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Problems : 1-27 odd, 45-48, 77-82, 95,96
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CHAPTER 4
Review Exercises
399
CHAPTER 4 Key Ideas
Inverse Tangent Function 381
Special Angles 330, 331, 346
Sinusoids 352
Properties, Theorems, and Formulas
Arc Length 322, 323
Right Triangle Trigonometric Ratios 329
Trigonometric Functions of Real Numbers 344
Inverse Sine Function 378
Inverse Cosine Function 380
Procedures
Angle Measure Conversion 322
Gallery of Functions
[–2π , 2π ] by [–4, 4]
f(x) ⫽ sin x
[–2π , 2π ] by [–4, 4]
f(x) ⫽ cos x
[⫺3∏/2, 3∏/2] by [–4, 4]
f(x) ⫽ tan x
[–2π , 2π ] by [–4, 4]
f(x) ⫽ cot x
[–2π , 2π ] by [–4, 4]
f(x) ⫽ sec x
[–2π , 2π ] by [–4, 4]
f(x) ⫽ csc x
2
–1
2
1
– 2
–1
[–1.5, 1.5] by [–1.7, 1.7]
f(x) ⫽ sin⫺1 x
– 2
1
[–2, 2] by [–1, 3.5]
f(x) ⫽ cos⫺1 x
[–4, 4] by [–2.8, 2.8]
f(x) ⫽ tan⫺1 x
CHAPTER 4 Review Exercises
Exercise numbers with a gray background indicate problems that
the authors have designed to be solved without a calculator.
The collection of exercises marked in red could be used as a chapter test.
In Exercises 1–8, determine the quadrant of the terminal side of the
angle in standard position. Convert degree measures to radians and
radian measures to degrees.
1.
5p
2
3. – 135°
2.
3p
4
4. – 45°
5. 78°
6. 112°
p
7.
12
8.
7p
10
In Exercises 9 and 10, determine the angle measure in both degrees
and radians. Draw the angle in standard position if its terminal side is
obtained as described.
9. A three-quarters counterclockwise rotation
10. Two and one-half counterclockwise rotations
400
CHAPTER 4 Trigonometric Functions
In Exercises 11–16, the point is on the terminal side of an angle in
standard position. Give the smallest positive angle measure in both degrees and radians.
11. 1 13, 12
12. 1-1, 12
15. 16, – 122
16. 12, 42
13. 1 – 1, 132
14. 1 – 3, – 32
In Exercises 17–28, evaluate the expression exactly without a calculator.
17. sin 30°
18. cos 330°
5p
21. sin
6
2p
22. csc
3
p
23. sec a – b
3
2p
b
24. tan a 3
25. csc 270°
26. sec 180°
19. tan 1 -135°2
20. sec 1 -135°2
27. cot 1 -90°2
28. tan 360°
In Exercises 29–32, evaluate exactly all six trigonometric functions of
the angle. Use reference triangles and not your calculator.
p
29. 6
19p
30.
4
31. – 135°
32. 420°
33. Find all six trigonometric functions of a in ^ ABC.
B
β
A
α
12 cm
38. Use a calculator in radian mode to solve sin x = 0.218 if
0 … x … 2p.
In Exercises 39–44, solve the right ^ ABC.
b
51. 1-5, -32
52. 14, 92
In Exercises 53–60, use transformations to describe how the graph of
the function is related to a basic trigonometric graph. Graph two periods.
53. y = sin 1x + p2
55. y = – cos 1x + p/2) + 4
54. y = 3 + 2 cos x
56. y = – 2 – 3 sin 1x – p2
58. y = – 2 cot 3x
57. y = tan 2x
59. y = – 2 sec
x
2
60. y = csc px
In Exercises 61–66, state the amplitude, period, phase shift, domain,
and range for the sinusoid.
61. ƒ1×2 = 2 sin 3x
62. g1x2 = 3 cos 4x
63. ƒ1×2 = 1.5 sin 12x – p/42
64. g1x2 = – 2 sin 13x – p/32
65. y = 4 cos 12x – 12
66. g1x2 = – 2 cos 13x + 12
In Exercises 67 and 68, graph the function. Then estimate the values
of a, b, and h so that ƒ1×2 L a sin 1b1x – h22.
68. ƒ1×2 = 3 cos 2x – 2 sin 2x
37. Use a calculator in radian mode to solve tan x = 1.35 if
p … x … 3p/2.
A
50. 112, 72
C
36. Use a calculator in degree mode to solve cos u = 3/7 if
0° … u … 90°.
α
49. 1-3, 62
67. ƒ1×2 = 2 sin x – 4 cos x
35. Use a right triangle to determine the values of all trigonometric functions of u, where tan u = 15/8.
β
48. sec x 6 0 and csc x 7 0
In Exercises 49–52, point P is on the terminal side of angle u. Evaluate the six trigonometric functions for u.
5 cm
34. Use a right triangle to determine the values of all trigonometric functions of u, where cos u = 5/7.
c
47. tan x 6 0 and sin x 7 0
In Exercises 69–72, use a calculator to evaluate the expression.
Express your answer in both degrees and radians.
69. sin-1 10.7662
70. cos -1 10.4792
72. sin-1 a
71. tan-1 1
In Exercises 73–76, use transformations to describe how the graph of
the function is related to a basic inverse trigonometric graph. State the
domain and range.
73. y = sin-1 3x
-1
75. y = sin
74. y = tan-1 2x
13x – 12 + 2 76. y = cos -1 12x + 12 – 3
In Exercises 77–82, find the exact value of x without using a calculator.
B
77. sin x = 0.5, p/2 … x … p
a
78. cos x = 13/2, 0 … x … p
C
79. tan x = – 1,
0 … x … p
39. a = 35°, c = 15
40. b = 8, c = 10
80. sec x = 2, p … x … 2p
41. b = 48°, a = 7
42. a = 28°, c = 8
81. csc x = – 1, 0 … x … 2p
43. b = 5, c = 7
44. a = 2.5, b = 7.3
82. cot x = – 13, 0 … x … p
In Exercises 45–48, x is an angle in standard position with
0 … x … 2p. Determine the quadrant of x.
45. sin x 6 0 and tan x 7 0
46. cos x 6 0 and csc x 7 0
13
b
2
In Exercises 83 and 84, describe the end behavior of the function.
83.
sin x
x
2
84.
3 -x/12
sin 12x – 32
e
5
CHAPTER 4
In Exercises 85–88, evaluate the expression without a calculator.
85. tan 1tan
87. tan 1sin
-1
-1
12
3/52
86. cos
-1
88. cos
-1
1cos p/32
1cos(-p/7)2
In Exercises 89–92, determine whether the function is periodic. State
the period (if applicable), the domain, and the range.
89. ƒ1×2 = ƒ sec x ƒ
90. g1x2 = sin ƒ x ƒ
91. ƒ1×2 = 2x + tan x
92. g1x2 = 2 cos 2x + 3 sin 5x
93. Arc Length Find the length of the arc intercepted by a
central angle of 2p/3 rad in a circle with radius 2.
94. Algebraic Expression Find an algebraic expression
equivalent to tan 1cos -1 x2.
95. Height of Building The angle of elevation of the top
of a building from a point 100 m away from the building on
level ground is 78°. Find the height of the building.
96. Height of Tree A tree casts a shadow 51 ft long when
the angle of elevation of the Sun (measured with the horizon)
is 25°. How tall is the tree?
97. Traveling Car From the top of a 150-ft building Flora
observes a car moving toward her. If the angle of depression
of the car changes from 18° to 42° during the observation,
how far does the car travel?
98. Finding Distance A lighthouse L stands 4 mi from the
closest point P along a straight shore (see figure). Find the
distance from P to a point Q along the shore if ∠ PLQ = 22°.
P
Q
4 mi
22°
L
101. Height of Tree Dr. Thom Lawson standing on flat
ground 62 ft from the base of a Douglas fir measures the angle of elevation to the top of the tree as 72°24¿ . What is the
height of the tree?
401
102. Storing Hay A 75-ft-long conveyor is used at the
Lovelady Farm to put hay bales up for winter storage. The
conveyor is tilted to an angle of elevation of 22°.
(a) To what height can the hay be moved?
(b) If the conveyor is repositioned to an angle of 27°, to what
height can the hay be moved?
103. Swinging Pendulum In the Hardy Boys Adventure
While the Clock Ticked, the pendulum of the grandfather
clock at the Purdy place is 44 in. long and swings through an
arc of 6°. Find the length of the arc that the pendulum
traces.
104. Finding Area A windshield wiper on a vintage 1994
Plymouth Acclaim is 20 in. long and has a blade 16 in. long. If
the wiper sweeps through an angle of 110°, how large an area
does the wiper blade clean? (See Exercise 71 in Section 4.1.)
105. Modeling Mean Temperature The average daily
air temperature 1°F2 for Fairbanks, Alaska, from 1971 to
2000, can be modeled by the equation
T1x2 = 37.3 sin c
2p
1x – 1142 d + 26,
365
where x is time in days with x = 1 representing January 1.
On what days do you expect the average temperature to be
32°F?
Source: National Climatic Data Center, as reported in the World Almanac
and Book of Facts 2009.
106. Taming The Beast The Beast is a featured roller
coaster at the King Island’s amusement park just north of
Cincinnati. On its first and biggest hill, The Beast drops from
a height of 52 ft above the ground along a sinusoidal path to a
depth 18 ft underground as it enters a frightening tunnel. The
mathematical model for this part of track is
h1x2 = 35 cos a
x
b + 17, 0 … x … 110,
35
where x is the horizontal distance from the top of the hill and
h1x2 is the vertical position relative to ground level (both in
feet). What is the horizontal distance from the top of the hill
to the point where the track reaches ground level?
99. Navigation An airplane is flying due east between two
signal towers. One tower is due north of the other. The bearing from the plane to the north tower is 23°, and to the south
tower is 128°. Use a drawing to show the exact location of the
plane.
100. Finding Distance The bearings of two points on the
shore from a boat are 115° and 123°. Assume the two points
are 855 ft apart. How far is the boat from the nearest point on
shore if the shore is straight and runs north-south?
Review Exercises
First hill of
The Beast
52ft
Ground level
18ft
Tunnel
402
CHAPTER 4 Trigonometric Functions
CHAPTER 4 Project
Modeling the Motion of a Pendulum
Explorations
As a simple pendulum swings back and forth, its displacement can be modeled using a standard sinusoidal equation of
the form
1. If you collected motion data using a CBL or CBR, a plot
y = a cos 1b1x – h22 + k
where y represents the pendulum’s distance from a fixed point
and x represents total elapsed time. In this project, you will
use a motion detection device to collect distance and time data
for a swinging pendulum, then find a mathematical model that
describes the pendulum’s motion.
Collecting the Data
To start, construct a simple pendulum by fastening about
1 meter of string to the end of a ball. Set up the Calculator
Based Laboratory (CBL) system with a motion detector or a
Calculator Based Ranger (CBR) system to collect time and
distance readings for between 2 and 4 seconds (enough time
to capture at least one complete swing of the pendulum).
See the CBL/CBR guidebook for specific setup instructions.
Start the pendulum swinging in front of the detector, then
activate the system. The data table below shows a sample set
of data collected as a pendulum swung back and forth in front
of a CBR.
Total Elapsed
Time (seconds)
Distance from the CBR
(meters)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
0.665
0.756
0.855
0.903
0.927
0.931
0.897
0.837
0.753
0.663
0.582
0.525
0.509
0.495
0.521
0.575
0.653
0.741
0.825
0.888
0.921
of distance versus time should be shown on your graphing calculator or computer screen. If you don’t have access to a CBL/CBR, enter the data in the sample table
into your graphing calculator/computer. Create a scatter
plot for the data.
2. Find values for a, b, h, and k so that the equation y = a
cos 1b1x – h22 + k fits the distance versus time data
plot. Refer to the information box on page 355 in this
chapter to review sinusoidal graph characteristics.
3. What are the physical meanings of the constants a and k
in the modeling equation y = a cos 1b1x – h22 + k?
[Hint: What distances do a and k measure?]
4. Which, if any, of the values of a, b, h, and/or k would
change if you used the equation y = a sin
1b1x – h22 + k to model the data set?
5. Use your calculator or computer to find a sinusoidal regression equation to model this data set (see your
grapher’s guidebook for instructions on how to do this).
If your calculator/computer uses a different sinusoidal
form, compare it to the modeling equation you found
earlier, y = a cos 1b1x – h22 + k.
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