Difference Formula
EXAMPLE 2
3
3π
12
Find sin (u º v) given that sin u = º with π < u < and cos v = with
5
2
13
π
0 < v < .
2
SOLUTION
4
5
5
13
Using a Pythagorean identity and quadrant signs gives cos u = º and sin v = .
sin (u º v) = sin u cos v º cos u sin v
Difference formula for sine
45 153
3 12
5 13
Substitute.
16
65
Simplify.
= º º º
= º Simplifying an Expression
EXAMPLE 3
Simplify the expression cos (x º π).
SOLUTION
cos (x º π) = cos x cos π + sin x sin π
= (cos x)(º1) + (sin x)(0)
Evaluate.
= ºcos x
Simplify.
Solving a Trigonometric Equation
EXAMPLE 4
Difference formula for cosine
π
4
π4 Solve sin x + + 1 = sin º x for 0 ≤ x < 2π.
SOLUTION
π4 π
4
π
4
π
4
π
4
π
4
π
4
Write original
equation.
sin x + + 1 = sin º x
π
4
sin x cos + cos x sin + 1 = sin cos x º cos sin x
π
4
Use formulas.
π
4
sin x cos + cos x sin + 1 = cos x sin º sin x cos Commutative property
π
4
2 sin x cos = º1
Simplify.
2 Evaluate.
2(sin x) 2 = º1
1
2
2
sin x = º = º
2
5π
4
Solve for sin x.
7π
4
In the interval 0 ≤ x < 2π, the solutions are x = and x = .
✓CHECK
You can check the solutions with a graphing calculator by graphing each
side of the original equation and using the Intersect feature to determine the x-values
for which the expressions are equal.
870
Chapter 14 Trigonometric Graphs, Identities, and Equations
Page 3 of 6
GOAL 2
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Biomechanics
SUM AND DIFFERENCE FORMULAS IN REAL LIFE
EXAMPLE 5
Simplifying a Real-Life Formula
The force F (in pounds) on a person’s back when he or she bends over at an angle † is
0.6W sin († + 90°)
sin 12°
F = where W is the person’s weight (in pounds).
Simplify this formula.
†
SOLUTION
Begin by expanding sin († + 90°).
0.6W sin († + 90°)
sin 12°
0.6W(sin † cos 90° + cos † sin 90°)
0.208
F = ≈ 0.6
0.208
= W [(sin †)(0) + (cos†)(1)] ≈ 2.88W cos †
EXAMPLE 6
Solving a Trigonometric Equation in Real Life
AUTOMOTIVE ENGINEERING The heights h (in inches) of
pistons 1 and 2 in an automobile engine can be modeled by
4π
3
1
2
3
4
5
6
h 1 = 3.75 sin 733t + 7.5 and h 2 = 3.75 sin 733 t + + 7.5
where t is measured in seconds. How often are these two
pistons at the same height?
FOCUS ON
CAREERS
SOLUTION
Let h1 = h2 and solve for t.
4π
3
3.75 sin 733t + 7.5 = 3.75 sin 733 t + + 7.5
2932π
3
2932π
3
sin 733t = sin 733t cos + cos 733t sin 12 32 sin 733t = (sin 733t) º + (cos 733t) º
3
3
sin 733t = º cos 733t
2
2
3
3
tan 733t = º
RE
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AUTO MECHANIC
INT
Auto mechanics
inspect and repair mechanical and electrical systems of
motor vehicles. They may
specialize in areas such as
transmission systems or
diagnostic services.
NE
ER T
CAREER LINK
www.mcdougallittell.com
3 733t = tanº1 º + nπ
3
π
6
π
nπ
t = º + 4398
733
733t = º + nπ
π
733
The heights are equal once every second. So in one second,
π
733
733
π
the heights are equal the following number of times: 1 = ≈ 233.3.
14.6 Using Sum and Difference Formulas
871
Page 4 of 6
GUIDED PRACTICE
✓
Concept Check ✓
Vocabulary Check
1. Give the sum and difference formulas for sine, cosine, and tangent.
2. Fill in the blanks for each of the following equations.
? cos 30° º cos 45° sin ?
a. sin (45° º 30°) = sin ? + tan 60°
b. tan (90° + 60°) = ?
1 º tan 90° 3. Explain how you can evaluate tan 105° using either the sum or difference
formula for tangent.
Skill Check
✓
Find the exact value of the expression.
4. cos 105°
5. sin 15°
6. tan 75°
11π
7. cos 12
23π
8. sin 12
7π
9. tan 12
Solve the equation for 0 ≤ x < 2π.
4π
π
13. sin x º = 2 sin x º 3
3
π
π
10. 2 sin x + = tan 3
3
π
π
11. tan x + = tan x + 6
4
π
π
12. cos x º = 1 + cos x + 6
6
π
14. 4 sin (x + π) = 2 cos x + + 2
2
16.
15. ºcos x = 1 + 2 cos (x º π)
AUTOMOTIVE ENGINEERING Look back at Example 6 on page 871.
The height h3 of piston 3 in the same engine can be modeled by
2π
h3 = 3.75 sin 733 t + + 7.5
3
where t is measured in seconds. How often is piston 3 the same height as piston 2?
PRACTICE AND APPLICATIONS
STUDENT HELP
FINDING VALUES Find the exact value of the expression.
Extra Practice
to help you master
skills is on p. 960.
17. cos 210°
18. tan 195°
19. tan 225°
20. sin (º15°)
21. cos (º225°)
22. sin 165°
11π
23. tan 12
17π
24. cos 12
11π
25. sin º
12
π
26. cos 12
5π
27. tan º
12
5π
28. sin 12
STUDENT HELP
HOMEWORK HELP
Example 1: Exs. 17–28
Example 2: Exs. 29–40
Example 3: Exs. 41–48
Example 4: Exs. 49–54
Examples 5, 6: Exs. 57–59
872
EVALUATING EXPRESSIONS Evaluate the expression given cos u = 47 with
π
2
9
10
3π
2
0 < u < and sin v = º with π < v < .
29. sin (u + v)
30. cos (u + v)
31. tan (u + v)
32. sin (u º v)
33. cos (u º v)
34. tan (u º v)
Chapter 14 Trigonometric Graphs, Identities, and Equations
Page 5 of 6
INT
STUDENT HELP
NE
ER T
HOMEWORK HELP
Visit our Web site
www.mcdougallittell.com
for help with problem
solving in Ex. 58.
EVALUATING EXPRESSIONS Evaluate the expression given that sin u = 35 with
π
5
3π
< u < π and cos v = º with π < v < .
2
6
2
35. sin (u + v)
36. cos (u + v)
37. tan (u + v)
38. sin (u º v)
39. cos (u º v)
40. tan (u º v)
SIMPLIFYING EXPRESSIONS Simplify the expression.
41. tan (x º 2π)
42. tan (x + π)
π
45. sin x º 2
3π
46. cos x + 2
43. sin (x + π)
44. cos (x + π)
π
47. cos x + 2
3π
48. sin x º 2
SOLVING TRIGONOMETRIC EQUATIONS Solve the equation for 0 ≤ x < 2π.
π
π
51. sin x + + sin x º = 0
6
6
π
π
52. cos x + + cos x º = 1
3
3
π
54. tan (x + π) + cos x + = 0
2
π
π
49. cos x + º 1 = cos x º 6
6
3π
3π
50. sin x + + sin x º = 1
4
4
53. tan (x + π) + 2 sin (x + π) = 0
GEOMETRY CONNECTION In Exercises 55
and 56, use the following information.
y
m 2 tan †2
In the figure shown, the acute angle of
intersection, †2 º †1, of two lines with
slopes m1 and m2 is given by:
†2 †1
m2 º m1
1 + m 1m 2
†1
tan (†2 º †1) = FOCUS ON
PEOPLE
m 1 tan †1
†2
x
55. Find the acute angle of intersection of the
1
lines y = x + 3 and y = 2x º 3.
2
56. Find the acute angle of intersection of the lines y = x + 2 and y = 3x + 1.
57.
SOUND WAVES The pressure P of sound waves on a person’s eardrum
can be modeled by
a
r
2πl r
P = cos º 1100t
where a is the maximum sound pressure (in pounds per square foot) at the
source, r is the distance (in feet) from the source, l is the length (in feet) of
the sound wave, and t is the time (in seconds). Simplify this formula when
r = 16 feet, l = 4 feet, and a = 0.4 pound per square foot.
58.
RE
FE
L
AL I
BARRIE
ROKEACH has
more than 20 years of
experience as an aerial
photographer. He has had
more than one dozen
individual exhibitions in
museums and galleries
around the country.
AERIAL PHOTOGRAPHY You are at a height h
taking aerial photographs. The ratio of the length WQ
of the image to the length NA of the actual object is
camera
†
WQ
ƒ tan († º t) + ƒ tan t
= NA
h tan †
where ƒ is the focal length of the camera, † is the angle
with the vertical made by the line from the camera to
point A and t is the tilt angle of the film. Use the
difference formula for tangent to simplify the
WQ
NA
ƒ
h
ratio. Then show that = when t = 0.
h
W
œ
t
N
A
Source: Math Applied to Space Science
14.6 Using Sum and Difference Formulas
873
Page 6 of 6
59.
FERRIS WHEEL The heights h (in feet) of two people in different seats on a
Ferris wheel can be modeled by
h1 = 28 cos 10t + 38
and
π
6
h2 = 28 cos 10 t º + 38
where t is the time (in minutes). When are the two people at the same height?
60. CRITICAL THINKING You can write the sum and difference formulas for cosine
as a single equation: cos (u ± v) = cos u cos v ‚ sin u sin v. Explain why the
symbol ± is used on the left side, but the symbol ‚ is used on the right side.
Then use the symbols ± and ‚ to write the sum and difference formulas for sine
and tangent as single equations.
Test
Preparation
MULTI-STEP PROBLEM Suppose two middle-A tuning forks are struck at
61.
different times so that their vibrations are slightly out of phase. The combined
pressure change P (in pascals) caused by the forks at time t (in seconds) is:
P = 3 sin 880πt + 4 cos 880πt
a. Graph the equation on a graphing calculator using a viewing window of
0 ≤ x ≤ 0.5 and º6 ≤ y ≤ 6. What do you observe about the graph?
b. Write the given model in the form y = a cos b(x º h).
c. Graph the model from part (b) to confirm that the graphs are the same.
★ Challenge
VERIFYING FORMULAS Use the difference formula for cosine to verify the
following formulas.
62. The sum formula for cosine, by replacing v with ºv
EXTRA CHALLENGE
www.mcdougallittell.com
sin †
63. The difference formula for tangent, by using the identity tan † = co s †
64. The difference formula for sine, by using a cofunction identity
MIXED REVIEW
SIMPLIFYING EXPRESSIONS Using the given matrices, simplify the
expression. (Review 4.2)
A=
1
3
2
3
º1
2 0
3 1 º1
2
,B=
,C=
,D=
, E = º1
º1 º2
3
º4 1
º2 0
2
0 º2
65. AD + D
66. 3(A + C)
67. º2AB + B
68. DE + AC
69. 5CD
70. AD º CD
SOLVING TRIANGLES Solve ¤ABC. (Review 13.5, 13.6)
71. A = 18°, B = 28°, b = 100
72. A = 60°, C = 95°, c = 5
73. a = 13, b = 4, c = 11
74. a = 2, b = 2.5, c = 3
C
a
b
A
c
SOLVING TRIGONOMETRIC EQUATIONS Solve the equation in the interval
0 ≤ x < 2π. (Review 14.4 for 14.7)
874
=0
75. tan x + 3
76. 4 cos2 x º 3 = 0
77. 8 tan x + 8 = 0
78. º5 + 8 cos x = º1
Chapter 14 Trigonometric Graphs, Identities, and Equations
B
Page 1 of 8
14.7
What you should learn
GOAL 1 Evaluate
expressions using doubleand half-angle formulas.
Use double- and
half-angle formulas to solve
real-life problems, such as
finding the mach number for
an airplane in Ex. 70.
GOAL 2
Why you should learn it
RE
FE
To model real-life
situations with double- and
half-angle relationships, such
as kicking a football
in Example 8.
AL LI
Using Double- and
Half-Angle Formulas
GOAL 1
DOUBLE- AND HALF-ANGLE FORMULAS
In this lesson you will use formulas for double angles (angles of measure 2u)
u
2
and half angles angles of measure . The three formulas for cos 2u below are
u
2
equivalent, as are the two formulas for tan . Use whichever formula is most
convenient for solving a problem.
DOUBLE-ANGLE AND HALF-ANGLE FORMULAS
DOUBLE-ANGLE FORMULAS
cos 2u = cos2 u º sin2 u
sin 2u = 2 sin u cos u
cos 2u = 2 cos2 u º 1
tan 2u = 2
2 tan u
1 º tan u
cos 2u = 1 º 2 sin2 u
HALF-ANGLE FORMULAS
sin = ± u
2
1º2cosu
tan = u
2
1+c2osu
tan = cos = ± u
2
u
2
1 º cos u
sin u
u
2
sin u
1 + cos u
u
2
u
2
The signs of sin and cos depend on the quadrant in which lies.
EXAMPLE 1
Evaluating Trigonometric Expressions
π
8
Find the exact value of (a) tan and (b) cos 105°.
SOLUTION
π
π
a. Use the fact that is half of .
8
4
STUDENT HELP
Study Tip
In Example 1 note that, in
u
1
general, tan ≠ tan u.
2
2
Similar statements can
be made for the other
trigonometric functions
of double and half
angles.
π
8
1 π
2 4
1 º cos 1º
sin 2
π
2
2 º 2
2
4 =
2 = = 2
tan = tan = º1
π
4
2
b. Use the fact that 105° is half of 210° and that cosine is negative in Quadrant II.
1
2
1+co2s210°
cos 105° = cos (210°) = º 3
1 + º
2 2 º 3
2
º
3
= º = º = º 2
4
2
14.7 Using Double- and Half-Angle Formulas
875
Page 2 of 8
STUDENT HELP
Study Tip
3π
Because π < u < in
2
Example 2, you can
multiply through the
1
inequality by to get
2
π u 3π
u
< < , so is in
2
2
4
2
Quadrant II.
Evaluating Trigonometric Expressions
EXAMPLE 2
3
5
3π
2
Given cos u = º with π < u < , find the following.
u
b. sin 2
a. sin 2u
SOLUTION
4
a. Use a Pythagorean identity to conclude that sin u = º.
5
sin 2u = 2 sin u cos u
45 35 24
25
= 2 º º = u
u
b. Because is in Quadrant II, sin is positive.
2
2
u
sin =
2
EXAMPLE 3
1 º co s u
=
2
3
1 º º
5
4
25
=
= 5
2
5
Simplifying a Trigonometric Expression
cos 2†
sin † + cos †
Simplify .
STUDENT HELP
Study Tip
Because there are three
formulas for cos 2u, you
will want to choose the
one that allows you to
simplify the expression in
which cos 2u appears, as
illustrated in Example 3.
SOLUTION
cos2 † º sin2 †
co s 2 †
= sin † + cos †
sin † + cos †
(cos † º sin †)(cos † + sin †)
sin † + cos †
EXAMPLE 4
Use a double-angle formula.
= Factor difference of squares.
= cos † º sin †
Simplify.
Verifying a Trigonometric Identity
Verify the identity sin 3x = 3 sin x º 4 sin3 x.
SOLUTION
sin 3x = sin (2x + x)
876
Rewrite sin 3x as
sin (2x + x).
= sin 2x cos x + cos 2x sin x
Use a sum formula.
= (2 sin x cos x) cos x + (1 º 2 sin2 x) sin x
Use double-angle
formulas.
= 2 sin x cos2 x + sin x º 2 sin3 x
Multiply.
= 2 sin x (1 º sin2 x) + sin x º 2 sin3 x
Use a Pythagorean
identity.
= 2 sin x º 2 sin3 x + sin x º 2 sin3 x
Distributive property
= 3 sin x º 4 sin3 x
Combine like terms.
Chapter 14 Trigonometric Graphs, Identities, and Equations
Page 3 of 8
EXAMPLE 5
INT
STUDENT HELP
NE
ER T
HOMEWORK HELP
Visit our Web site
www.mcdougallittell.com
for extra examples.
Solving a Trigonometric Equation
Solve tan 2x + tan x = 0 for 0 ≤ x < 2π.
SOLUTION
tan 2x + tan x = 0
Write original equation.
2 tan x
+ tan x = 0
1 º tan2 x
Use a double-angle formula.
2 tan x + tan x (1º tan2 x) = 0
Multiply each side by 1 º tan2 x.
2 tan x + tan x º tan3 x = 0
Distributive property.
3
3 tan x º tan x = 0
Combine like terms.
tan x (3 º tan2 x) = 0
Factor.
Set each factor equal to 0 and solve for x.
tan x = 0
3 º tan2 x = 0
or
3 = tan2 x
x = 0, π
±3 = tan x
π 2π 4π 5π
3 3 3 3
x = , , , ✓CHECK You can use a graphing calculator to check
the solutions. Graph the following function:
y = tan 2x + tan x
Then use the Zero feature to find the x-values for
which y = 0.
..........
Zero
X=1.0471976 Y=0
Some equations that involve double or half angles can be solved directly—without
resorting to double- or half-angle formulas.
EXAMPLE 6
Solving a Trigonometric Equation
x
2
Solve 2 cos + 1 = 0.
SOLUTION
x
2 cos + 1 = 0
2
Write original equation.
x
2
2 cos = º1
x
2
Subtract 1 from each side.
1
2
cos = º
Divide each side by 2.
2π
x
= + 2nπ
3
2
4π
3
x = + 4nπ
or
4π
+ 2nπ
3
General solution for }}
or
8π
+ 4nπ
3
General solution for x
x
2
14.7 Using Double- and Half-Angle Formulas
877
Page 4 of 8
GOAL 2
USING TRIGONOMETRY IN REAL LIFE
The path traveled by an object that is projected at an initial height of h0 feet, an initial
speed of v feet per second, and an initial angle † is given by
16
v cos †
x2 + (tan †)x + h0
y = º
2
2
where x and y are measured in feet. (This model neglects air resistance.)
Simplifying a Trigonometric Model
EXAMPLE 7
SPORTS Find the horizontal distance
traveled by a football kicked from ground
level (h 0 = 0) at speed v and angle †.
†
Not drawn to scale
SOLUTION
Using the model above with h 0 = 0, set y equal to 0 and solve for x.
16
v co s †
16
x º tan † = 0
(ºx) v2 c o s 2 †
16
x º tan † = 0
v2 cos2 †
x2 + (tan †)x = 0
º
2
2
Let y = 0.
Factor.
Zero product property
(Ignore ºx = 0.)
16
x = tan †
v2 c o s 2 †
1
x = v2 cos2 † tan †
16
Add tan † to each side.
1
16
Multiply each side by }}v 2 cos2 †.
1
16
Use cos † tan † = sin †.
1
32
Rewrite }} as }} • 2.
1
32
Use a double-angle formula.
x = v2 cos † sin †
FOCUS ON
1
16
x = v2 (2 cos † sin †)
APPLICATIONS
x = v2 sin 2†
EXAMPLE 8
1
32
Using a Trigonometric Model
SPORTS You are kicking a football from ground level with an initial speed of
80 feet per second. Can you make the ball travel 200 feet?
RE
FE
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FIELD GOALS The
SOLUTION
1
32
200 = (80)2 sin 2†
INT
longest professional
field goal was 63 yards,
made by Tom Dempsy in
1970. This record was tied
by Jason Elam during the
1998–1999 season.
1 = sin 2†
NE
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APPLICATION LINK
www.mcdougallittell.com
878
Substitute for x and v in the formula from Example 7.
1
32
Divide each side by }}(80)2 = 200.
90° = 2†
sinº1 1 = 90°
45° = †
Solve for †.
You can make the football travel 200 feet if you kick it at an angle of 45°.
Chapter 14 Trigonometric Graphs, Identities, and Equations
Page 5 of 8
GUIDED PRACTICE
✓
Concept Check ✓
Vocabulary Check
? formula for sine.
1. Complete this statement: sin 2u = 2 sin u cos u is called the cos 2† º cos †
2. Suppose you want to simplify . Which double-angle formula for
cos † º 1
cosine would you use to rewrite cos 2†? Explain.
3. ERROR ANALYSIS Explain what is wrong in the calculations shown below.
a.
b.
30°
tan 15° = tan 2
390°
2
1 º sin 390°
2
cos 195° = cos 1
= tan 30°
2
=
1 3
= 2 3
=
3
6
=
41
1
= 1 º 2
2
1
2
= Skill Check
✓
2
5
3π
2
Given tan u = with π < u < , find the exact value of the expression.
u
4. sin 2
u
5. cos º
2
u
6. tan 2
7. sin 2u
8. cos 2u
9. tan 2u
Simplify the expression.
10. cos2 x º 2 cos 2x + 1
x
11. sin 2x tan 2
12. sin2 x + sin 2x + cos2 x
tan 2x
13. s e c2 x
x
x
14. sin cos 2
2
15. cos 2x cot2 x º cot2 x
16.
FOOTBALL Look back at Example 8 on page 878. Through what range of
angles can you kick the football to make it travel at least 150 feet?
PRACTICE AND APPLICATIONS
STUDENT HELP
Extra Practice
to help you master
skills is on p. 960.
EVALUATING TRIGONOMETRIC EXPRESSIONS Find the exact value of the
expression.
17. tan 15°
18. sin 22.5°
19. tan (º22.5°)
20. cos 67.5°
21. tan (º75°)
7π
22. cos 8
7π
23. sin º
12
5π
24. cos º
12
π
25. sin 12
u
u
, cos , and tan .
HALF-ANGLE FORMULAS Find the exact values of sin u
2
2
2
3
π
26. cos u = , 0 < u < 5
2
2 3π
27. cos u = , < u < 2π
3 2
9 π
28. sin u = , < u < π
10 2
4 3π
29. sin u = º, < u < 2π
5 2
14.7 Using Double- and Half-Angle Formulas
879
Page 6 of 8
DOUBLE-ANGLE FORMULAS Find the exact values of sin 2x, cos 2x, and
STUDENT HELP
tan 2x.
HOMEWORK HELP
π
30. tan x = 2, 0 < x < 2
1
π
31. tan x = º, º < x < 0
2
2
1
3π
32. cos x = º, π < x < 3
2
3 3π
33. sin x = º, < x < 2π
5 2
Example 1: Exs. 17–25
Example 2: Exs. 26–33
Example 3: Exs. 34–42
Example 4: Exs. 43–50
Examples 5, 6: Exs. 51–65
Examples 7, 8: Exs. 69–73
SIMPLIFYING TRIGONOMETRIC EXPRESSIONS Rewrite the expression
π
without double angles or half angles, given that 0 < x < . Then simplify the
2
expression.
x
34. 2
+2o
csx cos 2
sin 2x
35. sin x
36. tan 2x (1 + tan x)
37. cos 2x º 3 sin2 x
cos 2x
38. cos2 x
sin x
x
39. tan 2
1 º c o s2 x
x
40. (1 + cos x) tan 2
1 + cos 2x
41. cot x
sin tan 2
2
42. 1 º cos x
x
2
x
VERIFYING IDENTITIES Verify the identity.
43. (sin x + cos x)2 = 1 + sin 2x
44. 1 + cos 10x = 2 cos2 5x
†
45. cos † + 2 sin2 = 1
2
†
†
1
2†
46. sin cos = sin 3
3
2
3
47. cos 3x = cos3 x º 3 sin2 x cos x
48. sin 4† = 4 sin † cos † (1 º 2 sin2 †)
49. cos2 2x º sin2 2x = cos 4x
50. cot † + tan † = 2 csc 2†
SOLVING TRIGONOMETRIC EQUATIONS Solve the equation for 0 ≤ x < 2π.
1
51. sin x = º1
2
1
52. cos x º cos x = 0
2
53. sin 2x cos x = sin x
54. cos 2x = º2 cos2 x
55. tan 2x º tan x = 0
x
56. sin + cos x = 1
2
cos 2x
57. tan 2x = 2
x
58. tan = sin x
2
cos 2x
59. =1
cos2 x
FINDING GENERAL SOLUTIONS Find the general solution of the equation.
60. cos 2x = º1
61. sin 2x + sin x = 0
62. cos 2x º cos x = 0
x
63. cos º sin x = 0
2
x
64. sin + cos x = 0
2
65. cos 2x = 3 sin x + 2
66. LOGICAL REASONING Show that the three double-angle formulas for cosine
are equivalent.
67. LOGICAL REASONING Show that the two half-angle formulas for tangent are
equivalent.
68.
Writing Use the formula at the top of page 878 to explain why the projection
angle that maximizes the distance a projectile travels is † = 45° when h0 = 0.
69.
PROJECTILE HEIGHT Find a formula for the maximum height of an object
projected from ground level at speed v and angle †. To do this, find half of the
1
32
horizontal distance v2 sin 2† and then substitute it for x in the general model
for the path of a projectile (where h0 = 0) at the top of page 878.
880
Chapter 14 Trigonometric Graphs, Identities, and Equations
Page 7 of 8
FOCUS ON
APPLICATIONS
70.
AERONAUTICS An airplane’s mach number M is the ratio of its speed to the
speed of sound. When an airplane travels faster than the speed of sound, the sound
waves form a cone behind the airplane (see Lesson 10.5, page 620).
†
1
The mach number is related to the apex angle † of the cone by sin = .
2
M
Find the angle † that corresponds to a mach number of 4.5.
INCA DWELLING In Exercises 71 and 72, use the following information.
Shown below is a drawing of an Inca dwelling found in Machu Picchu, about
50 miles northwest of Cuzco, Peru. All that remains of the ancient city today
are stone ruins.
RE
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71. Express the area of the triangular portion of
the side of the dwelling as a function of
MACHU PICCHU
is an ancient Incan
city discovered by Hiram
Bingham in 1911 and located
in the Andes Mountains near
Cuzco, Peru. The site consists of 5 square miles of
terraced gardens linked by
3000 steps.
18 ft
18 ft
†
†
†
sin and cos .
2
2
72. Express the area found in Exercise 71 as a
function of sin †. Then solve for † assuming
that the area is 132 square feet.
73.
OPTICS The index of refraction n of a transparent material is the ratio of the
speed of light in a vacuum to the speed of light in the material. Some common
materials and their indices are air (1.00), water (1.33), and glass (1.5). Triangular
prisms are often used to measure the index of refraction based on this formula:
† å
sin + 2
2
n = †
sin å
air
†
2
t
ligh
For the prism shown, å = 60°. Write the index
†
2
of refraction as a function of cot . Then find †
prism
if the prism is made of glass.
Test
Preparation
QUANTITATIVE COMPARISON In Exercises 74–76, choose the statement that
is true about the given quantities.
A
¡
B
¡
C
¡
D
¡
The quantity in column A is greater.
The quantity in column B is greater.
The two quantities are equal.
The relationship cannot be determined from the given information.
Column A
Column B
74. sinsin
x, x,
with
45°45°
≤<
x≤
90°90° sin 2x
with
x<
sin 2x
75. coscos
x, x,
with
90°90°
≤<
x≤
135°
with
x<
135° cos 2x
cos 2x
76. tantan
x, x,
with
45°45°
≤<
x≤
90°90° tan 2x
with
x<
tan 2x
★ Challenge
1
77. DERIVING FORMULAS Use the diagram
shown at the right to derive the formulas for
EXTRA CHALLENGE
†
2
†
2
†
2
†
2
sin , cos , and tan when † is an acute angle.
†
cos †
1
sin †
www.mcdougallittell.com
14.7 Using Double- and Half-Angle Formulas
881
Page 8 of 8
MIXED REVIEW
FUNCTION OPERATIONS Let ƒ(x) = 4x + 1 and g(x) = 6x. Perform the
indicated operation and state the domain. (Review 7.3)
78. ƒ(x) + g(x)
79. ƒ(x) º g(x)
80. ƒ(x) • g(x)
81. ƒ(x) ÷ g(x)
82. ƒ(g(x))
83. g(ƒ(x))
CALCULATING PROBABILITIES Calculate the probability of rolling a die
30 times and getting the given number of 4’s. (Review 12.6)
84. 1
85. 3
86. 5
87. 6
88. 8
89. 10
SIMPLIFYING EXPRESSIONS Simplify the expression. (Review 14.6)
3π
90. cos x º 2
91. sin (x º π)
96.
π
94. sin x + 2
93. cos (x º π)
π
95. tan x + 4
π
92. tan x º 3
MOVING The truck you have rented for moving your furniture has a ramp.
If the ramp is 20 feet long and the back of the truck is 3 feet above the ground, at
what angle does the ramp meet the ground? (Review 13.4)
QUIZ 3
Self-Test for Lessons 14.6 and 14.7
Find the exact value of the expression. (Lesson 14.6)
1. sin 105°
2. cos 285°
3. tan 165°
17π
4. sin 12
13π
5. cos 12
π
6. tan º
12
1
π
Given sin u = with < u < π, find the exact value of the expression.
3
2
(Lesson 14.7)
u
7. sin 2
u
8. cos 2
10. sin 2u
11. cos 2u
u 3+
2
2
9. tan 2
3 º2
2
12. tan 2u
Simplify the expression. (Lessons 14.6, 14.7)
13. sin (x + 3π)
14. cos (π º x)
sin 2x
16. 2 co s x
x
17. 2cos2 º cos x
2
1 º tan x
18. tan 2x
2
π
15. tan x + 4
Solve the equation. (Lessons 14.6, 14.7)
19. sin 3x = 0.5
20. cos 2x º cos2 x = 0
21. sin (2x º π) = sin x
22. tan (º2x) = 1
23.
1
GOLF Use the formula x = v2 sin 2† to find the horizontal distance x
32
(in feet) that a golf ball will travel when it is hit at an initial speed of 50 feet per
second and at an angle of 40°. (Lesson 14.7)
882
Chapter 14 Trigonometric Graphs, Identities, and Equations
Page 1 of 5
CHAPTER
14
Chapter Summary
WHAT did you learn?
WHY did you learn it?
Graph sine, cosine, and tangent functions. (14.1)
Graph the height of a boat moving over waves.
(p. 836)
Graph translations and reflections of sine, cosine,
and tangent graphs. (14.2)
Graph the height of a person rappelling down a cliff.
Use trigonometric identities to simplify
expressions. (14.3)
Simplify the parametric equations that describe a
carousel’s motion. (p. 854)
Verify identities that involve trigonometric
expressions. (14.3)
Show that two equations modeling the shadow of a
sundial are equivalent. (p. 853)
Solve trigonometric equations. (14.4, 14.6, 14.7)
Solve an equation that models the position of the sun
at sunrise. (p. 860)
Write sine and cosine models for graphs and data.
Write models for temperatures inside and outside an
igloo. (p. 866)
(14.5)
(p. 843)
Use sum and difference formulas. (14.6)
Relate the length of an image to the length of an
actual object when taking aerial photographs. (p. 873)
Use double- and half-angle formulas. (14.7)
Find the angle at which you should kick a football to
make it travel a certain distance. (p. 878)
Use trigonometric functions to solve real-life
problems. (14.1–14.7)
Model real-life patterns, such as the vibrations of a
tuning fork. (p. 833)
How does Chapter 14 fit into the BIGGER PICTURE of algebra?
In Chapter 14 you continued your study of trigonometry, focusing more on algebra
connections than geometry connections. You graphed trigonometric functions and
studied characteristics of the graphs, just as you have done with other types of
functions during this course.
In this chapter you saw how some algebraic skills are used in trigonometry, such as in
solving a trigonometric equation in quadratic form. If you go on to study higher-level
algebra, you will see how trigonometry is used in algebra, such as in finding the
complex nth roots of a real number.
STUDY STRATEGY
How did you use
multiple methods?
Here is an example of two
methods used for Example 2 on
page 849 following the Study
Strategy on page 830.
Multiple Methods
(1) sec † tan2 † + sec † = sec
† (sec2 † º 1) + sec †
= sec3 † º sec † + sec
= sec3 †
(2) sec † tan2 † + sec † = sec
†
† (tan2 † + 1)
= sec † (sec2 †)
= sec3 †
883
Page 2 of 5
CHAPTER
14
Chapter Review
VOCABULARY
• periodic function, p. 831
• cycle, p. 831
14.1
• period, p. 831
• amplitude, p. 831
• frequency, p. 833
• trigonometric identities, p. 848
Examples on
pp. 831–834
GRAPHING SINE, COSINE, AND TANGENT FUNCTIONS
EXAMPLES You can graph a trigonometric function by identifying the characteristics
and key points of the graph.
1
2
y = tan 3x
y = 2 sin 4x
π
2
2π
|4|
Period = = Amplitude = |2| = 2
π
|3|
π4 π2 π8, 2
38π, º2
Intercepts: (0, 0); , 0 ; , 0
Maximum:
π
3
Period = = Intercept: (0, 0)
π
6
π
6
Asymptotes: x = , x = º
1π2 12 Minimum:
1
2
π
12
Halfway points: , ; º, º
1
1
x
π
8
π
π
6
6
x
Draw one cycle of the function’s graph.
1
1. y = sin x
4
14.2
1
2. y = cos πx
2
3. y = tan 2πx
2
4. y = 3 tan x
3
Examples on
pp. 840–843
TRANSLATIONS AND REFLECTIONS OF TRIGONOMETRIC GRAPHS
EXAMPLE
π
4
To graph y = 2 º 4 cos 2 x + , start with the graph of y = 4 cos 2x.
π
Translate the graph left units and up 2 units, and reflect it in the line y = 2.
4
2π
Amplitude = |º4| = 4
Period = = π
6
|2|
π
On y = 2: (0, 2); , 2
2
π4 Maximum: , 6
884
π
4
34π Minimums: º, º2 ; , º2
Chapter 14 Trigonometric Graphs, Identities, and Equations
π
2
x
Page 3 of 5
Graph the function.
5. y = 5 sin (2x + π)
14.3
1
7. y = 2 + tan x + π
2
6. y = º4 cos (x º π)
Examples on
pp. 848–851
VERIFYING TRIGONOMETRIC IDENTITIES
EXAMPLE
You can verify identities such as sin x + cot x cos x = csc x.
csoins xx sin x + cot x cos x = sin x + cos x
Reciprocal identity
sin2 x + cos2 x
sin x
Write as one fraction.
= 1
sin x
Pythagorean identity
= csc x
Reciprocal identity
= Simplify the expression.
π
9. csc2 (ºx) cos2 º x
2
8. tan (ºx) cos (ºx)
π
10. sin2 º x º 2 sin2 x + 1
2
Verify the identity.
tan2 x
11. sin2 (ºx) = tan2 x + 1
14.4
12. 1 º cos2 x = tan2 (ºx) cos2 x
Examples on
pp. 855–858
SOLVING TRIGONOMETRIC EQUATIONS
You can find the general solution of a trigonometric equation or just the
solution(s) in an interval.
EXAMPLE
3 tan2 x º 1 = 0
Write original equation.
3 tan2 x = 1
Add 1 to each side.
1
3
tan2 x = Divide each side by 3.
3
3
tan x = ±
Take square roots of each side.
π
6
5π
6
There are two solutions in the interval 0 ≤ x < π: x = and x = . The general
π
6
5π
6
solution of the equation is: x = + nπ or x = + nπ where n is any integer.
Find the general solution of the equation.
13. 2 sin2 x tan x = tan x
14. sec2 x º 2 = 0
15. cos 2x + 2 sin2 x º sin x = 0
16. tan2 3x = 3
17. 2 sin x º 1 = 0
18. sin x (sin x + 1) = 0
Chapter Review
885
Page 4 of 5
14.5
Examples on
pp. 862–864
MODELING WITH TRIGONOMETRIC FUNCTIONS
34π , 4
EXAMPLE You can write a model for the
sinusoid at the right. Since the maximum and
minimum values of the function do not occur at
points equidistant from the x-axis, the curve has
a vertical shift. To find the value of k, add the
maximum and minimum values and divide by 2.
4 + (º2)
2
M+m
2
1
π
4
2
2
k = = = = 1
2π
b
x
π4 , 2
π
4
The period is = π, so b = 2. Because the minimum occurs at , the graph is a sine
curve that involves a reflection but no horizontal shift. The amplitude is
Mºm
2
4 º (º2)
2
|a| = = = 3. Since a < 0, a = º3. The model is y = 1 º 3 sin 2x.
Write a trigonometric function for the sinusoid with maximum at A and
minimum at B.
π
3π
π
20. A(0, 6), B(2π, 0)
21. A(0, 1), B , º1
19. A , 2 , B , º2
2
2
2
14.6
Examples on
pp. 869–871
USING SUM AND DIFFERENCE FORMULAS
EXAMPLE You can use formulas to evaluate trigonometric functions of the sum or
difference of two angles.
sin 105° = sin (45° + 60°) = sin 45° cos 60° + cos 45° sin 60°
2
1
2
3
2 + 6
=
2 •2+ 2 • 2 =
4
Find the exact value of the expression.
22. sin 150°
14.7
23. cos 195°
7π
25. tan 12
24. tan 15°
13π
26. cos 12
Examples on
pp. 875–878
USING DOUBLE- AND HALF-ANGLE FORMULAS
EXAMPLE
You can use formulas to evaluate some trigonometric functions.
3
π
1 º cos 1 º 2
6
1 π
π
= = 2 º 3
tan = tan • = π
2 6
12
1
sin 6
2
Find the exact value of the expression.
27. tan 165°
886
28. sin 67.5°
5π
29. cos 8
π
30. cos 12
Chapter 14 Trigonometric Graphs, Identities, and Equations
31. sin 6π
Page 5 of 5
Chapter Test
CHAPTER
14
Draw one cycle of the function’s graph.
1
1. y = 3 cos x
4
1
2. y = 4 sin πx
2
5
3. y = tan x
2
4. y = º2 tan 2x
5. y = º3 + 2 cos (x º π)
6. y = 1 º cos x
1
7. y = 5 + sin x
2
8. y = 5 + 2 tan (x + π)
Simplify the expression.
cos 2x + sin2 x
10. cos2 x
π
9. cos x º 2
tan 2x
sec2 x
11. º 2 tan x
1 º tan2 x
4 sin x cos x º 2 sin x sec x
12. 2 tan x
Verify the identity.
x
14. tan = csc x º cot x
2
13. º2 cos2 x tan (ºx) = sin 2x
15. cos 3x = cos3 x º 3 sin2 x cos x
Solve the equation in the interval 0 ≤ x < 2π. Check your solutions.
17. tan2 x º 2 tan x + 1 = 0
16. º6 + 10 cos x = º1
18. tan (x + π) + 2 sin (x + π) = 0
Find the general solution of the equation.
19. 4 º 3 sec2 x = 0
21. cos x csc2 x + 3 cos x = 7 cos x
20. cos x º sin x sin 2x = 0
Find the exact value of the expression.
22. sin 345°
23. tan 112.5°
13π
25. tan 12
24. cos 375°
π
26. sin 8
41π
27. cos 12
Find the amplitude and period of the graph. Then write a trigonometric
function for the graph.
28.
29.
2
30.
2
3
x
π
6
x
1
x
π
2
31.
TIDES The depth of the ocean at a swim buoy reaches a maximum of 6 feet
at 3 A.M. and a minimum of 2 feet at 9 A.M. Write a trigonometric function that
models the water depth y (in feet) as a function of time t (in hours). Assume that
t = 0 represents 12:00 A.M.
32.
TEMPERATURES The average daily temperature T (in degrees Fahrenheit)
in Baltimore, Maryland, is given in the table. The variable t is measured in
months, with t = 0 representing January 1. Use a graphing calculator to write a
trigonometric model for T as a function of t.
Source: U.S. National Oceanic and Atmospheric Administration
t
0.5
1.5
2.5
3.5
4.5
5.5
6.5
7.5
8.5
9.5
10.5
11.5
T
75
79
87
94
98
101
104
105
100
92
87
77
Chapter Test
887
Page 1 of 2
Chapter Standardized Test
CHAPTER
14
TEST-TAKING STRATEGY Long-term preparation for the SAT can be done throughout your
high school career and can improve your overall abilities. If you keep up with your homework,
both your problem-solving abilities and your vocabulary will improve. This type of long-term
preparation will definitely affect not only your SAT scores, but your overall future academic
performance as well.
1. MULTIPLE CHOICE Which function is graphed?
1
π
A
¡
C
¡
E
¡
x
π
B
¡
D
¡
y = tan 2x
1
y = tan x
2
5. MULTIPLE CHOICE What is the exact value of
5π
tan ?
12
A
¡
D
¡
3 º 23
B
¡
E
¡
A
¡
tan † = 3
4
B
¡
csc † = C
¡
sec † = 5
3
D
¡
tan † = º
E
¡
cot † = 2 + 3
2 º 3
C
¡
3 º 1
23 + 3
3
6. MULTIPLE CHOICE Given that cos † = º and
5
3π
π < † < , which of the following is true?
2
y = ºtan 2x
1
y = ºtan x
2
y = º2 tan x
2. MULTIPLE CHOICE Which function is graphed?
5
4
4
3
3
4
7. MULTIPLE CHOICE What trigonometric function
has a graph with maximum (π, 2) and minimum
(3π, º2)?
2
x
π
4
A
¡
C
¡
E
¡
B
¡
D
¡
1
4
y = 4 cos x
y = 3 + 4 cos 4x
1
4
y = 3 + 4 cos x
y = 4 cos 4x
y = 4 + 3 cos 4x
3. MULTIPLE CHOICE What is the simplified form of
π
sin2 º x tan2 x + cos2 (ºx) + tan2 x?
2
A
¡
D
¡
sec2 x
2
1 + sec x
B
¡
E
¡
2 tan2 x
2
C
¡
csc2 x
2
sin x + tan x
4. MULTIPLE CHOICE Which of the following is a
solution of the equation tan2 x cos x = cos x?
¡
x=0
D
¡
x = A
5π
6
¡
x = E
¡
x = B
π
6
5π
4
¡
C
π
2
x = A
¡
C
¡
E
¡
1
2
y = 2 sin x
y = 2 cos 2x
1
2
y = 2 cos x
π
2
y = 2 sin x
8. MULTIPLE CHOICE Which of the following is a
tan x sin 2x + 2 sin2 x
solution of the equation = 1?
º1 + 4 sin x
A
¡
D
¡
x=0
x=π
B
¡
E
¡
5π
6
x = C
¡
11π
6
x = There is no solution.
9. MULTIPLE CHOICE What is the simplified form of
x
2 sin x tan 2
the expression ?
π
cos º x sin (ºx) + 1
2
A
¡
B
¡
2 sec2 x
¡
D
¡
2
cos2 x
1 º co s x
cos2 x
2 º 2 cos x
C cos2 x
2 sin x
E cos2 x
¡
888
B
¡
D
¡
y = 2 sin 2x
Chapter 14 Trigonometric Graphs, Identities, and Equations
Page 2 of 2
QUANTITATIVE COMPARISON In Exercises 10 and 11, choose the statement
that is true about the given quantities.
A
¡
B
¡
C
¡
D
¡
The quantity in column A is greater.
The quantity in column B is greater.
The two quantities are equal.
The relationship cannot be determined from the given information.
Column A
Column B
10.
Amplitude of the graph of y = º5 + 4 sin 3πx
Amplitude of the graph of y = 3 º 5 sin 2πx
11.
Period of the graph of y = tan 4πx
Period of the graph of y = 3 tan 4x
12.
MULTI-STEP PROBLEM The average daily time R of the sunrise and the
average daily time S of the sunset for each month in Dallas, Texas, is given in
the table. The variable t is measured in months, with t = 0 representing January 1.
t
0.5
1.5
2.5
3.5
4.5
5.5
6.5
7.5
8.5
9.5
10.5
11.5
R
7:29
7:10
6:37
5:58
5:29
5:20
5:31
5:51
6:11
6:32
6:58
7:21
S
17:45
18:12
18:36
18:58
19:20
19:36
19:35
19:11
18:33
17:54
17:27
17:24
a. Use a graphing calculator to find trigonometric models for R and S as functions
of t. When entering the data into the calculator, you must convert the number of
minutes into a fraction of an hour. For example, enter 7:27 as 7 + (27 / 60).
b. Graph the functions you found in part (a). Use a viewing window of 0 ≤ x ≤ 48
and 0 ≤ y ≤ 24. Describe the periods, amplitudes, and locations of local
maximums and minimums. How are the functions alike? How are they different?
c. Let D = S º R. What does D represent?
d. Graph D in the same viewing window as R and S. How are the maximums and
minimums of the three functions related? Explain the real-life significance of
the relationships.
13.
MULTI-STEP PROBLEM The average number of daylight hours DM in
Great Falls, Michigan, is given in the table. The variable t is measured in
months, with t = 0 representing January 1.
t
0.5
1.5
2.5
3.5
4.5
5.5
6.5
7.5
8.5
9.5
10.5
11.5
DM
8:57
10:18
11:55
13:38
15:07
15:54
15:31
14:14
12:34
10:51
9:21
8:32
a. Use a calculator to find a trigonometric model for DM as a function of t.
b. Use the table given in Exercise 12. Subtract each R-value from its corresponding
S-value to find the average number of hours of sunlight a day for each month in
Dallas. Use a graphing calculator to find a trigonometric model for the data as a
function of t.
c. Graph the functions you found in parts (a) and (b). Use a viewing window of
0 ≤ x ≤ 48 and 0 ≤ y ≤ 24. Describe the periods, amplitudes, and locations
of local maximums and minimums of the functions. How are the functions
alike? How are they different? Do the graphs intersect? If so, where?
Chapter Standardized Test
889
Page 1 of 2
CHAPTER
14
Cumulative Practice
for Chapters 1–14
Write an equation of the line with the given characteristics. (2.4)
1. slope: º2, y-intercept: 7
2. points: (5, 0), (º3, 2)
3. vertical line through (4, 2)
Solve the system. (3.1, 3.2, 3.6, 4.3, 4.5, 10.7)
4. x º 2y = 6
3x + y = 4
x2 + y2 º 6x º 8y + 16 = 0
ºx + 2y º z = 2
3x º y + 4z = 10
Solve the matrix equation. (4.4)
7.
6. x2 + y2 = 16
5. x + y + z = 10
4
3
2 º5
X=
º1 º1
3 º1
8.
5 3
º1 6
X=
7 4
2 0
9.
8 º1
6
0
X=
º2
0
3 º2
Perform the indicated operations. (6.3, 6.5, 9.4, 9.5)
10. (º2x2 º x + 4) º (3x + 10)
11. (x º 4)(2x2 + 3x º 1)
12. (x3 º 5x + 6) ÷ (x º 2)
x2 º 36
x+6
13. ÷ 2x
8x + 10
6x
xº4
+ 14. xº2
x2 + 3x º 10
4x
1
15. º xº7
x+7
Evaluate the expression without using a calculator. (7.1, 8.4)
5
4
16. 82/3
17. 125º1/3
18. º93/2
19. º
1
20. 1
0,0
00
1
21. log2 16
22. log3 81
23. ln e7
24. log 0.01
25. log5 1
Find the distance between the two points. Then find the midpoint of the line
segment connecting the two points. (10.1)
26. (0, 0), (3, º8)
27. (º5, 0), (0, 2)
28. (º1, º4), (2, 3)
29. (7, 4), (0, º3)
Write the next term of the sequence. Then write a rule for the nth term.
(11.1º11.3)
30. 1, 4, 9, 16, . . .
31. 8, 4, 2, 1, . . .
32. 2, 6, 18, 54, . . .
33. º6, º1, 4, 9, . . .
Find the sum of the series. (11.1º11.4)
10
34.
5
∑ 16
35.
i=1
∑ 100012
4
∑ (3i º 1)
36.
i=1
∑ 2º13
‡
i
37.
i=0
nº1
n=1
Find the number of permutations or combinations. (12.1, 12.2)
38. 6P5
39. 10P2
40. 3P3
41. 8C1
42. 4C2
43. 7C4
Find the arc length and area of a sector with the given radius r and central
angle †. (13.2)
44. r = 11 cm, † = 80°
45. r = 6 in., † = 270°
46. r = 3 ft, † = 120°
Evaluate the function without using a calculator. (13.3)
47. tan 390°
890
48. sin (º45°)
Chapters 1–14
49. csc 90°
3π
50. cot º
4
5π
51. cos 3
Page 2 of 2
Evaluate the expression without using a calculator. Give your answer in both
radians and degrees. (13.4)
52. cosº1 0
1
53. sinº1 2
54. tanº1 1
2
55. cosº1 º
2
56. tanº1 (º3
)
Solve ¤ABC. (13.5, 13.6)
57. A = 65°, a = 7, b = 4
58. B = 110°, a = 3, c = 8
59. a = 10, b = 9, c = 4
61. C = 98°, a = 34, b = 20
62. a = 7, b = 4, c = 6
Find the area of ¤ABC. (13.5, 13.6)
60. A = 63°, c = 13, b = 20
Graph the parametric equations. Then write an xy-equation and state the
domain. (13.7)
1
63. x = t + 1, y = t º 3 for 0 ≤ t ≤ 4
4
64. x = º2t, y = t + 3 for 1 ≤ t ≤ 5
Graph the function. (14.1, 14.2)
1
66. y = 4 sin πx
3
65. y = 5 cos 2x
Simplify the expression. (14.3)
2
69. tan (ºx) + tan x sec x
67. y = 5 + sin 4x
π sin º x
2
70. sin x
1
68. y = º3 + tan x
2
71. tan x sec x º csc x sec2 x
Find the general solution of the equation. (14.4)
+ 5 sin x
72. 3 sin x = 3
x
73. 2 cos2 º 1 = 0
2
74. cos x sin2 x º cos x = 0
Find the exact value of the expression. (14.6, 14.7)
75. sin 255°
76. sin 157.5°
77. tan 105°
π
78. tan 12
13π
79. cos 12
80. FRACTAL GEOMETRY Tell whether c = 1 + i is in the Mandelbrot set. Use
absolute value to justify your answer. (5.4)
81.
GIRLS BASKETBALL The heights (in inches) of the girls chosen for the first
team on PARADE’s 23rd annual All-America High School Girls Basketball
Team are listed below. Find the mean, median, mode(s), range, and standard
deviation of the heights. Draw a box-and-whisker plot for the heights.
Source: Parade Magazine (7.7)
76, 74, 76, 71, 72, 78, 66, 68, 74, 69
82.
EQUAL GENDERS What is the probability that a family with four children
has exactly two girls and two boys in any order? Assume that having a girl and
having a boy are equally likely events. (12.6)
83.
RIALTO TOWER Suppose you are looking at the Rialto Tower in Melbourne,
Australia, which reaches a height of 794 feet. Your angle of elevation to the top
of the building is 39.8°. How far are you from the base of the building?
Source: Council on Tall Buildings and Urban Habitat (13.1)
84.
BICYCLING As you pedal up a hill, the pedals on your mountain bike make
one revolution every two seconds. The maximum height of the pedal is 19 inches
above the ground and the minimum height is 5 inches above the ground. Write a
trigonometric model for the height H of the pedal as a function of time t. (14.1, 14.2)
Cumulative Practice
891
Page 1 of 2
PROJECT
Applying Chapters
13 and 14
The Mathematics of Music
OBJECTIVE Explore the relationship between music and trigonometric
functions.
Materials: plastic or glass bottle, container of water, CBL, CBL microphone, TI-82
or TI-83 graphing calculator with cable to link to the CBL
INT
STUDENT HELP
NE
ER T
TECHNOLOGY
HELP
For suggestions about
doing this project see
www.mcdougallittell.com
Sound is a variation in pressure transmitted through air, water, or other matter. Sound
travels as a wave. The sound of a pure note can be represented using a sine wave (or a
cosine wave; recall that a cosine wave is just a sine wave shifted horizontally). More
complicated sounds can be modeled by the sum of several sine waves.
The pitch of a sound wave is determined by the wave’s frequency. The greater the
frequency, the higher the pitch.
Note
Frequency
(cycles/second)
middle
C
D
E
F
G
A
B
C
262
294
330
349
392
440
494
523
INVESTIGATION
1. Fill a 12–20 ounce plastic or glass bottle almost to the
top with water. Blow across the top and listen to the note
produced. Pour a small amount of water out and repeat.
Continue to remove water and blow notes until the
bottle is empty. What happens to the frequency of the
notes as the water level decreases? How can you tell?
2. Fill the bottle partway with water and blow across the
top to create a note with constant pitch and volume. If
you have trouble producing a steady stream of air, use a
straw to blow across the bottle top. Use the CBL and the
CBL microphone to collect the sound data and store it in
the graphing calculator. Use the graphing calculator to
graph the pressure of the sound as a function of time.
The graph should resemble a sine wave.
3. Use the graph of the sound data to calculate the frequency of the note—the
number of complete cycles in one second.
4. Write a sine function to describe the note.
5. Choose a note from the table. Try producing the note as follows: Adjust the
water level in the bottle, blow across the top, and use the CBL and graphing
calculator to find the frequency of the resulting note. Repeat this process until
the frequency of the note you produce is approximately equal to the frequency
of the note you chose from the table.
6. Write a sine function to describe the note you chose from the table.
892
Chapters 13 and 14
Page 2 of 2
PRESENT YOUR RESULTS
Write a report to present your results.
•
•
Explain how you used the CBL.
Explain how you found the frequency of a note from
the note’s sine wave.
•
Explain how you wrote the sine functions in
Exercises 4 and 6.
•
Include a sketch of the water level in your bottle
for both notes you produced in Exercises 2 and 5.
•
Include a graph of the sine wave for each of the
two notes.
•
Consider including a recording of the notes you
produced. You might repeat the experiment to
produce several different notes.
•
Describe how you used your knowledge of
trigonometric functions in this project.
EXTENSION
Choose a note to play and have a classmate also choose a note.
Find sine functions y = ƒ(x) and y = g(x) that model the two
notes (as you did in Exercises 4 and 6 of the investigation).
Then play the notes simultaneously and use the CBL and
graphing calculator to graph the resulting sound wave.
Compare this graph with the graph of y = ƒ(x) + g(x).
What do you notice?
MUSIC CONNECTION
Bring in musical instruments and play a pure
note on each. Use the CBL and graphing calculator
to find the sine function that corresponds to each
note. Observe what happens when you change
the volume of the notes. Also, compare the sine
waves for different instruments playing the
same note.
Project
893