Chapter Solutions

CHAPTER 4
Trigonometric Functions
Section 4.1 Radian and Degree Measure . . . . . . . . . . . . . . . . 272
Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . 281
Section 4.3 Right Triangle Trigonometry . . . . . . . . . . . . . . . . 289
Section 4.4 Trigonometric Functions of Any Angle . . . . . . . . . . 300
Section 4.5 Graphs of Sine and Cosine Functions . . . . . . . . . . . 317
Section 4.6 Graphs of Other Trigonometric Functions . . . . . . . . . 329
Section 4.7 Inverse Trigonometric Functions . . . . . . . . . . . . . . 339
Section 4.8 Applications and Models . . . . . . . . . . . . . . . . . . 350
Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
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CHAPTER 4
Trigonometric Functions
Section 4.1
Radian and Degree Measure
You should know the following basic facts about angles, their measurement, and their applications.
■ Types of Angles:
(a) Acute: Measure between 0 and 90.
(b) Right: Measure 90.
(c) Obtuse: Measure between 90 and 180.
(d) Straight: Measure 180.
■ and are complementary if They are supplementary if
■ Two angles in standard position that have the same terminal side are called coterminal angles.
■ To convert degrees to radians, use 1 180 radians.
■ To convert radians to degrees, use 1 radian 180.
1
■ one minute 160 of 1
1
■ one second 160 of 1 13600 of 1
■ The length of a circular arc is s r where is measured in radians.
■ Speed distancetime
■ Angular speed t srt
90.
180.
Vocabulary Check
1. Trigonometry 2. angle 3. standard position 4. coterminal
5. radian 6. complementary 7. supplementary 8. degree
9. linear 10. angular
1. The angle shown is
approximately 2 radians.
3
3. (a) Since lies in Quadrant IV.
2 < 7
4 < 2, 4
5
(b) Since
2 < 11 < 3,
4
Quadrant II.
7
11
4
lies in
2. The angle shown is
approximately 4 radians.
5
4. (a) Since
lies in
2 < 5
12 < 0, 2
Quadrant IV.
(b) Since 3
2 < 13 < ,
9
Quadrant II.
13
9
lies in
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272

Section 4.1 Radian and Degree Measure 273
5. (a) Since < 1 < 0; 1 lies in Quadrant IV. 6. (a) Since < 3.5 < 3 3.5 lies in Quadrant III.
2 2 ,
(b) Since < 2 < lies in
2 ; 2
Quadrant III.
(b) Since
2.25 lies in Quadrant II.
2 < 2.25 < ,
7. (a)
13
4
y
8. (a)
7
4
y
3π
4
x
x
−
7 π
4
(b)
4
3
y
(b)
5
2
y
4π
3
x
−
5π
2
x
9. (a)
11
6
y
11π
6
10. (a) 4
y
4
x
x
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(b)
2
3
6 2 13
6
6 2 11
6
y
2π
3
x
11. (a) Coterminal angles for : (b) Coterminal angles for
6 3 :
2
3 2 8
3
2
(b)
3 2 4
3
3
2
y
−3
x

274 Chapter 4 Trigonometric Functions
12. (a) Coterminal angles for
7
6 2 19
6
7
6 2 5
6
(b) Coterminal angles for
5
4 2 13
4
5
4 2 3
4
7
: 13. (a) Coterminal angles for
6
5
4 :
9
4 2
4
9
4 4 7
4
(b) Coterminal angles for
2
15 2 28
15
2
15 2 32
15
9 : 14. (a) Coterminal angles for
4
2
15 :
7
8 2 23
8
7
8 2 9
8
(b) Coterminal angles for
45 :
8
45 2 98
45
8
45 2 82
45
7
8 :
8
15. Complement:
Supplement:
17. Complement:
Supplement:
2 3 6
3 2
3
2 6 3
6 5
6
16. Complement: Not possible; is greater than
4
2 .
(a)Supplement:
3
4 4
3
18. Complement: Not possible; is greater than
3
2 .
Supplement:
2
3 3
2
19. Complement:
Supplement:
2 1 0.57 20. Complement: None 2 >
1 2.14
Supplement:
2 1.14
2
21.
22.
The angle shown is approximately 210.
23. (a) Since 90 < 150 < 180, 150 lies in
Quadrant II.
(b) Since 270 < 282 < 360, 282 lies in
Quadrant IV.
25. (a) Since 180 < 132 50 < 90,
132 50 lies in Quadrant III.
(b) Since 360 < 336 30 < 270,
336 30 lies in Quadrant I.
The angle shown is approximately 45.
24. (a) Since 0 < 87.9 < 90, 87.9 lies in
Quadrant I.
(b) Since 0 < 8.5 < 90, 8.5 lies in Quadrant I.
26. (a) Since 270 < 245.25 < 180, 245.25
lies in Quadrant II.
(b) Since 90 < 12.35 < 0, 12.35 lies
in Quadrant IV.
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Section 4.1 Radian and Degree Measure 275
27. (a)
30
y
(b)
150
y
150°
30
x
x
28. (a)
270
29. (a)
405
30. (a)
450
y
y
y
405°
− 450°
x
x
x
−270°
(b)
120
(b)
780
(b)
600
y
y
y
x
780°
x
−600
x
−120°
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31. (a) Coterminal angles for 52: 32. (a) Coterminal angles for 114: 33. (a) Coterminal angles for 300:
52 360 412
114 360 474
300 360 660
52 360 308
114 360 246
300 360 60
(b) Coterminal angles for 230:
(b) Coterminal angles for 36: (b) Coterminal angles for 390:
230 360 590
36 360 324
390 720 330
230 360 130
36 360 396
390 360 30
34. (a) Coterminal angles for 445: 35. Complement: 90 24 66 36. Complement: Not possible
445 720 275
Supplement: 180 24 156 Supplement: 180 129 51
445 360 85
(b) Coterminal angles for 740:
740 1080 340
740 720 20
37. Complement: 90 87 3 38. Complement: Not possible 39. (a) 30 30 180 6
Supplement: 180 87 93 Supplement: 180 167 13
(b) 150 150 180 5
6

276 Chapter 4 Trigonometric Functions
40. (a)
315 315 180 7
4
(b) 120 120 180 2
3
41. (a)
(b) 240 240 180 4
3
20 20 180 9
42. (a) 270 270
(b) 144 144
180 3 180 4
2
5
43. (a)
2 3
2 180
270
3
(b) 7 7
6 6 180
210
44. (a) 720
(b) 3 3 180
540
4 4 180
45. (a)
3 7
3 180
420
7
(b) 13 13
60 60 180
39
46. (a) 15 15
6 6 180
450
(b) 28
15 28
15 180
336
47. 115 115
2.007 radians 48. radians
180 83.7 83.7 180 1.461
49. 216.35 216.35
3.776 radians
180
51. 0.78 0.78 0.014 radians
180
50. 46.52 46.52
radians
180 0.812
52. 395 395
radians
180 6.894
53.
7 7 180
25.714 54.
55. 6.5 6.5 180
13 13 8 180
110.769
1170
8
56. 4.2 4.2 180
756
58. 0.48 0.48
180
27.502
60. 124 30 124.5
62. 408 16 25 408.274
64. 330 25 330.007
66. 115.8 115 48
67. 345.12 345 7 12
180
57. 2 2 114.592
59. 64 45 64
45
60 64.75
61. 85 18 30 85 18
60 3600 30 85.308
63. 125 36 125 3600 125.01
65. 280.6 280 0.660 280 36
68. 310.75 310 45
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Section 4.1 Radian and Degree Measure 277
69.
0.355 0.355
180
70.
20.34 20 20 24
0.7865 0.7865
180
45.0631
45 0.063160
45 3 0.78660
45 3 47
71.
s r
72.
s r
73.
3s r
6 5
6 5
radians
31 12
31
12 2 7 12 radians
32 7
3 32 7 44 7
radians
74.
s r
60 75
60
75 4 5 radians
Because the angle represented is clockwise, this angle is
4 5
radians.
75. The angles in radians are:
0 0
30
6
45
4
60
3
90
2
135 3
4
180
210 7
6
270 3
2
330 11
6
76. The angles in degrees are:
6 30
4 45
3 60
2
3 120
5
6 150
5
4 225
4
3 240
5
3 300
7
4 315
180
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77. s r
8 15
8 15
radians
s r 160
82. r 9 feet,
80 2 radians
60
s r 9 3 3 feet
3
s r 10
78. radian
22 5 11
80. r 80 kilometers, s 160 kilometers
81. s r, in radians
79. s r
35 14.5
70
radians
29 2.414
s 14180
inches
180 14 43.982
83. s r, in radians
s 27
2
meters 56.55 meters
3 18

278 Chapter 4 Trigonometric Functions
86. r s
3
43 9
4
3
84. r 12 centimeters,
4
s r 12
3
centimeters 28.27 cm
4 9
meters 0.72 meter
85. r s
36
2 72
feet 22.92 feet
87. r s 82
135180 328 miles 34.80 miles
3
88. r s 8
330180 48
11
inches 1.39 inches
89.
42 7 33 25 46 32
16 21 1 0.28537 radian
s r 40000.28537 1141.48 miles
90.
r 4000 miles
31 46 26 8 57 54 1.0105 rad
s r 40001.0105 4042.0 miles
91.
s r 450
0.07056 radian 4.04
6378
4 2 33.02
92.
r 6378, s 400
s r 400
0.0627 rad 3.59
6378
s r 2.5
93. 23.87
6 25
60 5 12 radian
94.
s r 24
5
4.8 rad 275.02
95. (a) single axel:
(b) double axel:
1 1 2
2 1 2
revolutions 360 180 540
2 3 radians
revolutions 720 180 900
4 5 radians
(c) triple axel: 3 1 2 revolutions 1260 7 radians
96. Linear speed s t r
t
97. (a)
6400 12502
436.967 kmmin
110
Revolutions
2400
Second 60 40 revsec 98. (a) Revolutions
4800 80 revsec
Second 60
Angular speed 240 80 radsec
Angular speed 280 160 radsec
(b) Radius of saw blade
Radius in feet
3.75
12
7.5
2
Speed s t r r rangular speed
t t
0.312580 78.54 ftsec
3.75 in.
0.3125 ft
(b) Radius of saw blade
Radius in feet
3.625
12
7.25
2
Speed s t r r rangular speed
t t
0.3021160 151.84 ftsec
3.625 in.
0.3021 ft
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Section 4.1 Radian and Degree Measure 279
99. (a)
Revolutions
Hour
Angular speed 228,800 57,600 radhr
252
Radius of wheel
12 in.ft5280 ftmi 5
25,344 miles
Speed
480 28,800 revhr
160
s t r r
t
5
25,344 57,600 125 35.70 mileshr
11
(b) Let x spin balance machine rate.
70 rangular speed r 2
rangular speed
t
x
160 5
25,344 120x ⇒ x 941.18 revmin
100. (a)
400 radmin ≤ angular speed ≤ 1000 radmin
(b) Speed
200 ≤ revolutions
minute
2 200 ≤ angular speed ≤ 2500
s t r r
t
6400 ≤ linear speed ≤ 61000
For the outermost track, 6000 cmmin
≤ 500
rangular speed 6angular speed
t
180
101. False, 1 radian 57.3, so one radian
is much larger than one degree.
102. No, 1260 is coterminal with 180.
103. True: 2
3
4 12 8 3
180
12
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104. (a) An angle is in standard position when the origin is the vertex and the initial side
coincides with the positive x-axis.
(b) A negative angle is generated by a clockwise rotation.
(c) Angles that have the same initial and terminal sides are coterminal angles.
(d) An obtuse angle is between 90 and 180.
105. If is constant, the length of the arc is
proportional to the radius s r, and
hence increasing.
107. A 1 square meters
2 r 2 1 2 102
3 50
3
106. Let A be the area of a circular sector of radius r
and central angle . Then
A
r 2
108. Because
2
⇒ 1 2 r 2.
s r, 12
15 .
Hence, A 1 2 r 2 1 2 12
15 152 90 ft2 .

280 Chapter 4 Trigonometric Functions
109.
A 1 2 r2, s r
(a)
0.8
Domain:
r > 0
8
A
s
⇒ 2 s r r0.8 Domain: r > 0
A 1 2 r 2 0.8 0.4r
0
The area function changes more rapidly for r > 1 because
it is quadratic and the arc length function is linear.
0
12
(b)
r 10
⇒
A 1 210 2 50
s r 10
Domain: 0 <
Domain: 0 <
< 2
< 2
320
A
s
0
0
2
110. If a fan of greater diameter is installed, the angular speed does not change.
111. Answers will vary. 112. Answers will vary.
113.
y
4
3
2
1
−2 2 3 4 5 6
−2
x
114.
y
2
1
−4 −3 −2 1 2 3 4
−2
−3
x
−3
−4
−6
115.
y
116.
y
5
4
4
3
3
2
1
1
−4 −3 −2 1 2 3 4
x
−5 −4 −3
1 2 3
−2
x
−3
−3
−4
117.
y
3
2
1
−4 −3 −2 1 2 3 4
−2
−4
−5
x
118.
y
4
3
2
1
−2 1 2 3 5 6
−2
−3
−4
x
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Section 4.2 Trigonometric Functions: The Unit Circle 281
Section 4.2
Trigonometric Functions: The Unit Circle
■
■
■
■
You should know how to evaluate trigonometric functions using the unit circle.
You should know the definition of a periodic function.
You should be able to recognize even and odd trigonometric functions.
You should be able to evaluate trigonometric functions with a calculator in both radian and degree mode.
Vocabulary Check
1. unit circle 2. periodic 3. odd, even
1.
sin y 15
17
2.
x, y
12
13 , 5 13
cos x 8
17
sin y 5 13
tan y x 15 8
cos x 12
13
cot x y 8 15
tan y x 513
1213 5 12
sec 1 x 17 8
csc 1 y 17
15
csc 1 y 1
513 13 5
sec 1 x 1
1213 13
12
cot x y 1213
513 12 5
3.
sin y 5
13
4.
x, y 4 5 , 3 5
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cos x 12
13
tan y x 5 12
cot x y 12 5
sec 1 x 13
12
csc 1 y 13 5
sin y 3 5
cos x 4 5
tan y x 35
45 3 4
csc 1 y 1
35 5 3
sec 1 x 1
45 5 4
cot x y 45
35 4 3
5. t 2
corresponds to
2 , 2
4
2
.
6. t
3 ⇒ 1 2 , 3
2

282 Chapter 4 Trigonometric Functions
7. t 7 corresponds to 3
6
2 , 1 2 .
9. t 2 corresponds to 1 2 , 3
3
2 .
8. t 5
4 ⇒ 2 2 , 2 2
10. t 5 corresponds to
1
3
2 , 3 2 .
11. t 3 corresponds to 0, 1.
2
12. t ⇒ 1, 0
13. t 7 corresponds to
2
2 , 2
4
2 .
14. t 4 corresponds to 1 2 , 3
3
2 .
15. t 3 corresponds to 0, 1.
2
16. t 2 corresponds to 1, 0.
17. t corresponds to
4
sin t y 2
2
cos t x 2
2
tan t y x 1
19. t 7 corresponds to
6
sin t y 1 2
cos t x 3
2
tan t y x
1 3 3
3
21. t 2 corresponds to
3
sin t y 3
2
cos t x 1 2
tan t y x 3
2
2 , 2
2 .
3 2 , 1 2 .
1 2 , 3
2 .
18. t corresponds to
3
sin
3 y 3
2
cos
3 x 1 2
1 2 , 3
2 .
tan
3 y x 32
12
3
20. t 5 corresponds to
4
sin t y 2
2
cos t x 2
2
tan t y x 1
22. t 5 corresponds to
3
sin t y 3
2
cos t x 1 2
tan t y x 3
2 2 , 2
2 .
1 2 , 3 2 .
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Section 4.2 Trigonometric Functions: The Unit Circle 283
23. t 5 corresponds to
3
sin t y 3
2
cos t x 1 2
tan t y x 3
1 2 , 3
2 .
24. t 11 corresponds to
6
sin t y 1 2
cos t x 3
2
tan t y x 3 3
3
2 , 1 2 .
25. t corresponds to
6
sin t y 1 2
cos t x 3
2
tan t y x 3 3
27. t 7 corresponds to
4
sin t y 2
2
cos t x 2
2
tan t y x 1
3
2 , 1 2 .
2
2 , 2
2 .
26. t 3 corresponds to
4
sin 3
4 y 2 2
cos 3
4 x 2 2
tan 3
4 y x 1
28. t 4 corresponds to
3
sin t y 3
2
cos t x 1 2
tan t y x 3
2 2 , 2 2 .
1 2 , 3
2 .
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29. t 3 corresponds to 0, 1.
2
sin t y 1
cos t x 0
tan t y is undefined.
x
31. t 3 corresponds to
4
2
sin t y
2
cos t x 2
2
2 2 , 2
2 .
csc t 1 y 2
sec t 1 x 2
30. t 2 corresponds to 1, 0.
sin2 y 0
cos2 x 1
tan2 y x 0 1 0
tan t
y
x 1
cot t x y 1

284 Chapter 4 Trigonometric Functions
32. t 5 corresponds to
6
sin t y 1 2
3 2 , 1 2 .
33. t corresponds to 0, 1.
2
sin t y 1
csc t 1 y 1
cos t x 3
2
cos t x 0
sec t 1 is undefined.
x
tan t y x 1 3 3 3
tan t y is undefined. cot t x x
y 0
csc t 1 y 2
sec t 1 x 2 3 23 3
cot t x y 3
34. t 3 corresponds to 0, 1.
2
sin 3
2 y 1
cos 3
2 x 0
tan 3
2 y x 1
0 ⇒
csc 3
2 1 y 1
1 1
sec 3
2 1 x 1 0 ⇒
cot 3
2 x y 0
1 0
36. t 7 corresponds to
4
sin 7
4 y 2
2
cos 7
4 x 2
2
tan 7
4 y x 1
undefined
undefined
2
2 , 2
2 .
sec 7
4 1 x 2
cot 7
4 x y 1
35. t 2 corresponds to
3
sin t y 3
2
1 2 , 3 2 .
csc t 23
3
cos t x 1 sec 2
2
tan t y x 3
37. sin 5 sin 0
csc 7
4 1 y 2
cot 3
3
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Section 4.2 Trigonometric Functions: The Unit Circle 285
38. Because
40. Because
42. Because
7 6 :
cos 7 cos6 cos 1
9
4 2
4 :
sin 9
4 sin 2
sin
5
6 1 2
19
6 4 5
6 :
4 sin 4 2
2
sin 19
6 sin 4 5
6
39. cos 8 2
cos
3 3 1 2
41. cos 13
43. sin 9
6 cos 6 cos 11
4 sin
4 2 2
6 3
2
44. Because 8
:
3 4 4
3
cos 8
3 cos 4 4
3
cos 4
3 1 2
45.
sin t 1 3
(a) sint sin t 1 3
(b) csct csc t 3
46. cos t 3 4
47.
cost 1 5
(a)
cost cos t 3 4
(a)
cos t cost 1 5
(b) sect 1
cost 1
cos t 4 3
(b) sect 1
cost 5
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48.
50.
sint 3 8
(a)
sin t sint 3 8
(b) csc t 1
sint 1
sint 8 3
cos t 4 5
(a)
cos t cos t 4 5
49.
sin t 4 5
(a)
sin t sin t 4 5
(b) sint sin t 4 5
51. sin 7
9 0.6428
(b) cost cos t 4 5

286 Chapter 4 Trigonometric Functions
52. tan 2
5
11
11
3.0777 53. cos 0.8090 54. sin
5 9 0.6428
55. csc 1.3 1.0378 56. cot 3.7 1
tan 3.7 1.6007 57. cos1.7 0.1288
58. cos2.5 0.8011 59. csc 0.8 1
sin 0.8 1.3940 60. sec 1.8 1
cos 1.8 4.4014
61. sec 22.8
1
cos 22.8 1.4486
62. sin13.4 0.7404
63. cot 2.5 1
tan 2.5 1.3386
64. tan 1.75 5.5204 65. csc1.5
1
sin1.5 1.0025
66. tan2.25 1.2386 67. sec4.5
1
cos4.5 4.7439
68. csc5.2
1
sin5.2 1.1319
69. (a)
sin 5 1
(b) cos 2 0.4
70. (a)
sin 0.75 y 0.7
(b) cos 2.5 x 0.8
71. (a)
(b)
sin t 0.25
t 0.25 or 2.89
72. (a) sin t 0.75
t 4.0 or t 5.4
73.
cos t 0.25
t 1.82 or 4.46
(b) cos t 0.75
t 0.72 or t 5.56
I 5e 2t sin t
I0.7 5e 1.4 sin 0.7
0.79 amperes
74. At t 1.4, 75. yt 1 4 cos 6t
I 5e 21.4 sin 1.4 0.30 amperes.
(a) y0 1 4 cos 0 0.2500 ft
(b)
0.0177 ft
76.
yt 1 4 et cos 6t
(a)
(b)
y0 1 4 e0 cos0 1 4 foot
t
y
0.50 1.02 1.54 2.07 2.59
0.15 0.09 0.05 0.03 0.02
y 1 4 1 4 cos 3 2
(c) y 1 2 1 4 cos 3 0.2475 ft
(c) The maximum displacements are decreasing
because of friction, which is modeled by the
e t term.
(d)
1
4 et cos 6t 0
cos 6t 0
6t
2 , 3
2
t
12 , 4
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Section 4.2 Trigonometric Functions: The Unit Circle 287
77. False. sin
4
3 3
2 > 0
78. True
79. False. 0 corresponds to 1, 0.
81. (a) The points have y-axis symmetry.
(b) sin t 1 sin t 1 since they have the same
y-value.
(c) cos t 1 cos t 1 since the x-values have
opposite signs.
80. True
82. (a) The points x 1 , y 1 and x 2 , y 2 are symmetric
about the origin.
(b) Because of the symmetry of the points, you can
make the conjecture that sint 1 sin t 1 .
(c) Because of the symmetry of the points, you can
make the conjecture that cost 1 cos t 1 .
83.
cos 1.5 0.0707, 2 cos 0.75 1.4634
84.
sin0.25 sin0.75 0.2474 0.6816 0.9290
Thus, cos 2t 2 cos t.
sin 1 0.8415
Therefore, sin t 1 sin t 2 sint 1 t 2 .
85.
cos x cos
86.
sin y sin
sec 1 1
cos cos sec
csc 1 y csc
y
tan y x tan
(x, y)
θ
−θ
x
cot x y cot
(x, −y)
© Houghton Mifflin Company. All rights reserved.
87. ht f tgt is odd.
88. ft sin t and gt tan t
ht f tgt f tgt ht
Both f and g are odd functions.
ht ftgt sin t tan t
ht sint tant
sin ttan t
sin t tan t ht
The function ht ftgt is even.

288 Chapter 4 Trigonometric Functions
89.
f x 1 23x 2
y 1 23x 2
−9 9
x 1 23y 2
6
f
f −1
90.
fx 1 4 x3 1
y 1 4 x3 1
x 1 4 y3 1
6
f
f −1
−9 9
2x 3y 2
−6
x 1 1 4 y3
−6
2x 2 3y
4x 1 y 3
2
3x 1 y
f 1 x 2 3x 1
y 4x 3 1
f 1 x 4x 3 1
91. f x x 2 4, x ≥ 2, y ≥ 0
7
y x 2 4
x y 2 4
x 2 y 2 4
x 2 4 y 2
−1
x 2 4 y, x ≥ 0
f 1 x x 2 4, x ≥ 0
f −1
−2 10
f
92.
fx
y
x
xy x 2y
2x
x 1 , x > 1
2x
−3 9
x 1 , x > 1
2y
y 1
4
−4
x 2y xy
x y2 x
x
2 x y, x < 2
f 1 x
x
2 x , x < 2
93.
f x
2x
x 3
94.
f x
5x
x 2 x 6 5x
x 3x 2
Asymptotes: x 3, y 2
Asymptotes:
x 3, x 2, y 0
95.
8
7
6
5
4
3
1
−4 −3 −2 −1
−2
−3
−4
y
2 4 5 6 7 8
f x x2 3x 10
2x 2 8
x 5
2x 2 ,
Asymptotes: x 2, y 1 2
x
x 2
x 2x 5
2x 2x 2
y
10
8
6
4
−6
−8
−7 −6 −5 −2 −1
−1
1
−2
4
3
2
1
y
x
4
6
8 10
x
© Houghton Mifflin Company. All rights reserved.
−3
−4

Section 4.3 Right Triangle Trigonometry 289
96.
f x x3 6x 2 x 1
2x 2 5x 8
x 2 7 4
Slant asymptote: y x 2 7 4
Vertical asymptotes: x 3.608, x 1.108
15x 4
42x 2 5x 8
10
8
6
4
2
y
−8 −6 −4 −2
6 8 10
x
−8
−10
97.
y x 2 3x 4 98.
Domain: all real numbers
0 x 2 3x 4 x 4x 1 ⇒ x 1, 4
Intercepts: 0, 4, 1, 0, 4, 0
No asymptotes
y ln x 4
Domain: all x 0
ln x 4 0 ⇒ x 4 1 ⇒ x ±1
Intercepts: 1, 0, 1, 0
Asymptote: x 0
99.
fx 3 x1 2 100.
Domain: all real numbers
Intercept: 0, 5
Asymptote: y 2
x 7
fx
x 2 4x 4 x 7
x 2 2
Domain: all real numbers x 2
Intercepts:
7, 0, 0, 7 4
Asymptotes: x 2, y 0
Section 4.3
Right Triangle Trigonometry
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■ You should know the right triangle definition of trigonometric functions.
(a) sin opp
(b) cos adj
(c) tan opp
hyp
hyp
adj
(d) csc hyp
(e) sec hyp
(f) cot adj
opp
adj
opp
■ You should know the following identities.
θ
Adjacent side
(a) sin 1
(b) csc 1
(c) cos 1
csc
sin
sec
(d) sec 1
(e) tan 1
(f) cot 1
cos
cot
tan
(g) (h) cot cos
tan sin
cos
sin
(i) sin 2 cos 2 1
(j) 1 tan 2 sec 2 (k) 1 cot 2 csc 2
■ You should know that two acute angles are complementary if and cofunctions of
complementary angles are equal.
■ You should know the trigonometric function values of 30, 45, and 60, or be able to construct triangles from
which you can determine them.
and
90,
Hypotenuse
Opposite side

290 Chapter 4 Trigonometric Functions
Vocabulary Check
1. (a) iii (b) vi (c) ii (d) v (e) i (f) iv
2. hypotenuse, opposite, adjacent 3. elevation, depression
1.
5
3
2.
θ
13
b
5
θ
b 13 2 5 2 169 25 12
adj 5 2 3 2 16 4
sin opp
hyp 3 5
cos adj
hyp 4 5
tan opp
adj 3 4
csc hyp
opp 5 3
sec hyp
adj 5 4
cot adj
opp 4 3
sin
cos
tan
csc
sec
opp
adj
opp
hyp
hyp
hyp 5 13
hyp 12
13
adj 5 12
opp 13 5
adj 13
12
cot adj
opp 12
5
3.
4.
8
18 c
θ
15
θ
12
hyp 8 2 15 2 17
sin opp
hyp 8 17
cos adj
hyp 15
17
tan opp
adj 8
15
csc hyp
opp 17
8
sec hyp
adj 17
15
C 18 2 12 2 468 613
sin
cos
tan
csc
sec
18
12
18
613 313
13
613 213
13
12 3 2
13
3
13
2
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cot adj
opp 15
8
cot 2 3

Section 4.3 Right Triangle Trigonometry 291
5.
opp 10 2 8 2 6
sin opp
hyp 6 10 3 5
10
opp 2.5 2 2 2 1.5
sin opp
hyp 1.5
2.5 3 5
2.5
θ
2
cos adj
hyp 8 10 4 5
θ
8
cos adj
hyp 2
2.5 4 5
tan opp
adj 6 8 3 4
tan opp
adj 1.5
2 3 4
csc hyp
opp 10 6 5 3
csc hyp
opp 2.5
1.5 5 3
sec hyp
adj 10 8 5 4
sec hyp
adj 2.5
2 5 4
cot adj
opp 8 6 4 3
cot adj
opp 2
1.5 4 3
The function values are the same since the triangles are similar and the corresponding sides are proportional.
© Houghton Mifflin Company. All rights reserved.
6.
adj 15 2 8 2 161
sin
cos
tan
csc
sec
opp
adj
opp
hyp
hyp
hyp 8 15
hyp 161
15
adj 8
161 8161
161
opp 15 8
adj 15
161 15161
161
cot adj
opp 161
8
θ
15
8
adj 7.5 2 4 2 161
2
sin
cos
tan
csc
sec
opp
hyp 4
7.5 8 15
adj
opp
hyp
hyp
hyp 161
2 7.5 161
15
adj 4
1612 8
161 8161
161
opp 7.5
4 15 8
adj 7.5
1612 15
161 15161
161
cot adj
opp 161
2 4 161
8
The function values are the same because the triangles are similar, and corresponding sides are proportional.
7.5
θ
4

292 Chapter 4 Trigonometric Functions
7.
adj 3 2 1 2 8 22
adj 6 2 2 2 32 42
sin opp
hyp 1 3
cos adj
hyp 22
3
θ
3
1
sin opp
hyp 2 6 1 3
cos adj
hyp 42 22
6 3
θ
6
2
tan opp
adj 1
22 2
4
tan opp
adj 2
42 1
22 2
4
csc hyp
opp 3
csc hyp
opp 6 2 3
sec hyp
adj 3
22 32
4
sec hyp
adj 6
42 3
22 32
4
cot adj
opp 22
cot adj
opp 42 22
2
The function values are the same since the triangles are similar and the corresponding sides are proportional.
8.
hyp 1 2 2 2 5
sin
cos
opp
hyp
adj
hyp
1 5 5
5
2
5 25
5
θ
2
1
hyp 3 2 6 2 35
sin
cos
3
6
35
35
1 5 5
5
2
5 25
5
θ
6
3
tan
csc
sec
opp
hyp
hyp
adj 1 2
opp 5
1 5
adj 5
2
tan
csc
sec
3 6 1 2
35
3
35
6
5
5
2
cot adj
opp 2 1 2
cot 6 3 2
The function values are the same because the triangles are similar, and corresponding sides are proportional.
© Houghton Mifflin Company. All rights reserved.

Section 4.3 Right Triangle Trigonometry 293
9. Given:
sin 5 6 opp
hyp
5 2 adj 2 6 2 6
5
adj 11
cos adj
hyp 11
6
tan opp
adj 5
11 511
11
cot adj
opp 11
5
sec hyp
adj 6
11 611
11
θ
11
10.
hyp 5 2 1 2 26
sin
cos
tan
csc
opp
hyp 1
26 26
26
adj
opp
hyp
hyp 5
26 526
26
adj 1 5
opp 26 26
1
sec hyp
adj 26
5
θ
26
5
1
csc hyp
opp 6 5
11. Given:
sin 15
4
cos 1 4
sec 4 4 1 hyp
adj
opp 2 1 2 4 2 4
opp 15
15
θ
1
12.
opp 7 2 3 2 40 210
sin
tan
csc
210
7
210
3
7
210 710
20
7
θ
3
2
10
tan 15
sec
7 3
cot 1
15 15
15
csc 4
15 415
15
cot 3
210 310
20
© Houghton Mifflin Company. All rights reserved.
13. Given:
sin 310
10
cos 10
10
sec 10
cot 1 3
csc 10
3
tan 3 3 1 opp
adj
3 2 1 2 hyp 2 10
3
hyp 10
θ
1
14.
adj 17 2 4 2 273
sin
cos
tan
sec
opp
adj
opp
1
hyp 4 17
hyp 273
17
adj 4
273 4273
273
cos
cot 1 273
tan 4
17
273 17273
273
θ
17
273
4

294 Chapter 4 Trigonometric Functions
15. Given: cot 9 4 adj
opp
97
4 2 9 2 hyp 2
hyp 97
sin 4
97 497
97
cos 9
97 997
97
θ
9
4
16.
sin 3 8
adj 8 2 3 2 55
cos
tan
csc
adj
opp
1
hyp 55
8
adj 3
55 355
55
sin
8 3
θ
8
55
3
tan 4 9
sec 97
9
csc 97
4
sec
1
cos
cot 1 55
tan 3
8
55 855
55
17. sin
19. tan
21. cot
23. cos
Function (deg) (rad) Function Value
30
60
60
30
25. cot 45
1
4
6
3
3
6
1
2
3
3
3
3
2
18. cos
24. sin
26. tan
Function (deg) (rad) Function Value
45
20. sec 45 2
4
22. csc 45 2
4
45
30
4
4
6
2
2
2
2
1
3
27. sin 1
csc
28. cos 1
sec
29. tan 1
cot
30. csc 1
sin
31. sec 1
cos
32. cot 1
tan
35. sin 2 cos 2 1 36. 1 tan 2 sec 2
39. tan90 cot
43.
sin 60 3
2 , cos 60 1 2
(a)
tan 60
sin 60
cos 60 3
(c) cos 30 sin 60 3
2
40. cot90 tan
(b)
33. tan sin
cos
37. sin90 cos
41. sec90 csc
sin 30 cos 60 1 2
(d) cot 60
cos 60
sin 60
1 3 3
3
34. cot cos
sin
38. cos90 sin
42. csc90 sec
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Section 4.3 Right Triangle Trigonometry 295
44.
sin 30 1 , tan 30 3
2 3
45.
csc 3, sec 32
4
(a)
csc 30 1
sin 30 2
(a)
sin 1
csc
1 3
(b)
(c)
cot 60 tan90 60 tan 30 3
3
cos 30
(d) cot 30 1
tan 30
sin 30
tan 30 12
33 3
23 3
2
3
3 33
3
3
(b)
(c)
cos 1 22
sec 3
tan sin
13
cos 223 2
4
(d) sec90º csc 3
46.
sec 5, tan 26
(a)
cos 1
sec
1 5
(b) cot 1 1
tan 26 6
12
(c)
cot90 tan 26
(d) sin tan cos 26 1 5 26
5
47.
cos 1 4
(a)
sec 1 4
cos
(c)
cot cos
sin
(b)
sin 2 cos 2 1
sin 2
1
4 2 1
sin 2 15
16
sin 15
4
14
154
1
15
15
15
(d) sin90 cos 1 4
© Houghton Mifflin Company. All rights reserved.
48.
tan 5
(a)
(b)
(c)
cot 1
tan
1 5
lies in Quadrant I or III.
sec 2 1 tan 2 ⇒ cos
tan90 cot 1 5
1
1 tan 2
(d) csc 1 cot 2 1 1 25 26
5
49. tan cot tan 1
tan 1
1
1 25 1
26 26
26
50. csc tan 1 sin
1 sec
sin cos cos
51. tan cos
sin
cos cos sin
52. cot sin cos
sin cos
sin

296 Chapter 4 Trigonometric Functions
53.
1 cos 1 cos 1 cos 2
54.
csc cot csc cot csc 2 cot 2
sin 2 cos 2 cos 2
1
sin 2
55.
sin
cos
sin2 cos 2
cos sin sin cos
1
sin cos
1 1
csc sec
sin cos
56.
tan cot
tan
tan
cot
tan tan
1 cot
1cot
1 cot 2 csc 2
57. (a)
sin 41 0.6561
(b) cos 87 0.0523
58. (a)
tan 18.5 0.3346
(b) cot 71.5
1
tan 71.5 0.3346
59. (a)
sec 42 12 sec 42.2
(b) csc 48º 7
1
cos 42.2 1.3499
1
sin48 7 60º 1.3432
60. (a)
cos8 50 25 cos 8 50
60 25
3600
(b) sec8 50 25
cos8.840278 0.9881
1
cos8 50 25 1.0120
61. Make sure that your calculator is in radian mode.
(a) cot
16 1
tan16 5.0273
(b) tan
8 0.4142
1
62. (a) sec1.54
cos1.54 32.4765
(b) cos1.25 0.3153
63. (a)
65. (a)
67. (a)
sin 1 2 ⇒
(b) csc 2 ⇒
sec 2 ⇒
(b) cot 1 ⇒
csc 23
3
(b) sin 2
2
30
6
30
6
60
3
45
⇒
⇒
4
60
45
4
3
64. (a)
66. (a)
68. (a)
cos 2
2
(b) tan 1 ⇒
(b) cos 1 2 ⇒
(b)
⇒
tan 3 ⇒
cot 3
3
45
45
4
60
tan 3 3 3 ⇒
sec 2
cos 1 2 2
2
3
4
60
⇒
3
60
3
45
4
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Section 4.3 Right Triangle Trigonometry 297
69.
tan 30
y
105
cos 30 105
r
⇒ y 105 tan 30 105
3
3 353
⇒ r 105
cos 30 105
32 210
3 703
70.
71.
cos 30 x
15
3
⇒ x 15 cos 30 15
2 153
2
sin 30 y
15 ⇒ y 15 sin 30 15 1 2 15 2
cos 60 x
16 ⇒ x 16 cos 60 16 1 2 8
sin 60 y
16 ⇒ y 16 sin 60 16 3
2 83
72.
cot 60 x
38 ⇒ x 38 cot 60 38 1 3 383
3
sin 60 38 r
⇒ r 38
sin 60 38
32 763
3
73.
tan 45 20 ⇒ 1 20 ⇒ x 20 74.
x
x
r 2 20 2 20 2 ⇒ r 202
tan 45 y
10 ⇒ 1 y
10 ⇒ y 10
r 2 10 2 10 2 ⇒ r 102
75.
tan 45 25
x
⇒ 1 25
x
⇒ x 25
r 2 25 2 25 2 20 20 40 ⇒ r 210
© Houghton Mifflin Company. All rights reserved.
76.
77. (a)
tan 45
y
46 ⇒ 1 y
46
⇒ y 46
r 2 46 2 46 2 96 96 192 ⇒ r 83
6
(b) tan 6 and tan h Thus,
5 h 5
21 21 .
(c)
h
16
h 621
5
6
θ
5
25.2 feet
78. (a)
30
75°
x
(b) sin 75 x
30
(c) x 30 sin 75 28.98 meters

298 Chapter 4 Trigonometric Functions
79.
tan opp
adj
tan 58 w
100
80.
cot 9 c h
cot 3.5 13 c
h
3.5° 9°
13 c
h
w 100 tan 58 160 feet
not drawn to scale
Subtracting,
13
h
cot 3.5 cot 9
h
13
cot 3.5 cot 9
13
16.3499 6.3138
1.295 1.3 miles.
81. (a)
(b)
(c)
tan 50
50 1 ⇒ 4582. (a)
L 2 50 2 50 2 2 50 2 ⇒ L 502 feet
502
6
252
3
50
6 25 3 ftsec
ftsec
rate down the zip line
vertical rate
(b)
(c)
sin 35.4
1693.5 519.3 1174.2 feet above sea level
896.5
300
Vertical rate
x
896.5
x 896.5 sin 35.4 519.3 feet
minutes to reach top
519.3
896.5300
173.8 feet per minute
83.
( x1, y1)
( x2, y2)
56
30°
56
sin 30 y 1
56
1
y 1 sin 3056 56 28
2
cos 30 x 1
56
x 1 cos 3056 3 56 283
2
x 1 , y 1 283, 28
60°
sin 60º y 2
56
3
y 2 sin 6056 2 56 283
cos 60 x 2
56
1
x 2 cos 6056 56 28
2
x 2 , y 2 28, 283
© Houghton Mifflin Company. All rights reserved.

Section 4.3 Right Triangle Trigonometry 299
84.
x 9.397, y 3.420
y
sin 20
10 0.34
cos 20 x
10 0.94
85. True
sin 60 csc 60 sin 60
1
sin 60 1
tan 20 y cot 20 x x 0.36
y 2.75
sec 20 10 x 1.06
csc 20 10 y 2.92
86. False
sin 45cos 45 2
2 2
2
2 1
87. True
1 cot 2 csc 2 for all
88. No. tan 2 1 sec 2 , so you can find ± sec .
89. (a)
sin
cos
0 20 40 60 80
0 0.3420 0.6428 0.8660 0.9848
1 0.9397 0.7660 0.5000 0.1736
(b) Sine and tangent are increasing, cosine is
decreasing.
(c) In each case, tan sin
.
cos
tan
0 0.3640 0.8391 1.7321 5.6713
90.
cos
0 20 40 60 80
1 0.9397 0.7660 0.5000 0.1736
cos sin90
and 90
are complementary angles.
sin90
1 0.9397 0.7660 0.5000 0.1736
91.
7 92.
1 93.
7
94.
−3 3
−6 6
2
−6 6
−6 6
© Houghton Mifflin Company. All rights reserved.
95.
−1
−3
−1
4 96. 4 97.
4
98.
−2 10
−2 10
−2 10
−4
−4
−4
−6
4
−2 10
−4

300 Chapter 4 Trigonometric Functions
Section 4.4
Trigonometric Functions of Any Angle
■
Know the Definitions of Trigonometric Functions of Any Angle.
If is in standard position, x, y a point on the terminal side and r x 2 y 2 0, then:
sin y r
csc r y , y 0
cos x sec r r
x , x 0
tan y cot x x , x 0 y , y 0
■
■
■
■
■
■
You should know the signs of the trigonometric functions in each quadrant.
You should know the trigonometric function values of the quadrant angles 0,
You should be able to find reference angles.
You should be able to evaluate trigonometric functions of any angle. (Use reference angles.)
You should know that the period of sine and cosine is 2.
You should know which trigonometric functions are odd and even.
Even: cos x and sec x
Odd: sin x, tan x, cot x, csc x
2 , , and 3
2 .
Vocabulary Check
y
y
1. 2. csc
3. 4.
r
x
5. cos
6. cot
7. reference
r
x
1. (a)
x, y 4, 3
(b)
x, y 8, 15
r 16 9 5
r 64 225 17
sin y r 3 5
cos x r 4 5
tan y x 3 4
csc r y 5 3
sec r x 5 4
cot x y 4 3
sin y r 15 17
cos x r 8 17
tan y x 15 8
csc r y 17 15
sec r x 17 8
cot x y 8 15
© Houghton Mifflin Company. All rights reserved.

Section 4.4 Trigonometric Functions of Any Angle 301
2. (a)
x 12, y 5
(b)
x 1, y 1
r 12 2 5 2 13
r 1 2 1 2 2
sin y r 5
13 5 13
sin y r 1 2 2
2
cos x r 12
13
cos x r 1
2 2 2
tan y x 5
12 5 12
tan y x 1
1 1
csc r y 13
5 13 5
csc r y 2
1 2
sec r x 13
12
sec r x 2
1 2
cot x y 12
5 12 5
cot x y 1
1 1
3. (a)
x, y 3, 1
(b)
x, y 2, 2
r 3 1 2
r 4 4 22
sin y r 1 2
csc r y 2
sin y r 2
2
csc r y 2
cos x r 3
2
tan y x 3
3
sec r x 23
3
cot x y 3
cos x sec r r 2 2
x 2
tan y cot x x 1 y 1
4. (a)
x 3, y 1
(b)
x 2, y 4
r 3 2 1 2 10
r 2 2 4 2 25
sin y r 1
10 10
10
sin y r 4
25 25 5
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cos x r 3
10 310
10
tan y x 1 3
csc r y 10
1
sec r x 10
3
cot x y 3 1 3
10
cos x r 2
25 5
5
tan y x 4
2 2
csc r y 25
4 5 2
sec r x 25
2
5
cot x y 2
4 1 2

302 Chapter 4 Trigonometric Functions
5.
x, y 7, 24
6.
x 8, y 15,
r 49 576 25
r 8 2 15 2 17
sin y r 24
25
csc r y 25
24
sin
y r 15
17
cos
x r 8 17
cos x r 7 25
sec r x 25 7
tan
y x 15 8
csc
r y 17
15
tan y x 24 7
cot x y 7 24
r x 17 8
sec cot x y 8 15
7.
x, y 5, 12
8.
x 24, y 10
r 5 2 12 2 25 144 169 13
r 24 2 10 2 26
sin y r 12 13
sin
y r 10
26 5 13
cos x r 5 13
cos
x r 24
26 12
13
tan y x 12 5
tan
y x 10
24 5 12
csc r y 13 12
csc
r y 13 5
sec r x 13 5
sec
r x 13
12
cot x y 5 12
cot x y 12 5
9.
x, y 4, 10 10.
x 5, y 6
r 16 100 229
r 5 2 6 2 61
sin y r 529
29
sin
y r 6
61 661
61
cos x r 229 29
tan y x 5 2
csc r y 29
5
sec r x 29 2
cot x y 2 5
cos
tan
csc
sec
x r 5
y x 6
r y 61
r x 61
cot x y 5 6
61 561
61
5 6 5
6
5
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Section 4.4 Trigonometric Functions of Any Angle 303
11.
x, y 10, 8 12.
r 10 2 8 2 164 241
sin y r 8
241 441
41
cos x r 10
241 541 41
tan y x 8
10 4 5
csc r y 41
4
sec r x 41 5
x 3, y 9
r 3 2 9 2 90 310
sin
cos
tan
csc
sec
y r 9
x r 3
y x 9
r y 10
3
r x 10
310 310
10
310 10
10
3 3
cot x y 5 4
cot x y 1 3
13. sin < 0 ⇒ lies in Quadrant III or Quadrant IV.
cos < 0 ⇒ lies in Quadrant II or Quadrant III.
sin < 0
and
cos < 0 ⇒
lies in Quadrant III.
14. sec > 0 and cot < 0
r x
and
x > 0 y < 0
Quadrant IV
15. cot > 0 ⇒
cos > 0 ⇒
lies in Quadrant I or Quadrant III.
lies in Quadrant I or Quadrant IV.
cot > 0 and cos > 0 ⇒ lies in Quadrant I.
16. tan > 0 and csc < 0
y r
and
x > 0 y < 0
Quadrant III
17.
sin y r 3 5 ⇒ x2 25 9 16
18.
cos x r 4
5
⇒ y 3
in Quadrant II
⇒ x 4
in Quadrant III
⇒ y 3
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sin y r 3 5
cos x r 4 5
tan y x 3 4
csc r y 5 3
sec r x 5 4
cot x y 4 3
sin
cos
y r 3 5
x r 4 5
y x 3 4
csc
sec
5 3
5 4
tan cot 4 3

304 Chapter 4 Trigonometric Functions
19.
sin < 0 ⇒ y < 0 20. csc
tan y x 15
8
sin y r 15 17
cos x r 8 17
⇒ r 17
csc r y 17 15
sec r x 17 8
cot
sin
cos
r y 4 1 ⇒ x ±15
< 0 ⇒ x 15
y r 1 4
x r 15
4
csc
sec
4
415
15
cot x y 8 15
y x 15
tan cot 15
15
21.
sec r x 2
1 ⇒ y2 4 1 3
sin ≥ 0 ⇒ y 3
sin y r 3
2
csc r y 23
3
cos x sec r r 1 2
x 2
tan y x 3
cot x y 3 3
22. sin 0 and
2 ≤
cos cos 1
tan sin
0
cos
csc is undefined.
sec 1
cot is undefined.
≤ 3
2 ⇒
23. cot is undefined ⇒
2 ≤
≤ 3
2 ⇒
n.
, y 0, x r
sin y csc r is undefined.
r 0
y
cos x sec r r r
r 1 x 1
tan y x 0 x 0
cot is undefined.
24. tan is undefined and
sin 1
cos 0
tan is undefined.
csc 1
sec is undefined.
cot 0
25. To find a point on the terminal side of , use any point on the line y x that lies in
Quadrant II. 1, 1 is one such point.
x 1, y 1, r 2
sin 1 2 2
2
cos 1 2 2 2
tan 1
csc 2
sec 2
cot 1
≤
≤ 2 ⇒
3
2 .
© Houghton Mifflin Company. All rights reserved.

Section 4.4 Trigonometric Functions of Any Angle 305
x, 1 3 x
26. Quadrant III, x > 0
r x 2 1 9 x 2 10x
3
sin
cos
tan
csc
sec
y r x3
x r x
10x3 10 10
y x 13x
10x3 310 10
x
r y 10x3
r x 10x3
13x 10
x
1 3
cot x y x
13 x 3
10
3
27. To find a point on the terminal side of , use any
point on the line y 2x that lies in Quadrant III.
1, 2 is one such point.
x 1, y 2, r 5
sin 2 5 25 5
cos 1 5 5 5
tan 2
1 2
csc 5
2 5 2
sec 5
1 5
cot 1
2 1 2
28.
4x 3y 0 ⇒ y 4 3 x
x, 4 3 x
Quadrant IV, x > 0
r x 2 16 9 x2 5 3 x
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sin
cos
y r 43x
x r x
53 x 4 5
53x 3 5
y x 43 x
csc
sec
5 4
5 3
tan 4 cot 3 x 3
4
29. x, y 1, 0 30. tan
undefined 31. x, y 0, 1
2 y x 1 0 ⇒
sec r x 1
1 1
cot 3 0
2 x since corresponds to 0, 1.
y 1 0
2
35.
x, y 1, 0
cot x y 1 0 ⇒
undefined
32. csc 0 r undefined 33. x, y 1, 0
y 1 3
⇒ 34. csc
0 2 1
sec 0 r x 1 sin 3 1
1 1
1 1
2
36. csc
2 1
sin
2
1 1 1

306 Chapter 4 Trigonometric Functions
37.
120
180 120 60
y
38.
225 180 45
y
θ ′ = 60
θ =120
θ = 225
x
x
θ ′ = 45
135
39. is coterminal with 225.
225 180 45
y
40.
330 360 30
y
θ ′ = 55
θ = −330
θ ′= 30
x
x
θ = −235
41.
5
3 , 2 5
3 3
42.
3
4 4
y
y
θ =
5π
3
π
θ ′ = 3
x
θ ′ =
4
π
θ =
3 π 4
x
5
7
43. is coterminal with
6
6 .
44.
2
3
3
45.
7
6
6
208 180 28
y
x
π
θ ′ =
6
θ = −
5π
6
20846.
y
θ = 208°
θ ′ = 28°
x
θ ′ =
3
π
322
y
θ =
−
2π
3
360 322 38
x
θ = 322°
y
θ ′ = 38°
x
© Houghton Mifflin Company. All rights reserved.

Section 4.4 Trigonometric Functions of Any Angle 307
47.
29248. 165 lies in
360 292 68
θ = −292°
y
θ ′ = 68°
x
Quadrant III.
Reference angle:
180 165 15
θ ′ = 15
y
x
θ = −165
11
49. is coterminal with
5
5 .
50.
17
7 14
7 3
7
y
5
y
θ ′ =
3π
7
θ ′ =
π
5
x
θ =
17π
7
x
θ =
11π
5
1.8
51. lies in Quadrant III.
Reference angle:
y
1.8 1.342
52. lies in Quadrant IV.
Reference angle: 4.5 1.358
y
θ ′ = 1.342
θ = −1.8
x
θ′= 1.358
θ = 4.5
x
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45,
53. Quadrant III
sin 225 sin 45 2
2
cos 225 cos 45 2
2
tan 225 tan 45 1
750
55. is coterminal with 330, Quadrant IV.
360 330 30
sin750 sin 30 1 2
cos750 cos 30 3
2
tan750 tan 30 3
3
300, 360 300 60,
54. Quadrant IV
sin
300 sin 60 3
2
cos 300 cos 60 1 2
tan 300 tan 60 3
495, 45,
56. Quadrant III
sin495 sin 45 2
2
cos495 cos 45 2
2
tan495 tan 45 1

308 Chapter 4 Trigonometric Functions
5
57. Quadrant IV
3 , 58. 4 , 3
4 4
3
sin
3
4 2
2
2 5
sin
5
cos
5
tan
5
3 sin
3 cos
3 tan
3 3 2
3 1 2
3 3
3
cos
3
3
4 2 2
tan 3
4 1
sin 11
4 sin
cos 11
4 cos
tan 11
4 tan
4 2
2
4 2 2
4 1
4
59. Quadrant IV
6 ,
60.
3
,
3
61. Quadrant II
4 , 62.
3
is coterminal with
3 .
sin
6 sin 6 1 sin
2
4
3 3
2
cos
6 cos 6 3
cos
2
4
3 1
2
tan
6 tan 6 3
4
tan
3
3 3
17
7
63. is coterminal with
6
6 .
Quadrant III
6
6 ,
7
sin 17
cos 17
tan 17
6 sin
6 cos
6 tan
6 1 2
6 3 2
6 3
3
64.
10
4
sin
cos
10
in Quadrant III.
3
3
3 sin 3 3 2
10
3 cos 3 1 2
tan 10
3 tan
20
3
3 3
,
3
sin 20
3 3
2
cos 20
3 1
2
tan
20
3 3
4
© Houghton Mifflin Company. All rights reserved.

Section 4.4 Trigonometric Functions of Any Angle 309
65.
sin 3 5
66.
cot 3
sin 2 cos 2 1
cos 2 1 sin 2
cos 2 1 3 5 2
1 cot 2 csc 2
1 3 2 csc 2
10 csc 2
csc > 0 in Quadrant II.
10 csc
cos > 0
cos 2 1 9 25
cos 2 16
25
in Quadrant IV.
csc 1
sin
sin 1 1
csc 10 10
10
cos 4 5
67.
csc 2
68.
cos 5 8
1 cot 2 csc 2
cot 2 csc 2 1
cot 2 2 2 1
cot 2 3
cos 1 ⇒ sec 1
sec
cos
sec 1
58 8 5
cot < 0 in Quadrant IV.
cot 3
69.
sec 9 4
70.
tan 5 4 ⇒ cot 4 5
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1 tan 2 sec 2
tan 2 sec 2 1
tan 2 9 4 2 1
tan 2 65
16
tan > 0 in Quadrant III.
tan 65
4
1 cot 2 csc 2
1 16
25 csc2
csc ± 41
5
Quadrant IV ⇒ csc 41
5

310 Chapter 4 Trigonometric Functions
71. sin 2 is in Quadrant II.
5 and cos < 0 ⇒
cos 1 sin 2 1 4
25 21 5
tan sin
25
cs 215 2
21 221
21
csc 1
sin
5 2
sec 1 5
cos 21 521
21
cot 1 21
tan 2
72. cos 3 is in Quadrant III.
7 and sin < 0 ⇒
sin 1 cos 2 1 9 49 40
7
tan sin
2107 210
cs 37 3
cot 1 3
tan 210 310
20
csc 1
7
sin 210 710
20
sec 1
cos
7 3
210
7
73. tan 4 and cos < 0 ⇒ is in Quadrant II.
sec 1 tan 2 1 16 17
cos 1 1
sec 17 17
17
sin tan cos 4 17
17 417
17
csc 1 17
sin 417 17
4
cot 1
tan
1 4
74. cot 5 and sin > 0 ⇒ is in Quadrant II.
tan 1 1
cot 5
sec 1 tan 2 1 1 25 26
5
cos 1 5
sec 26 526
26
sin tan cos 1 5 526
26 26
26
csc 1 26
sin 26 26
© Houghton Mifflin Company. All rights reserved.

Section 4.4 Trigonometric Functions of Any Angle 311
75. csc 3 is in Quadrant IV.
2 and tan < 0 ⇒
sin 1
csc
2 3
cos 1 sin 2 1 4 9 5
3
sec 1
cos
tan sin
23
cos 53 2
5 25
5
cot 1 5
tan 2
3
5 35
5
76. sec 4 is in Quadrant III.
3 and cot > 0 ⇒
cos 1
sec
3 4
sin 1 cos 2 1 9
16 7 4
csc 1 4 sin 7 47
7
tan sin
74
cos 34
7
3
cot 1
tan
3 7 37
7
1
77. sin 10 0.1736 78. sec 235
cos 235 1.7434 79. tan 245 2.1445
1
80. csc 320
81. cos110 0.3420
sin 320 1.5557 82.
cot220
1
tan220
1.1918
83.
sec280
1
cos280
84. csc 0.33 1
sin 0.33 3.0860
85. tan
2
9 0.8391
5.7588
86. tan 11
9 0.8391 87. csc 8
88. cos 15
9 1
14 0.9749
sin 8
9
2.9238
© Houghton Mifflin Company. All rights reserved.
89. (a) sin 1 reference angle is 30 or
2 ⇒ 90. (a) cos ⇒ reference angle is 45º or
2
and is in Quadrant I or Quadrant II.
6
Values in degrees: 30, 150
Values in radian:
(b) sin 1 reference angle is 30 or
2 ⇒
6 , 5
6
and is in Quadrant III or Quadrant IV.
6
Values in degrees: 210, 330
Values in radians: 7
6 , 11
6
and is in Quadrant I or IV.
4
Values in degrees: 45, 315
Values in radians:
(b) cos ⇒ reference angle is 45 or
2
2
2
4 , 7
4
and is in Quadrant II or III.
4
Values in degrees: 135, 225
Values in radians: 3
4 , 5
4

312 Chapter 4 Trigonometric Functions
91. (a) csc 23 ⇒ reference angle is 60 or
3
and
3
is in Quadrant I or Quadrant II.
Values in degrees:
Values in radians:
60, 120
3 , 2
3
(b) cot 1 ⇒ reference angle is 45 or
and is in Quadrant II or Quadrant IV.
4
Values in degrees: 135, 315
Values in radians: 3
4 , 7
4
92. (a) csc
Reference angle is 45 or
4 .
Values in degrees: 225, 315
(b) csc
2 ⇒
Values in radians:
2 ⇒
Reference angle is or 30.
6
Values in degrees: 30, 150
Values in radians:
5
4 , 7
4
sin 1 2
sin 1
2
6 , 5
6
93. (a) sec 23 reference angle is or
3
6
30, and is in Quadrant II or Quadrant III.
Values in degrees: 150, 210
5
Values in radians:
6 , 7
6
(b) cos 1 reference angle is or 60,
2 ⇒ 3
and is in Quadrant II or Quadrant III.
Values in degrees: 120, 240
Values in radians: 2
3 , 4
3
⇒ 94. (a)
cot 3 ⇒ cos
3
sin
Reference angle is or 30.
6
Values in degrees: 150, 330
Values in radians:
6 , 11
6
(b) Values in degrees: 45 or 315
5
Values in radians:
4
or
7
4
95. (a)
(b)
96. (a)
f g sin 30 cos 30 1 2 3
2 1 3
2
cos 30 sin 30 3 1
2
(c) cos 30 2
3
2 2 3 4
(b)
f g sin 60 cos 60 3
2 1 2 3 1
2
cos 60 sin 60 1 2 3
2 1 3
2
(c) cos 60 2
1
2 2 1 4
(d)
(e)
sin 30 cos 30
1
2 3
2 3
4
2 sin 30 2
1
2 1
(f) cos30 cos 30 3
2
(d)
(e)
sin 60 cos 60
3
2 1 2 3
4
2 sin 60 2 3
2 3
(f) cos60 cos 60 1 2
© Houghton Mifflin Company. All rights reserved.

97. (a)
(b)
98. (a)
f g sin 315 cos 315 2
2 2
2 0
cos 315 sin 315 2
2 2
2 2
(c) cos 315 2
2
2 2 1 2
(b)
(c)
99. (a)
f g sin 225 cos 225 2
2 2
cos 225 sin 225 2
2
cos 225 2
2
2 2 1 2
2
2 0
(d) sin 225 cos 225
2
2 2
2 1 2
(b)
(c)
f g sin 150 cos 150 1 2 3
2
cos 150 sin 150 3
2
cos 150 2
3
2 2 3 4
(d) sin 150 cos 150 1 2 3
2
1 2
1 3
2
3
4
Section 4.4 Trigonometric Functions of Any Angle 313
2 2 (d)
(e)
1 3
2
sin 315 cos 315
2
2 2
2 1 2
2 sin 315 2
2
2 2
(f) cos315 cos315 2
2
(e)
2 sin 225 2
2
2 2
(f) cos225 cos225 2
2
(e)
2 sin 150 2
1
2 1
(f ) cos150 cos150 3
2
© Houghton Mifflin Company. All rights reserved.
100. (a)
(b)
(c)
101. (a)
f g sin 300 cos 300 3
2
cos 300 sin 300 1 2 3
2 1 3
2
cos 300 2
1
2 2 1 4
(d) sin 300 cos 300
3
2 1 2 3
4
(b)
(c)
1 2 1 3
2
f g sin 7 7
cos
6 6 1 2 3 1 3
2 2
cos 7 7
sin
6 6 3
2 1 2 1 3
2
7
cos 6 2
3
2 2 3 4
(d) sin 7 7
cos
6 6 1 2 3 2 3
4
(e)
2 sin 300 2
3
2 3
(f) cos300 cos300 1 2
(e)
2 sin 7
6 2 1 2 1
(f) cos
7
6 cos 7
6 3
2

314 Chapter 4 Trigonometric Functions
102. (a)
(b)
(c)
f g sin 5 5
cos
6 6 1 2 3 1 3
2 2
cos 5 5
sin
6 6 3 1 1 3
2 2 2
5
cos 6 2
3
2 2 3 4
(d) sin 5 5
cos
6 6 1 2 3
2 3
4
(e)
2 sin 5
6 2 1 2 1
(f) cos
5
6 cos 5
6 3
2
103. (a)
(b)
(c)
f g sin 4 4
cos
3 3 3 1 1 3
2 2 2
cos 4 4
sin
3 3 1 2 3
2 3 1
2
4
cos 3 2 1 2 2 1 4
(d) sin 4 4
cos
3 3 3
2 1 2 3
4
(e)
2 sin 4
3 2 3
2 3
(f) cos
4
3 cos 4
3 1 2
104. (a)
(b)
(c)
f g sin 5 5
cos
3 3 3 1 2 2 1 3
2
cos 5 5
sin
3 3 1 2 3
2 1 3
2
5
cos 3 2
1
2 2 1 4
(d) sin 5 5
cos
3 3 3
2 1 2 3
4
(e)
2 sin 5
3 2 3
2 3
(f) cos
5
3 cos 5
3 1 2
105. (a)
(b)
106. (a)
f g sin 270 cos 270 1 0 1(d)
cos 270 sin 270 0 1 1
(c) cos 270 2 0 2 0
(b)
107. (a)
(c) cos 180 2 1 2 1
(b)
f g sin 180 cos 180 0 1 1(d)
cos 180 sin 180 1 0 1
f g sin 7
2
cos 7 7
sin
2 2 0 1 1
(c) cos 7
2 2 0 2 0
(e)
sin 270 cos 270 10 0
2 sin 270 21 2
(f) cos270 cos270 0
(e)
7
cos 1 0 1 (d)
2
sin 180 cos 180 01 0
2 sin 180 20 0
(f) cos180 cos180 1
(e)
sin 7 7
cos
2 2 10 0
2 sin 7 21 2
2
(f) cos
7
2 cos 7
2 0
© Houghton Mifflin Company. All rights reserved.

Section 4.4 Trigonometric Functions of Any Angle 315
108. (a)
(b)
f g sin 5 5
cos
2 2 1 0 1 (d)
cos 5 5
sin
2 2 0 1 1
(c) cos 5
2 2 0 2 0
(e)
sin 5 5
cos
2 2 10 0
2 sin 5
2 21 2
(f) cos
5
2 cos 5
2 0
109.
110.
T 49.5 20.5 cos
t
6 7
6
(a) January:
(b) July:
t 1 ⇒ T 49.5 20.5 cos
1
7
6 6 29
t 7 ⇒ T 70
(c) December: t 12 ⇒ T 31.75
S 23.1 0.442t 4.3 sin
t
6 , t 1 ↔ Jan. 2004
(a) S1 23.1 0.4421 4.3 sin thousand
6 25.7
(b) S14 33.0 thousand
(c) S5 27.5 thousand
(d) S6 25.8 thousand
Answers will vary.
© Houghton Mifflin Company. All rights reserved.
111. sin
(a)
(b)
(c)
6 d ⇒ d 6
30
d 6
sin 30 6
12 12
90
d 6
sin 90 6 1 6
120
d
sin
6
sin 120 6.9
miles
miles
miles
113. True. The reference angle for is
and sine is positive
in Quadrants I and II.
180 151 29,
151
112. As increases from 0 to 90, x decreases from
12 cm to 0 cm and y increases from 0 cm to
12 cm. Therefore, sin y12 increases from 0
to 1, and cos x12 decreases from 1 to 0.
Thus, tan yx begins at 0 and increases
without bound. When the tangent
is undefined.
90,
114. False. cot
3
and
4 1 1
cot
4 1

316 Chapter 4 Trigonometric Functions
115. (a)
sin
sin180
0 20 40 60 80
0 0.3420 0.6428 0.8660 0.9848
0 0.3420 0.6428 0.8660 0.9848
(b) It appears that sin sin180 .
116.
Function
Domain
Range
Evenness No Yes No
Oddness Yes No Yes
Period
sin x cos x tan x
, ,
1, 1 1, 1
2 2
n
Zeros n
2 n
All reals except
2 n
,
Function csc x
sec x
cot x
Domain All reals except All reals except All reals except
2 n
Range , 1 1, , 1 1, ,
Evenness No Yes No
Oddness Yes No Yes
Period
2
n
Zeros None None
Patterns and conclusions may vary.
2
2 n
n
117.
119.
3x 7 14
3x 21
x 7
x 2 2x 5 0
x
2 ±4 20
2
1 ±6
x 3.449, x 1.449
118.
120.
44 9x 61
x
9x 17
x 17
9 1.889
2x 2 x 4 0
1 ±1 442
4
x 1.186, 1.686
1 ±33
4
© Houghton Mifflin Company. All rights reserved.

Section 4.5 Graphs of Sine and Cosine Functions 317
121.
3
x 1 x 2
9
122.
5
x x 4
2x
27 x 1x 2
10x x 2 4x
x 2 x 29 0
x 2 6x 0
x
1 ±1 429
2
x 5.908, 4.908
1 ±117
2
xx 6 0
x 6
( x 0 extraneous)
123.
4 3x 726
3 x log 4 726
x 3 log 4 726 3
ln 726
ln 4 1.752
124.
4500
4 e2x 50
90 4 e 2x
86 e 2x
2x ln 86
x 1 ln 86 2.227
2
125.
ln x 6
x e 6 0.002479 0.002
126. ln x 10 1 2 lnx 10 1 ⇒ lnx 10 2 ⇒ x 10 e 2 ⇒ x e 2 10 2.611
Section 4.5
Graphs of Sine and Cosine Functions
© Houghton Mifflin Company. All rights reserved.
■ You should be able to graph y a sinbx c and y a cosbx c.
■ Amplitude:
■
Period:
2
b
a
■ Shift: Solve bx c 0 and bx c 2.
1
■ Key increments: (period)
4
Vocabulary Check
1. amplitude 2. one cycle 3. 4. phase shift
b
2

318 Chapter 4 Trigonometric Functions
1.
2.
fx sin x
(a)
(b)
x-intercepts:
2, 0, , 0, 0, 0, , 0, 2, 0
y-intercept:
0, 0
(c) Increasing on: 2, 3
Decreasing on: 3
2 ,
(d) Relative maxima: 3
(a)
(b)
Relative minima:
fx cos x
x-intercepts:
3
y-intercept:
0, 1
2 ,
2 ,
2 , 1 ,
2 , 1
2 , 1 , 3
2 ,
2 , 3
2
2 , 1
(c) Increasing on: , 0, , 2
Decreasing on: 2, , 0,
(d) Relative maxima: 2, 1, 0, 1, 2, 1
Relative minima: , 1, , 1
2 , 3
2 , 2
2 , 0 , 2 , 0 , 2 , 0 , 3
2 , 0
3.
y 3 sin 2x
Period:
2
2
Amplitude: 3 3
Xmin = -2
Xmax = 2
Xscl = 2
Ymin = -4
Ymax = 4
Yscl = 1
4.
y 2 cos 3x
Period:
2
b 2
3
Amplitude: a 2
Xmin = -
Xmax =
Xscl = 4
Ymin = -3
Ymax = 3
Yscl = 1
5.
7.
y 5 2 cos x 2
Period:
Amplitude:
5 2 5 2
y 2 3 sin x
Period:
2
12 4
2
2
Xmin = -4
Xmax = 4
Xscl =
Ymin = -3
Ymax = 3
Yscl = 1
6.
8.
y 3 sin x 3
Period:
2
b 2
13 6
Amplitude: a 3 3
y 3 2 cos
Period:
2
x
2
b 2
2 4
6
6
Xmin = -
Xmax =
Xscl =
Ymin = -4
Ymax = 4
Yscl = 1
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Amplitude:
2 3 2 3
Amplitude: a 3 2

Section 4.5 Graphs of Sine and Cosine Functions 319
9.
y 2 sin x 10.
Period:
2
1 2
Amplitude: 2 2
y cos 2x
5
2
Period:
b 2
25 5
Amplitude: a 1 1
11.
y 1 4
Period:
cos
2x
3
2
23 3
Amplitude:
1 4 1 4
12.
y 5 2 cos x 4
13.
y 1 sin 4x
3
14.
y 3 4 cos
x
12
Period:
2
b 2
14 8
Period:
2
4
1 2
Period: 2
b 2
12 24
Amplitude: a 5 2
Amplitude:
1 3 1 3
Amplitude: a 3 4
15.
f x sin x
gx sinx
The graph of g is a horizontal shift to the right
units of the graph of f (a phase shift).
16.
fx cos x, gx cosx
g is a horizontal shift of f
units to the left.
17.
f x cos 2x
gx cos 2x
The graph of g is a reflection in the x-axis of the
graph of f.
18. fx sin 3x, gx sin3x
g is a reflection of f about the y-axis.
(or, about the x-axis)
19.
fx cos x
gx 5 cos x
The graph of g has five times the amplitude of f,
and reflected in the x-axis.
20. fx sin x, gx 1 2 sin x
The amplitude of g is one-half that of f. g is a
reflection of f in the x-axis.
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21. f x sin 2x
gx 5 sin 2x
The graph of g is a vertical shift upward of five
units of the graph of f.
23. The graph of g has twice the amplitude as the graph
of f. The period is the same.
25. The graph of g is a horizontal shift units to the
right of the graph of f.
22.
fx cos 4x, gx 6 cos 4x
g is a vertical shift of f six units downward.
24. The period of g is one-half the period of f.
26. Shift the graph of f two units upward to obtain the
graph of g.

320 Chapter 4 Trigonometric Functions
27.
f x sin x
28.
fx sin x
2
Period:
Amplitude: 1
gx sin x 3
gx 4 sin x
2
Period:
Amplitude:
4 4
3
2
x 0
2 2
sin x 0 1 0 1 0
4
y
g
sin x 3
1 3
0 1
2 2
3
2
3π
−
2
1
−2
−3
f
π
2
x
2
y
g
f
−4
6π
x
–2
29.
fx cos x
30.
fx 2 cos 2x
2
Period:
Amplitude: 1
gx 4 cos x is a vertical shift of the graph of
fx four units upward.
y
gx cos 4x
x 0
4 2 4
2 cos 2x 2 0 2 0 2
3
6
4
3
g
cos 4x
y
1
1 1 1 1
31.
− 2π
Period:
2
−1
−2
fx 1 2 sin x 2
4
Amplitude:
f
1
2
2π
gx 3 1 is the graph of fx
2 sin x 2
shifted vertically three units upward.
x
4
3
2
1
−4π
−π
−1
−2
−3
−4
y
g
f
2π
3π 4π
2
–2
x
f
g
π
x
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Section 4.5 Graphs of Sine and Cosine Functions 321
32.
fx 4 sin x
33.
fx 2 cos x
gx 4 sin x 2
1 3
x 0 1 2
2 2
f x 0 4 0 4 0
Period:
Amplitude: 2
gx 2 cosx is the graph of fx shifted
units to the left.
2
y
gx
2
2 2 6 2
3
f
y
4
3
2
−2 −1 1
2
−1
−2
1
f
g
x
–3
g
π
2π
x
34.
fx cos x
gx cos x
x 0
2
2
3
2
2
35.
f x sin x, gx cos x
sin x cos x
Period:
2
Amplitude: 1
2
2
cos x
cos x
2
1
0 1 0
0 1 0 1 0
1
−2
2
f = g
2
y
−2
f
1
g
− 2π
2π
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−π
−1
36. fx sin x, gx cos x
2
x 0
2 2
sin x
0 1 0 1 0
cos x
2
π
x
3
0 1 0 1 0
2
2
f = g
−2
2
−2
Conjecture: sin x cos x
2

322 Chapter 4 Trigonometric Functions
37.
fx cos x
38.
fx cos x, gx cosx
gx sin x
Thus, f x gx.
2
f = g
2 sin
2 x cos x
x 0
2 2
cos x 1 0 1 0 1
cosx
3
1 0 1 0 1
2
−2
2
2
−2
−2
f = g
2
−2
Conjecture: cos x cosx
39.
y 3 sin x
40.
y 1 4 cos x
2
Period:
Amplitude: 3
Key points:
0, 0,
, 0,
2 , 3 ,
3
2 , 3 ,
2, 0
Period:
2
Amplitude:
1
4
y
y
4
3
1
2
1
3π
π π 3π
− −
2 2 2 2
x
0.5
π
−
2
−0.5
π
2
π
2π
x
−4
−1
41.
y cos x 2
42.
y sin 4x
4
Period:
Amplitude: 1
Key points: 0, 1, , 0, 2, 1, 3, 0, 4, 1
3
2
−1
−2
−3
y
2π
4π
x
Period:
2
4 2
Amplitude: 1
y
2
1
3π
π 3π
5π
−
8
8 8 8
−2
x
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Section 4.5 Graphs of Sine and Cosine Functions 323
43.
y sin x
2
Period:
Amplitude: 1
Shift: Set x and x
4 0 4 2
Key points:
4
; a 1, b 1, c
4
x
4
x 9
4
4 , 0 , 3
4 , 1 , 5
4 , 0 , 7
4 , 1 , 9
4 , 0
−π
y
3
2
1
–2
–3
π
x
44.
y sinx
Horizontal shift
units to the right
2
1
y
π
−
2
−1
π
2
3π
2
x
−2
45.
y 8 cosx
Period:
2
10
8
y
Amplitude: 8
Key points: , 8,
2 , 0 , 0, 8,
2 , 0 , , 8
2
− 2π
−π
−2
−4
−6
−8
−10
π 2π
x
46.
y 3 cos x
47.
2
y 2 sin 2x
3
Amplitude: 3
Amplitude: 2
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Horizontal shift units to the left
2
Note: 3 cos x
4
3
1
π
−
2
−2
−3
−4
y
π
2
3π
2
2
x
3 sin x
Period:
23 3
−6
2
4
−4
6

324 Chapter 4 Trigonometric Functions
48.
y 10 cos
x
6
49.
y 4 5 cos
t
12
50.
y 2 2 sin 2x
3
Amplitude: 10
2
Period:
6 12
Amplitude: 5
2
Period:
12 24
Amplitude: 2
2
Period:
23 3
12
20
6
−18
18
−30
30
−6
6
−12
−20
−2
51.
y 2 3 cos x 2
52.
4
Amplitude:
2
2
3
Period:
12 4
y 3 cos6x 53.
Amplitude: 3
Period:
2
6 3
6
y 2 sin4x
Amplitude: 2
Period:
2
4
2
π
−
2
π
2
−
− 4π 5π
−6
−4
−2
54. 2
y 4 sin 55. y cos 2x
3 x
2 1
56.
3
Amplitude: 4
Amplitude: 1
Period:
3
Period: 1
3
8
x
y 3 cos 2 2 3
Amplitude: 3
Period: 4
1
−6
6
−12
12
−3
3
57.
−1
−8
y 5 sin 2x 1059.
58. y 5 cos 2x 6
Amplitude: 5
Amplitude: 5
Period:
Period:
20
12
−
−
−4
0
−7
y 1
100 sin 120t
Amplitude:
Period:
−
180
1
60
0.02
−0.02
1
100
180
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Section 4.5 Graphs of Sine and Cosine Functions 325
60.
y 1
100 cos50 t 61.
Amplitude:
Period:
−0.06
1
25
0.04
−0.04
1
100
0.06
fx a cos x d
1
Amplitude: 8 0 4
2
Since f x is the graph of
gx 4 cos x reflected about the
x-axis and shifted vertically four
units upward, we have a 4
and d 4. Thus,
fx 4 cos x 4
4 4 cos x.
62.
fx a cos x d
2 4
Amplitude:
1
2
Reflected in the x-axis:
a 1
4 1 cos 0 d
d 3
y 3 cos x
63. f x a cos x d 64. y a cos x d
65.
1
Amplitude: 7 5 6
2
Graph of f is the graph of
gx 6 cos x reflected about the
x-axis and shifted vertically one
unit upward. Thus,
f x 6 cos x 1.
Amplitude: 1 2
Period: 2
Reflected in x-axis,
a 1 2
d 4
y 4 1 2 cos x
f x a sinbx c
Amplitude:
Since the graph is reflected
about the x-axis, we have
a 3.
2
a 3
Period:
b ⇒ b 2
Phase shift: c 0
Thus, f x 3 sin 2x.
66.
y a sinbx c
67.
f x a sinbx c
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68.
Amplitude: 2 ⇒ a 2
Period:
2
4
b 4 ⇒ b 1 2
Phase shift:
y 2 sin
x
2
c 0
y a sinbx c 69.
Amplitude: 2 ⇒ a 2
Period: 4
2
b 4 ⇒ b 2
Phase shift:
c
b
x
y 2 sin 2
1 ⇒ c
2
2
Amplitude: a 1
Period:
Phase shift:
2 ⇒ b 1
Thus, f x sin x
y 1 sin x
y 2 1 2
bx c 0 when x
4 .
1
4 c 0 ⇒ c
4 .
In the interval 2, 2, sin x 1 2 when
x 5
6 , 6 , 7
6 , 11
6 .
−2
2
y 2
y 1
2
4
−2

326 Chapter 4 Trigonometric Functions
70.
y 1 cos x
y 2 1
y 1 y 2 when x , .
71.
v 0.85 sin
(a)
2
t
3
2
0
4
y 1
−2
2
−2
−2
y 2
(b)
Time for one cycle one period 2
3 6 sec
(c) Cycles per min 60 cycles per min
6 10
(d) The period would change.
72.
S 74.50 43.75 cos
(a)
120
t
6
73.
h 25 sin t 75 30
15
(a)
60
1
0
12
0
0
135
(b) Maximum sales: December t 12
Minimum sales: June t 6
(b) Minimum:
30 25 5 feet
Maximum: 30 25 55 feet
74.
75.
P 100 20 cos 8
3 t
130
0
60
1.5
Period:
83 3 4
1 heartbeat
34
C 30.3 21.6 sin
2t
365 10.9
2
(a) Period: 2
b 2 365 days
2365
This is to be expected: 365 days 1 year
(b) The constant 30.3 gallons is the average daily
fuel consumption.
⇒ 4 heartbeatssecond 80 heartbeatsminute
3
(c)
60
0
0
365
Consumption exceeds 40 gallonsday when
124 ≤ x ≤ 252. (Graph C together with y 40.)
(Beginning of May through part of September)
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Section 4.5 Graphs of Sine and Cosine Functions 327
76. (a) Yes, y is a function of t because for each
value of t there corresponds one and only
one value of y.
(b) The period is approximately
20.375 0.125 0.5 seconds.
(c) One model is
(d) 4
y 0.35 sin 4 t 2.
The amplitude is approximately
1
22.35 1.65 0.35 centimeters.
0
0
0.9
77. (a)
y
(c)
y
2.0
2.0
1.5
1.5
1.0
1.0
0.5
−10 10
20 30 40 50 60 70
x
−10 10
20 30 40 50 60 70
x
−0.5
−0.5
(b) y 0.506 sin0.209x 1.336 0.526
The model is a good fit.
2
(d) The period is
0.209 30.06.
(e) June 29, 2007 is day 545. Using the model,
y 0.2709 or 27.09%.
© Houghton Mifflin Company. All rights reserved.
78. (a) At 19.73 sin0.472t 1.74 70.2
(b) 80
(c)
0
0
The model somewhat fits the data.
95
0
45
12
12
The model is a good fit.
79. True. The period is 2
310 20
3 .
81. True
(d) Nantucket: 58
Athens: 70.2
The constant term (d) gives the average daily high
temperature.
(e) Period for
Nt 2
211 11
Period for At 2
0.472 13
You would expect the period to be 12 (1 year).
(f) Athens has greater variability. This is given by the
amplitude.
1
80. False. The amplitude is that of y cos x.
2
82. Answers will vary.
83. The graph passes through 0, 0 and has period . 84. The amplitude is 4 and the period 2. Since
Matches (e).
0, 4 is on the graph, matches (a).
4
85. The period is and the amplitude is 1. Since
0, 1 and , 0 are on the graph, matches (c).
86. The period is . Since 0, 1 is on the graph,
matches (d).

328 Chapter 4 Trigonometric Functions
87. (a) hx cos 2 x is even. (b) hx sin 2 x is even.
(c) hx sin x cos x is odd.
h(x) = cos 2 x
2
2
h(x) = sin 2 x
2
h(x) = sin x cos x
−
−
−
−2
−2
−2
88. (a) In Exercise 87, fx cos x is even and we saw that hx cos 2 x is even.
Therefore, for fx even and hx fx 2 , we make the conjecture that hx is even.
(b) In Exercise 87, gx sin x is odd and we saw that hx sin 2 x is even.
Therefore, for gx odd and hx gx 2 , we make the conjecture that hx is even.
(c) From part (c) of 87, we conjecture that the product of an even function and an odd
function is odd.
89. (a)
x
1
0.1
0.01
0.001
(b)
1.1
sin x
x
0.8415 0.9983 1.0 1.0
−1 1
0.8
x
sin x
x
0 0.001 0.01 0.1 1
Undef. 1.0 1.0 0.9983 0.8415
As x → 0, fx sin x approaches 1.
x
sin x
(c) As x approaches 0, approaches 1.
x
90. (a)
x 1 0.1 0.01 0.001
(b)
1 cos x
x
0.4597 0.05 0.005 0.0005
−1
0.5
1
91. (a)
(b)
−2π
x
−2π
1 cos x
x
2
−2
2
0 0.001 0.01 0.1 1
Undef. 0.0005 0.005 0.05 0.4597
2π
2π
(c) Next term for sine approximation:
Next term for cosine approximation:
−2π
2
2π
−0.5
As x → 0, 1 cos x approaches 0.
x
1 cos x
(c) As x approaches 0, approaches 0.
x
−2π
x7
7!
x6
6!
2
2π
© Houghton Mifflin Company. All rights reserved.
−2
−2
−2

Section 4.6 Graphs of Other Trigonometric Functions 329
92. (a) sin1 0.8415 (d)
93.
(b)
(e)
(c) sin
(f )
8 0.3827
cos
4 0.7071
sin 1 2 0.4794 cos1 0.5403
cos 1 2 0.8776
In all cases, the approximations are very accurate.
y
7 (2, 7)
6
5
4
3
2
1 (0, 1)
x
−4 −3 −2 −1 1 2 3 4
−1
Slope 7 1
2 0 3
94. y
(−1, 4) 4
3
2
1
−2
−3
−4
m 2 4
−4 −3 −2 −1 1 2 3 4
−1
(3, −2)
x
95. 8.5 8.5
180
487.014
3 1 6
4 3
2
96. 0.48 0.48
180
27.502
97. Answers will vary. (Make a Decision)
Section 4.6
Graphs of Other Trigonometric Functions
© Houghton Mifflin Company. All rights reserved.
■ You should be able to graph:
y a tanbx c y a cotbx c
y a secbx c y a cscbx c
■ When graphing y a secbx c or y a cscbx c you should know to first graph y a cosbx c
or y a sinbx c since
(a) The intercepts of sine and cosine are vertical asymptotes of cosecant and secant.
(b) The maximums of sine and cosine are local minimums of cosecant and secant.
(c) The minimums of sine and cosine are local maximums of cosecant and secant.
■ You should be able to graph using a damping factor.
Vocabulary Check
1. vertical 2. reciprocal 3. damping

330 Chapter 4 Trigonometric Functions
1.
2.
fx tan x
(a) x-intercepts: 2, 0, , 0, 0, 0, , 0, 2, 0
(c) Increasing on:
Never decreasing
2, 3
2 , 3
2 ,
(e) Vertical asymptotes: x 3
2 ,
2 , 2 , 3
2
2 ,
2 ,
fx cot x
(a) x-intercepts: 3
2 , 0 , 2 , 0 , 2 , 0 , 3
2 , 0
(c) Decreasing on: 2, , , 0, 0, , , 2
Never increasing
(e) Vertical asymptotes: x 2, , 0, , 2
2 ,
2 ,
3
2 , 3
2 , 2
(b) y-intercept:
0, 0
(d) No relative extrema
(b) No y-intercepts
(d) No relative extrema
3. fx sec x 4.
(a) No x-intercepts
(b) y-intercept: 0, 1
(c) Increasing on intervals:
2, 3
2 , 3
2 , , 0,
Decreasing on intervals:
,
2 ,
2 , 0 , , 3
2 , 3
2 , 2
(d) Relative minima: 2, 1, 0, 1, 2, 1,
Relative maxima: , 1, , 1
(e) Vertical asymptotes: x 3
2 ,
2 , 2 , 3
2
5. y 1 2 tan x
Period:
Two consecutive asymptotes:
0
x
y
4
1 2
0
4
1
2
2 ,
2 ,
x and x
2 2
π
−
2
4
3
2
1
y
π π
2
x
6.
fx csc x
(a) No x-intercepts
(b) No y-intercepts
(c) Increasing on intervals:
3
2 , , ,
2 , 2 , , ,
2
Decreasing on intervals:
2, 3
2 , 2 , 0 , 0, 2 , 3
2 , 2
(d) Relative minima: 3
2 , 1 , 2 , 1
Relative maxima: 2 , 1 , 3
2 , 1
(e) Vertical asymptotes: x ±, ±2
y 1 4 tan x
Period:
Asymptotes:
π
3
2
1
−1
−2
−3
y
x ±
2
π
x
3
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Section 4.6 Graphs of Other Trigonometric Functions 331
7.
y 2 tan 2x
8.
y 3 tan 4x
y
Period:
2
Period:
4
Two consecutive asymptotes:
2x
2 ⇒ x 4
2x
2 ⇒ x 4
6
y
Asymptotes:
4x ⇒ x
2
8
4x
2 ⇒ x 8
−4
−8
π
2
x
x
8
0
8
−
3π
4
−
π
4
−4
π 3π
4 4
x
y
2 0 2
−6
9. y 1 2 sec x
Graph first.
y
10.
y 1 4 sec x
y
Period:
y 1 2 cos x 1
2
One cycle: 0 to 2
−π
3
2
π
x
Period: 2
3
2
1
− 2π
−π
π 2π
x
11.
y sec x 3
y
12.
y 2 sec 4x 2
y
Period: 2
Shift graph of sec x
down three units.
−2
−1
1
−3
−4
−5
1 2
x
Period:
Asymptotes:
2
4 2
x
8 , x 8
−
π
4
6
4
2
π
4
π
2
x
© Houghton Mifflin Company. All rights reserved.
13.
y 3 csc x 2
Graph y 3 sin x first.
2
Period:
2
12 4
One cycle: 0 to 4
10
8
6
4
2
−3π
−2π
−π π
−8
−10
y
2π
3π 4π
x
14.
x
16
0
16
y 0.828 0 0.828
y csc x 3
2
Period:
13 6
Asymptotes:
x 0, x 3
2
4
−4π
x
y 1.155 1.155 1.155
3
y
−π
−1
−2
−3
π
4π
x

332 Chapter 4 Trigonometric Functions
15.
y 1 2 cot x 2
y
16.
y 3 cot x
y
Period:
12 2
Two consecutive
asymptotes:
x
2 0 ⇒ x 0
x
2 ⇒ x 2
x
− π
2
6
−4
−6
π
3
2
x
Period:
1
Asymptotes:
x 0, x 1
3
2
1
−3 −2 −1 1 2 3
−3
x
y
1
2
0 1 2
17.
y 2 tan
Period:
4 4
Two consecutive
asymptotes:
x
4 2 ⇒ x 2
x
4
x
4
2 ⇒
x 2
−6
y
6
4
2
−2 2 4 6
x
18.
y 1 2 tan x
Period: 1
Asymptotes:
x 1 2 , x 1 2
x 1 0
4
1
4
1
y 0 1 2 2
−1
y
3
2
1
−1
−2
−3
1
x
x
y
1
2
0 1
0 2
19.
21.
y
y 1 2 sec2x 20.
Period:
2
2
Asymptotes: x ±
4
y csc x
Graph y sin x first.
Period:
2
Asymptotes: Set
−
π
2
4
3
2
1
π
2
x 0 and x 2
x x
x
−2π
−π
6
4
2
y
y secx
Period:
2
Asymptotes: x ±
2
π
2π
x
−2π
y
8
6
−2
−4
−6
−8
2π
x
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Section 4.6 Graphs of Other Trigonometric Functions 333
22.
y csc2x
Period:
−π
π
−
2
2
2
5
4
3
−1
−2
−3
−4
−5
y
π
2
π
x
23.
y 2 cot x
Period:
Two consecutive
asymptotes:
x
2 0 ⇒ x 2
2
x
2 ⇒ x 3
2
x
3
4
5
4
−π
y
−2
−4
−6
π
x
y
2 0 2
24.
y 1 4 cotx
Period:
y
25.
y 2 csc 3x 2
sin3x
Period:
2
3
26.
y csc4x
y
1
sin4x
5
4
3
2
1
3π
π π 3π
− −
2 2 2 2
−2
−3
−4
−5
x
−
10
−10
10
−
2
3
−3
3
2
−
−
2
2
−10
−3
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27.
y 2 sec 4x
y
− 2
− 2
2
cos 4x
4
−4
4
−4
2
2
28.
y 1 4 sec x
−2
−2
2
−2
2
−2
1
4 cos x
2
2
29.
y 1 3 sec x
2
Period: 4
−6
−6
1
3 cos x
2
2
−2
2
2
2
6
6
−2

334 Chapter 4 Trigonometric Functions
30.
y 1 2 csc2x 31.
3
3
tan x 1
x 7
4 , 3
4 ,
4 , 5
4
−
−
−3
−3
32.
cot x 3
33.
sec x 2
The solutions appear to be:
x 7
6 , 6 , 5
6 , 11
6
(or in decimal form: 3.665, 0.524, 2.618, 5.760)
x ± 2
3 , ±4
3
34.
csc x 2
The solutions appear to be:
7
4 , 5
4 ,
4 , 3
4
(or in decimal form: 5.498, 3.927, 0.785, 2.356)
35. The graph of fx sec x has y-axis symmetry.
Thus, the function is even.
36.
fx tan x
tanx tan x
Thus, the function is odd and the graph of
is symmetric with the origin.
y tan x
37. The function
f x csc 2x 1
sin 2x
has origin symmetry. Thus, the function is odd.
38.
40.
f x cot 2x 39. y 1 sin x csc x and y 2 1
f x cot2x
Odd
cos2x
sin2x cos2x f x
sin2x
y 1 sin x sec x, y 2 tan x 41. y 1 cos x and
sin x
4
Equivalent
−2
−4
It appears that y 1 y 2 .
sin x sec x sin x
2
1
cos x sin x
cos x tan x
−3
2
−2
3
Not equivalent because is not defined at 0.
1
sin x csc x sin x 1, sin x 0
sin x
cot x cos x
sin x
y 1
y 2 cot x 1
tan x
−2
4
−4
2
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Section 4.6 Graphs of Other Trigonometric Functions 335
42.
y 1 sec 2 x 1, y 2 tan 2 x 43.
3
−
2
−1
It appears that y 1 y 2 .
1 tan 2 x sec 2 x
3
3
2
tan 2 x sec 2 x 1
fx x cos x
As x → 0,
Odd function
f
3
2 0 f x → 0.
Matches graph (d).
44.
fx x sin x 45.
gx x sin x
gx x cos x
46.
Even function
Matches graph (a) as x → 0,
fx → 0, and f x ≥ 0 for all x.
As x → 0, gx → 0.
Odd function
g2 0
Matches graph (c) as
x → 0, gx → 0.
Matches graph (b).
47.
f x sin x cos x
2 , gx 0 48.
fx gx
2
fx sin x cos x
gx 2 sin x
2
−2
3
f = g
2
The graph is the line y 0.
−2
−2
f = g
2
It appears that
That is,
sin x cos x
fx gx.
2
2 sin x.
−3
49.
f x sin 2 x, gx 1 1 cos 2x
2 50.
fx cos 2
x
2
2
f = g
f x gx
2
f = g
gx 1 2 1 cos x
−2
2
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51.
53.
fx e x cos x
Damping factor:
e x
x → , f x → 0
hx e x2 4
cos x
Damping factor:
h → 0 as x →
e x2 4
−2
−3
−3
−2
3
−3
2
2
6
3
52.
54.
It appears that
fx gx.
That is, that
x
cos 2
2 1 2 1 cos x.
fx e 2x sin x
Damping factor:
f x → 0 as x →
e 2x
gx e x 2 2
sin x
Damping factor: y e x 2 2
e x 2 2 ≤ gx ≤ e x 2 2
−3
−8
−2
2
−2
1
3
8
−2
x → ±, gx → 0
−1

336 Chapter 4 Trigonometric Functions
55. (a)
(b)
(c)
As x →
As x →
2 , fx → . 56.
2 , fx → .
As x →
2 , fx → .
(d) As x →
2 , fx → .
fx sec x
(a)
(b)
(c)
As x →
As x →
2 , fx → .
2 , fx → .
As x →
2 , fx → .
(d) As x →
2 , fx → .
57.
fx cot x 58.
(a) As x → 0 , fx → .
(b) As x → 0 , fx → .
(c) As x →
, fx → .
(d) As x →
, fx → .
fx csc x
(a) As x → 0 , fx → .
(b) As x → 0 , fx → .
(c) As x →
, fx → .
(d) As x →
, fx → .
59. As the predator population increases, the number of prey decreases. When the number
of prey is small, the number of predators decreases.
60.
Ht 54.33 20.38 cos
t
15.69 sin
6
t
6
Lt 39.36 15.70 cos
t
14.16 sin
6
t
6
61.
(a)
90
0
0
Period of cos
t
Period of sin
6 : 2
6 12
Period of Ht: 12
Period of Lt: 12
tan x 5 d
H
L
t
6 : 2
6 12
d 5
tan x 5 cot x 0
12
36
−36
(b) From the graph, it appears that the greatest difference
between high and low temperatures occurs in summer.
The smallest difference occurs in winter.
(c) The highest high and low temperatures appear to occur
around the middle of July, roughly one month after the
time when the sun is northernmost in the sky.
62.
cos x 36 d
d 36 36 sec x
cos x
−
2
72
36
2
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Section 4.6 Graphs of Other Trigonometric Functions 337
63. (a)
0.6
(b) The displacement function is not periodic, but damped.
It approaches 0 as t increases.
0
4
−0.6
1700
64. (a) 850 revmin
2
(b) The direction of the saw is reversed.
(c) L 60 2 cot , 0 < <
2
(d)
0.3 0.6 0.9 1.2 1.5
L 306.2 217.9 195.9 189.6 188.5
(e) Straight line lengths change faster.
(f) 500
0
0
2
65. True. 3
and x is a vertical
2
2 2
66. True. For x → , 2 x → 0.
asymptote for the tangent function.
67.
f x 2 sin x, gx 1 2 csc x
(a)
(b) f x > gx for
6 < x < 5 (c) As x → , 2 sin x → 0 and
6
1
since gx is the
2 csc x → ,
3
f
g
0
reciprocal of f x.
−1
68. (a)
2
(b)
y 3
4 sin x 1 3 sin 3x 1 5 sin 5x 1 7 sin 7x
−3
3
2
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69. (a)
−3
x
tan x
x
x
tan x
x
−2
2
−2
1
3
0.1
0.01
1.5574 1.0033 1.0 1.0
0.001
0 0.001 0.01 0.1 1
Undef. 1.0 1.0 1.0033 1.5574
−3
−2
(c) y 4 y 3 4
1 9 sin 9x
3
(b)
−1.1
2
0
As x → 0, fx tan x → 1.
x
(c) The ratio approaches 1 as x approaches 0.
1.1

338 Chapter 4 Trigonometric Functions
70. (a)
x
1
0.1
0.01
0.001
(b)
1.1
tan 3x
3x
0.0475
1.0311 1.0003 1.0
−1
1
−1.1
x
tan 3x
3x
0 0.001 0.01 0.1 1
Undef. 1.0 1.0003 1.0311 0.0475
As x → 0, fx
tan 3x
3x
→ 1.
(c) The ratio approaches 1 as x approaches 0.
71. Period is and graph is increasing on
2
4 ,
Matches (a).
4 .
72. Period is and is on the graph.
2 8 , 1
Matches (b).
73.
y = tan(x)
2
2x
y = x + 3
3!
+
16x 5
5!
74.
x 2 5x
y = 1 + +
4
2! 4!
4 y = sec(x)
−3
3
−2
The graphs of y 1 tan x and
are similar on the interval
2 ,
y 2 x 2x3
3! 16x5
5!
2 .
−3 3
0
The graphs of y 1 sec x and
are similar on the interval
2 ,
y 2 1 x2
2! 5x 4
4!
2 .
7 1 7 1
y 3x 14, x ≥ 14 3 , y ≥ 0 81.
82. One-to-one
75. Distributive Property 76.
77. Additive Identity Property
Multiplicative Inverse Property
78. a b 10 a b 10 79. Not one-to-one
80. Not one-to-one
Associative Property of Addition
x 3y 14, y ≥ 14 3 , x ≥ 0
y x 5 13
x 2 3y 14
x y 5 13
y 1 3x 2 14
x 3 y 5
f 1 x 1 3x 2 14, x ≥ 0
f 1 x x 3 5
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Section 4.7 Inverse Trigonometric Functions 339
Section 4.7
Inverse Trigonometric Functions
■ You should know the definitions, domains, and ranges of y arcsin x, y arccos x, and y arctan x.
Function Domain Range
y arcsin x ⇒ x sin y 1 ≤ x ≤ 1
y arccos x ⇒ x cos y 1 ≤ x ≤ 1 0 ≤ y ≤
y arctan x ⇒ x tan y < x <
2 ≤ y ≤ 2
2 < y < 2
■
You should know the inverse properties of the inverse trigonometric functions.
sinarcsin x x, 1 ≤ x ≤ 1 and arcsinsin y y,
2 ≤ y ≤ 2
cosarccos x x, 1 ≤ x ≤ 1 and arccoscos y y, 0 ≤ y ≤
tanarctan x x and arctantan y y,
2 < y <
2
■
You should be able to use the triangle technique to convert trigonometric functions or inverse trigonometric
functions into algebraic expressions.
Vocabulary Check
1. y sin 1 x, 1 ≤ x ≤ 1
2.
3. y tan 1 x, < x < ,
2 < y <
2
y arccos x, 0 ≤ y ≤
© Houghton Mifflin Company. All rights reserved.
1. (a) arcsin 1 (b) arcsin 0 0
2 6
2. (a) y arccos 1 2 ⇒ cos y 1 2 for 0 ≤ y ≤
(b) y arccos 0 ⇒ cos y 0 for 0 ≤ y ≤
(b) arccos 1 0 because cos 0 1 and 0 ≤ 1 ≤ .
⇒ y
3
⇒ y
3. (a) arcsin 1 because and
2
sin 2 1 2 ≤ 2 ≤ 2 .
4. (a) arctan 1 because and
4
tan 4 1 2 < 4 < 2 .
(b) arctan 0 0 because tan 0 0 and
2 < 0 < 2 .
2

340 Chapter 4 Trigonometric Functions
5. (a)
arctan 3
3
6
(b) arctan1
4
6. (a)
arccos 2
2 3
4
(b) arcsin 2
2 4
7. (a) y arctan3 ⇒ tan y 3 for
2 < y <
(b) y arctan 3 ⇒ tan y 3 ⇒ y
3
2 ⇒ y 3
8. (a)
y arccos 1 2 ⇒ cos y 1 2 for 0 ≤ y ≤
(b) y arcsin 2
2
(b)
y tan 1 3
3
⇒ sin y 2
2
⇒ tan y
3
3
⇒ y 2
for
2 ≤ y ≤ 2 ⇒ y 4
3
9. (a) y sin1 ⇒ sin y 3 for
2
2 2 ≤ y ≤ 2 ⇒ y 3
⇒ y
6
3
10. (a)
x 1.0 0.8 0.6 0.4 0.2
(b)
y 1.5708 0.9273 0.6435 0.4115 0.2014
x 0 0.2 0.4 0.6 0.8 1
y 0 0.2014 0.4115 0.6435 0.9273 1.5708
y
2
1
–1 1
–1
x
(c)
2
–2
−1
1
−2
11.
(d) 0, 0, symmetric to the origin
y arccos x
(a)
(c)
x 1 0.8 0.6 0.4 0.2
y 3.1416 2.4981 2.2143 1.9823 1.7722
x 0 0.2 0.4 0.6 0.8 1.0
y 1.5708 1.3694 1.1593 0.9273 0.6435 0
4
(b)
y
4
3
1
–2 –1 1 2
x
© Houghton Mifflin Company. All rights reserved.
−1
0
1
(d) Intercepts: 0,
2 ,
1, 0, no symmetry

Section 4.7 Inverse Trigonometric Functions 341
12. (a)
x 10 8 6 4 2
(b)
y 1.4711 1.4464 1.4056 1.3258 1.1071
x 0 2 4 6 8 10
y 0 1.1071 1.3258 1.4056 1.4464 1.4711
y
2
1
–10 –5 5 10
x
(c)
2
–2
−10
10
−2
(d) Horizontal asymptotes: y ±
2
13.
y arctan x ↔ tan y x
14.
y arccos x
3,
3 , 3
3 ,
6 , 1, 4
x 1 ⇒ y ,
2
x 1 cos 2 ⇒ y 2
3 1 2
y cos 6 3
6 ⇒ x 3
2 , 2
3 , cos 1
15. cos 1 0.75 0.72
16. sin 1 0.56 0.59
17. arcsin0.75 0.85
18. arccos0.7 2.35 19. arctan6 1.41
20. tan 1 5.9 1.40
21.
tan x 8
x
22.
cos 4 x
x
arctan x 8
θ
8
arccos 4 x4
θ
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23.
sin x 2
5
arcsin
x 2
5
25. Let y be the third side. Then
y 2 2 2 x 2 4 x 2 ⇒ y 4 x 2
sin x 2 ⇒
cos
tan
4 x2
2
x
4 x 2
arcsin x 2
⇒
⇒
θ
arctan
5
arccos
4 x2
2
x
4 x 2
x + 2
24.
tan x 1
10
arctan
x 1
10
10
26. Let y be the third side. Then
y 2 3 2 x 2 9 x 2 ⇒ y 9 x 2 .
sin x 3 ⇒
cos
tan
9 x2
3
x
9 x 2
arcsin x 3
⇒
⇒
θ
arccos
arctan
9 x2
3
x
9 x 2
x + 1

342 Chapter 4 Trigonometric Functions
27. Let y be the hypotenuse. Then y 2 x 1 2 2 2 x 2 2x 5 ⇒ y x 2 2x 5.
sin
cos
tan x 1
2
x 1
x 2 2x 5
2
x 2 2x 5
⇒
⇒
⇒
arcsin
arccos
arctan x 1
2
x 1
x 2 2x 5
2
x 2 2x 5
28. Let y be the hypotenuse. Then y 2 x 2 2 3 2 x 2 4x 13 ⇒ y x 2 4x 13.
sin
cos
tan x 2
3
x 2
x 2 4x 13
3
x 2 4x 13
⇒
⇒
⇒
arcsin
arccos
arctan x 2
3
x 2
x 2 4x 13
3
x 2 4x 13
29. sinarcsin 0.7 0.7 30. tanarctan 35 35
31. cosarccos0.3 0.3
32. sinarcsin0.1 0.1
33.
arcsinsin 3 arcsin0 0
Note:
3
is not in the range of the arcsine function.
34. arccos cos 7
2 arccos0
36. arcsin sin 7
4 arcsin 2
3
38. cos 1 cos
2 cos1 0
2
2 4
3
40. cos 1 tan
4 cos1 1
42. cosarctan1 cos
4 2
2
44. sinarctan1 sin
4 2 2
46. arccos sin
6 arccos 1 2 2 3
2
35. arctan tan 11
6 arctan 3
5
37. sin 1 sin
2 sin1 1
5
39. sin 1 tan
4 sin1 1
41. tanarcsin 0 tan 0 0
2
43. sinarctan 1 sin 4 2
2
45. arcsin cos
6 arcsin 3
47. Let y arctan 4 Then
3 .
2
3 6
2 3
tan y 4 and
3 , 0 < y < 2 , 5
4
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sin y 4 5 .
y
3

Section 4.7 Inverse Trigonometric Functions 343
u arcsin 3 5 ,
49. Let y arcsin 24 Then
25 .
48. Let
sin u 3 5 , 0 < u <
hyp
2 . 5
u
4
3
24
sin y
and cos y 7
25 .
25
24
25 , 7
sec arcsin 3 5 sec u
y
adj 5 4
50. Let
y arctan 3 5 .
u arctan 12 5 , 51. Let Then, and
tan u 12
5 , 2 < u < 0.
y
tan y 3 5 , 2 < y < 0
sec y 34
5 .
13
5
u
−12
y
34
5
−3
x
csc arctan 12 5
hyp
csc u
opp 13 12
52. Let
u arcsin 3 4 ,
y arccos 2 3 .
53. Let Then,
sin u 3 4 , 2 < u < 0.
cos y 2 3 ,
2 < y <
y
and
sin y 5
3 .
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tan arcsin 3 4
u
4
7
−3
tan u
3
7 37
7
54. Let u arctan 5 8 ,
tan u 5 8 , 0 < u < 2 .
89
u
8
cot arctan 5 8 cot u 8 5
5
5
−2
55. Let y arctan x. Then,
3
y
tan y x and cot y 1 x .
x 2 + 1
x
y
1
x

344 Chapter 4 Trigonometric Functions
56. Let
u arctan x,
tan u x x 1 .
Let y arccosx 2, cos y x 2.
Opposite side: 1 x 2 2
x 2 + 1
x
sin y
1 x 22
1
x 2 4x 3
u
1
sinarctan x sin u opp
hyp x
x 2 1
y
1
x + 2
1 − (x + 2) 2
58. Let
57.
u arcsinx 1,
sin u x 1 x 1
1 .
59. Let y arccos x Then cos y x and
5 .
5 ,
1
x −1
tan y
25 x2
.
x
u
2x − x 2
5
25 − x 2
secarcsin x 1 sec u hyp
adj 1
2x x 2
y
x
60. Let u arctan 4 tan u 4 x , x .
x
61. Let y arctan Then tan y
x and
7 . 7
x 2 + 16
4
csc y
7 x2
.
x
u
62. Let
x
cot arctan 4 x
u
u arcsin x h ,
r
r
r 2 − ( x − h) 2
cot u
adj
opp x 4
x − h
sin u x h .
r
7 + x 2
y
7
x
63. Let y arctan 14 Then tan y 14 and
x .
x
sin y 14
Thus,
196 x 2.
196 + x 2 14
y
x
y arcsin
14
196 x 2 .
© Houghton Mifflin Company. All rights reserved.
cos arcsin x h
r cos u r 2 x h 2
r

Section 4.7 Inverse Trigonometric Functions 345
64. If then sin u 36 x 2
arcsin 36 x 2
u,
.
6
6
36 − x 2
6
u
x
3
65. Let y arccos
Then,
x 2 2x 10 .
cos y
and
sin y
y arcsin
3
x 2 2x 10 3
x 1 2 9
x 1
x 1 2 9 .
Thus,
x 1
x 1 2 9 arcsin
x 1
x 2 2x 10 .
36 x2
arcsin arccos x 6
6
66. If arccos x 2 u, 2 < x < 4 then cos u x 2
2
2 . 67.
2
4x − x
2
y 2 arccos x
Domain: 1 ≤ x ≤ 1
Range: 0 ≤ y ≤ 2
Vertical stretch of fx arccos x
2
u
x − 2
arccos x 2
2
arctan 4x x 2
x 2 ,
since 2 < x < 4
−2
0
2
68.
y arcsin x 2
Domain: 2 ≤ x ≤ 2
Range:
2 ≤ y ≤ 2
2
69. The graph of fx arcsinx 2 is a horizontal
translation of the graph of y arcsin x by two units.
0
2
4
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70.
−3
3
−
2
gt arccost 2 71.
Domain: 3 ≤ t ≤ 1
This is the graph of y arccos t shifted two units
to the left.
− 2
f x arctan 2x
Domain: all real numbers
−3
2
3
Range:
2 < y < 2
−4
0
0
− 2

346 Chapter 4 Trigonometric Functions
72.
fx arccos x 4
Domain:
4, 4
Range: 0,
−5
0
5
73.
ft 3 cos 2t 3 sin 2t 74.
3 2 3 2 sin 2t arctan
32 sin2t arctan 1
32 sin 2t
6
4
3
3
ft 4 cos t 3 sin t
4 2 3 2 sin t arctan 4 3
5 sin t arctan 4 3
The graph suggests that
A cos t B sin t A 2 B 2 sin t arctan A B
−2
2
is true.
6
−6
−9
9
The graphs are the same.
−6
75. As x → 1 , arcsin x →
2 . 76. As x → 1 , arccos x → 0. 77. As x →, arctan x →
2 .
78. As x → 1 , arcsin x →
2 . 79. As x → 1 , arccos x → . 80. As x → , arctan x →
2 .
81. (a)
sin 10
s
⇒
arcsin
10
s
(b)
s 52: arcsin
10
52 0.1935,
11.1
82. (a)
(b)
(c)
s 26: arcsin
10
26 0.3948,
34 ft
tan 11
17 ⇒ 0.5743
tan0.5743 h
20
11 ft
or 32.9
22.6
⇒ h 20 tan0.5743
83. (a)
tan s
750
arctan
s
(b) When s 400,
arctan
400
When s 1600,
750
0.49 radian,
750
1.13 radians,
28.
65.
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12.94 feet

Section 4.7 Inverse Trigonometric Functions 347
84.
arctan
(a)
0
1.5
−0.5
3x
x 2 4 , x > 0 (b) is maximum when x 2 feet.
(c) The graph has a horizontal asymptote at
As the camera moves further from the picture, the
6
angle subtended by the camera approaches 0.
0.
85. (a) tan 6 x
arctan
6
(b) When x 10,
arctan
6
When x 3,
arctan
6
0.54 radian, 31.
10
3
87. False. arcsin 1 2 6
x
1.11 radians, 63.
86. (a)
tan x
20
arctan x
(b) When x 5,
arctan 5 20 14.0,
When x 12,
arctan 12
88. False. tan x sin x
cos x
20
(0.24 rad).
31.0, 0.54 rad.
20
89. y arccot x if and only if cot y x,
< x < and 0 < y < .
90. y arcsec x if and only if sec y x where
x ≤ 1 x ≥ 1 and 0 ≤ y < 2 and
2 < y ≤ . The domain of y arcsec x
y
is , 1 1, and the range is
0, 2 2, .
π
y
π
2
π
© Houghton Mifflin Company. All rights reserved.
x
−2 −1 1 2
91. y arccsc x if and only if csc y x.
Domain: , 1 1,
Range:
2 , 0 0, 2
π
2
−2 −1 1 2
−
π
2
y
π
2
−2 −1 1 2
x
x

348 Chapter 4 Trigonometric Functions
92. (a) y arccot x if and only if x cot y, < x < and
Thus,
0 < y <
1
x tan y and y arctan 1 x .
Hence, graph
y
arctan1x,
2,
arctan1x,
(b) y arcsec x if and only if x sec y, x ≤ 1 or x ≥ 1, and 0 ≤ y < or
2 < y < .
2
1
Thus, and y arccos 1 Hence graph y arccos 1 x , 1 1,
x cos y
x .
x ,
.
(c) y arccsc x if and only if x csc y, x ≤ 1 or x ≥ 1, and or 0 < y ≤
2 ≤ y < 0
2 .
x < 0
x 0.
x > 0
1
Thus, Hence, graph y arcsin
1
x , x , 1 1,
x sin y and y arcsin 1 x .
.
.
93. y arcsec 2 ⇒ sec y 2 and 0 ≤ y <
2 2 < y ≤
⇒ y
4
94. y arcsec 1 ⇒ sec y 1 and 0 ≤ y <
2 2 < y ≤
⇒ y 0
95. y arccot3 ⇒ cot y 3 and 0 < y <
⇒ y 5
6
96. y arccsc 2 ⇒ csc y 2 and
2 ≤ y < 0 0 < y ≤
2 ⇒ y 6
97. Let y arcsinx. Then,
98.
sin y x
sin y x
siny x
y arcsin x
y arcsin x.
Therefore, arcsinx arcsin x.
99. arcsin x arccos x
2
Let
Let
arcsin x ⇒ sin x.
arccos x ⇒ cos x.
y arctanx
tan y x
arctantany arctan x
y arctan x
y arctan x
tan y x,
2 < y < 2
tany x,
2 < y < 2
Hence, sin a cos ⇒ a and are complementary angles ⇒ a ⇒ arcsin x arccos x
2
2 .
© Houghton Mifflin Company. All rights reserved.

Section 4.7 Inverse Trigonometric Functions 349
100.
Area arctan b arctan a
(a)
a 0, b 1
(b)
a 1, b 1
Area arctan 1 arctan 0
Area arctan 1 arctan1
(c)
4 0 4 0.785
a 0, b 3
Area arctan 3 arctan 0
1.25 0 1.25
1.25
(d)
4
4
a 1, b 3
Area arctan 3 arctan1
2 1.571
1.25
4 2.03
101.
4
42
1 2 2
2
102.
2
3 23
33 23
3
103.
23
6
3
3
104.
55
210 55
225 5
22 52
4
105.
sin 5 6
106.
tan 2, 0 <
<
2
Adjacent side:
cos 11
6
tan 5
11 511
11
6 2 5 2 11
6
θ
11
5
sin 2 5 25
5
cos 5
5
csc 5
2
5
θ
1
2
csc 6 5
sec 5
sec 6
11 611
11
cot 1 2
© Houghton Mifflin Company. All rights reserved.
107.
cot 11
5
sin 3 4
Adjacent side: 16 9 7
cos 7
4
tan 3 7 37
7
csc 4 3
sec 4 7 47
7
θ
4
7
3
108.
sec 3, 0 <
sin 22
3
cos 1 3
tan 22
<
csc 3
22 32
4
cot 1
22 2
4
2
3
θ
1
2 2
cot 7
3

350 Chapter 4 Trigonometric Functions
Section 4.8
Applications and Models
■ You should be able to solve right triangles.
■ You should be able to solve right triangle applications.
■ You should be able to solve applications of simple harmonic motion: d a sin wt or d a cos wt.
Vocabulary Check
1. elevation, depression 2. bearing 3. harmonic motion
1. Given: A 30, b 10
B 90 30 60
tan A a b
⇒ a b tan A 10 tan 30 5.77
cos A b c ⇒ c b
cos A 10
cos 30 11.55
2. Given: B 60, c 15
A 90 60 30
sin B b c
⇒ b c sin B 15 sin 60 153 12.99
2
cos B a c ⇒ a c cos B 15 cos 60 15 2 7.50
3. Given: B 71, b 144. Given: A 7.4, a 20.5
tan B b a ⇒ a b
tan B 14
tan 71 4.82
sin B b c ⇒ c b
sin B 14
sin 71 14.81
A 90 71 19
B 90 7.4 82.6
5. Given: a 6, b 126. Given: a 25, c 45
c 2 a 2 b 2 ⇒ c 36 144 13.42
tan A a b 6 12 1 2 ⇒ A arctan 1 2 26.57
B 90 26.57 63.43
tan A a b ⇒ b a
tan A 20.5
tan 7.4 157.84
sin A a c ⇒ c a
sin A 20.5
sin 7.4 159.17
b c 2 a 2 1400 1014 37.42
sin A a c ⇒ A arcsin 25
45 33.75
cos B a c ⇒ B arccos 25
45 56.25
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Section 4.8 Applications and Models 351
7. Given: b 16, c 548. Given: b 1.32, c 18.9
a c 2 b 2 2660 51.58
cos A b c 16
54 ⇒ A arccos 16
54 72.76
B 90 72.76 17.24
a c 2 b 2 355.4676 18.85
cos A b c 1.32
18.9 ⇒ A arccos 1.32
18.9 86.00
sin B b c ⇒ B arcsin 1.32
18.9 4.00
9.
A 12 15, c 430.5
B 90 12 15 77 45
sin 12 15 a
430.5
a
C
B
c = 430.5
12°15′
b
A
a 430.5 sin 12 15 91.34
cos 12 15 b
430.5
b 430.5 cos 12 15 420.70
10. Given: B 65 12, a 145.5
A 90 65 12 24 48
cos B a c ⇒ c
a
cos B 145.5
cos65 12 346.88
tan B b a ⇒ b a tan B 145.5 tan65 12 314.89
11.
tan h
b2
12.
tan h
b2
h 1 2 b tan
h 1 2 b tan
h 1 12 tan 18 1.95 meters
2
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13.
15. (a)
h 1 8 tan 52 5.12 in.
2
tan h
b2
h 1 2 b tan
h 1 18.5 tan 41.6 8.21 ft
2
θ
L
60 ft
(b)
tan 60
L
L 60
tan
60 cot
14.
tan h
b2
h 1 2 b tan
h 1 3.26 tan 72.94 5.31 cm
2
(c)
L
10
20
30
40
50
340 165 104 72 50
(d) No, the shadow lengths do not increase in
equal increments. The cotangent function
is not linear.

352 Chapter 4 Trigonometric Functions
16. (a)
(c)
850 ft
θ
L
10 20 30 40 50
L 4821 2335 1472 1013 713
(b)
tan 850
L
L 850
tan
850 cot
(d) No, the cotangent function is not a
linear function.
17.
sin 80 h 20
18.
tan13
h
58.2
h 20 sin 80
20 h
h 58.2 tan13 13.44 meters
19.70 feet
80°
13°
58.2 meters
h
19.
sin 50
h
100
h 100 sin 50
20.
sin 31.5
x
4000
x 4000 sin 31.5
31.5°
4000
x
76.6 feet
100
h
2089.99 feet
50°
21. (a)
h
22.
tan 28
a
100
⇒ a 100 tan 28
tan 39.75 a s
100
(b) Let the height of the church x and the height
of the church and steeple y. Then:
(c)
35°
47° 40′
50 ft
tan 35 x
50
and
x 50 tan 35 and y 50 tan 47 40
h y x 50tan 47 40 tan 35
h 19.9 feet
tan 47 40 y
50
s
a
a s 100 tan 39.75
s 100 tan 39.75 a
100 tan 39.75 100 tan 28
30 feet
100
39.75°
28°
© Houghton Mifflin Company. All rights reserved.

Section 4.8 Applications and Models 353
23. (a)
l 2 100 2 20 3 h 2
24. (a)
sin 60 h 30
⇒ h 30 sin 60
l 100 2 17 h 2
h 2 34h 10,289
θ
153 25.98 ft
30
h
l
h
60°
d
θ
20 ft
θ
3 ft 3 ft
100 ft
(b) cos 100 ⇒
l
l
(c) If then l 100
Then
cos 35 122.077.
35,
17 h 2 100 2 l 2
17 h 70.02
arccos
100
17 h 2 l 2 100 2 4902.906
h 53.02 feet
(b)
(c)
tan h d 153
d
25 ≤
tan 25 ≤ tan ≤ tan 30
tan 25 ≤ h d
h
tan 25 ≥ d ≥
≤ 30
≤ tan 30
h
tan 30
55.72 ≥ d ≥ 45.00
25.98
d
d is in the interval 45.0, 55.72.
53 feet, 1 4 inch.
25.
tan 75
95
arctan 15
19 38.29
26. (a)
1
12 ft
2
θ
27.
sin 4000
16,500 ⇒ 14.03
90 75.97
1
17 ft
3
75
(b)
tan 121 2
17 1 3
4000
16,500
α
θ
© Houghton Mifflin Company. All rights reserved.
28.
2.5 miles
θ
95
tan 250
2.55280
arctan
250
θ
2.55280 1.09
250 feet
not drawn to scale
(c)
arctan 121 2
17 1 3
35.8
29. Since the airplane speed is
ft sec
ft
275 60 16,500
sec min min ,
after one minute its distance travelled is 16,500 feet.
sin 18
a
16,500
a 16,500 sin 18
5099 ft
18°
16500
a

354 Chapter 4 Trigonometric Functions
30.
sin 18
If
s
h
275s
h
275 sin 18
275( s)
h 10,000, s 10,000 117.7 seconds.
275 sin 18
18°
h
31.
sin 9.5 x 4
x 4 sin 9.5
0.66 mile
9.5°
4
x
If h 16,000, s 16,000 188.3 seconds.
275 sin 18
32.
sin25.2 1808
L
⇒ L 1808
1808
sin25.2 4246.3 feet 25.2°
L
33.
90 29 61
N
206 120 nautical miles
sin 61
cos 61
a
120
b
120
⇒ a 120 sin 61 104.95 nautical miles
⇒ b 120 cos 61 58.18 nautical miles
W
a
b
61°
120
S
E
34.
c 6001.5 900
y 900 sin 38 554.1 miles
x 900 cos 38 709.2 miles
c
52°
38°
x
y
32, 68
35. Note: ABC forms a right triangle.
(a)
(b)
90 32 58
Bearing from A to C: N 58 E
32
90 22
C
54
tan C d
50 ⇒ tan 54 d 50 ⇒ d 68.82 m
W
B
d
N
φ
γ
θ β
α
50
β
A
S
C
E
© Houghton Mifflin Company. All rights reserved.

Section 4.8 Applications and Models 355
36.
tan 14 d x
⇒ x d cot 14
37.
tan 45
30 3 2 ⇒
56.31
tan 34 d y ⇒
d
30 x d
30 d cot 14
Bearing:
N 56.31 W
N
cot 34
30 d cot 14
d
d cot 34 30 d cot 14
W
Port
45
30
θ
Ship
E
d
30
cot 34 cot 14
5.46 kilometers
S
38.
tan 85
160 ⇒
27.98
39.
tan 6.5 350
d
⇒ d 3071.91 ft
Bearing: S 27.98 W
N
tan 4 350
D
⇒ D 5005.23 ft
W
160
θ
Airport
85
θ
S
Plane
E
Distance between ships:
350 4°
d
D
D d 1933.3 ft
6.5°
S 1 S 2
© Houghton Mifflin Company. All rights reserved.
40.
42.
10 km
28°
55°
55°
d
T 1
cot 55 d 10 ⇒ d 7 kilometers
cot 28 D 10
D
28°
⇒ D 18.8 kilometers
Distance between towns:
D d 18.8 7 11.8 kilometers
tan 2.5 h , tan 10
h
x x 18
x
h
tan 2.5 h
h 18 tan 10
18
tan 10 tan 10
h tan 10 h tan 2.5 18tan 10tan 2.5
h
T 2
h
tan 2.5 , x h
tan 10 18
18tan 10tan 2.5
tan 10 tan 2.5
h
2.5°
x
not drawn to scale
1.04 miles 5515 feet
41.
tan 57 a x
tan 16
tan 16
cot 16
a
x 556
a
a cot 57 556
a cot 57 556
a
a cot 16 a cot 57 55 6
h
10°
x − 18
not drawn to scale
⇒ x a cot 57
57°
H
x
P 1 P 2
16°
550
60
⇒ a 3.23 miles 17,054 feet
a

356 Chapter 4 Trigonometric Functions
43. (a)
44. (a) Using the two triangles with acute angle ,
50 + 6 = 56
cos 3 and sin 3 .
L 1
L 2
θ
40
arctan 56
40 54.5
Hence, L L 1 L 2 3 sec 3 csc .
3
β
L 1
L 2
Similarly for the back row,
arctan 56
150 20.5.
(b) For 45, you need to be 56 feet away since
(b)
20
β
3
arctan 56
56 45.
0
0
2
(c) The least value is 0.785.
4
45.
L 1 : 3x 2y 5 ⇒ y 3 2 x 5 2 ⇒ m 1 3 2
46.
L 1 2x y 8 ⇒ m 1 2
L 2 : x y 1 ⇒ y x 1 ⇒ m
1 32
tan
1 132
2 1
52
12 5
arctan 5 78.7
L 2 x 5y 4 ⇒ m 2 1 5
tan m 2 m 1
1 1 m 2 m
1 1 m 2 m
arctan m 2 m 1
arctan
1
5 2
1 1 52
arctan3 2 3 74.7
47. The diagonal of the base has a length of
a 2 a 2 2a. Now, we have:
tan a
2a
arctan
49. cos 30 b r
b cos 30r
b 3r
2
1 2
1
2 35.3
θ
b
2a
r
2
30°
r
a
y
48.
50.
tan a2
a
arctan 2 54.7º
c 35 2 17.5
sin 15º a c
2
15°
a c sin 15 17.5 sin 15 4.53
θ
a
c
a 2
a
© Houghton Mifflin Company. All rights reserved.
y 2b
x
Distance 2a 9.06 centimeters
2
3r
2 3r

51. d 0 when t 0, a 8, period 2
52. Displacement at t 0 is
Use d a sin t since d 0 when t 0.
Amplitude:
2
2 ⇒
Thus, d 8 sin t.
Period:
Section 4.8 Applications and Models 357
2
t
d 3 sin 3
a 3 0 ⇒ d a sin t.
6 ⇒
3
53. d 3 when t 0, a 3, period 1.5
Use d a cos t since d 3 when t 0.
2
1.5 ⇒ 4 3
Thus, d 3 cos
4
t
3 .
54. Displacement at t 0 is
Amplitude:
Period:
2
t
d 2 cos 5
a 2 2 ⇒ d a cos t
10 ⇒
5
55.
d 4 cos 8t
(a) Maximum displacement amplitude 4
(b)
Frequency
2
8
2
4 cycles per unit of time
(c) d 4 cos85 4
(d) 8t
2 ⇒ t 1
16
56.
d 1 cos 20t
2
57.
d 1 sin 140t
16
© Houghton Mifflin Company. All rights reserved.
(a) Maximum displacement:
(b) Frequency:
2
(c) When t 5, d 1 2 cos205 1 2 .
(d) Least positive value for t for which d 0:
1
2 cos 20t 0
cos 20t 0
20t
2
20 10
2
t
2 1
20
1 40
a 1 2 1 2
(a)
(b)
(c)
Maximum displacement amplitude 1 16
Frequency
d 0
2
140
2
70 cycles per unit of time
(d) 140t ⇒ t 1
140

358 Chapter 4 Trigonometric Functions
58.
d 1 sin 792t
64
(a) Maximum displacement:
(b) Frequency:
2
792 396
2
a 1 64 1 64
(c) When t 5, d 1 64 sin7925 0.
(d) Least positive value for t for which d 0:
1
64 sin 792t 0
sin 792t 0
792t
t
792
1
792
59.
d a sin t60. At t 0, buoy is at its high point ⇒ d a cos t.
Period 2 1
frequency
2
1
264
2264 528
Distance from high to low 2 a 3.5
Returns to high point every 10 seconds:
Period 2 10 ⇒
5
d 7 4 cos
t
5
a 7 4
61.
y 1 4 cos 16t, t > 0
(a)
0.5
(b) Period: 2
16 8 seconds (c) 1
cos 16t 0 when
4
0
16t
2 ⇒ t 32 seconds.
−0.5
62. (a)
L 1
L 2
2 3
0.1 23.0
sin 0.1 cos 0.1
2 3
0.2 13.1
sin 0.2 cos 0.2
2 3
0.3 9.9
sin 0.3 cos 0.3
2 3
0.4 8.4
sin 0.4 cos 0.4
(c) L L 1 L 2 2 3
sin cos
L 1 L 2
(b)
(d)
2 3
0.5 7.6
sin 0.5 cos 0.5
2 3
0.6 7.2
sin 0.6 cos 0.6
2 3
0.7 7.0
sin 0.7 cos 0.7
2 3
0.8 7.1
sin 0.8 cos 0.8
The minimum length of the elevator is 7.0 meters.
12
From the graph, it appears
that the minimum length is
7.0 meters, which agrees
with the estimate of part (b).
0
0
2
© Houghton Mifflin Company. All rights reserved.

Section 4.8 Applications and Models 359
63. (a), (b)
Base 1 Base 2 Altitude Area
8 8 16 cos 10 8 sin 10 22.1
8 8 16 cos 20 8 sin 20 42.5
8 8 16 cos 30 8 sin 30 59.7
8 8 16 cos 40 8 sin 40 72.7
8 8 16 cos 50 8 sin 50 80.5
8 8 16 cos 60 8 sin 60 83.1
8 8 16 cos 70 8 sin 70 80.7
(c)
A 1 2b 1 b 2 h
(d) Maximum area is approximately 83.1 square feet
for
100
0
0
1 28 8 16 cos 8 sin
641 cos sin
60.
90
Maximum 83.1 square feet
64. (a)
S
(b)
a 1 14.30 1.70 6.3
2
S
Average sales
(in millions of dollars)
15
12
9
6
3
2 4 6 8 10 12
Month (1 ↔ January)
t
2
b 12 ⇒ b 6
Shift:
d 14.3 6.3 8
S d a cos bt
Average sales
(in millions of dollars)
15
12
9
6
3
2 4 6 8 10 12
Month (1 ↔ January)
t
2
(c) Period:
6 12
This corresponds to the 12 months in a year.
Since the sales of outerwear is seasonal, this
is reasonable.
t
S 8 6.3 cos 6
The model is a good fit.
(d) The amplitude represents the maximum
displacement from the average sale of
8 million dollars. Sales are greatest in
December (cold weather holidays) and
least in June.
© Houghton Mifflin Company. All rights reserved.
65.
St 18.09 1.41 sin
t
6 4.60
(a)
22
0
14
66. False
It means 24 degrees east of north.
12
2
(b) The period is
months, which is 1 year.
6 12
(c) The amplitude is 1.41. This gives the maximum change
in time from the average time 18.09 of sunset.
67. False. The other acute angle is
Then
tan41.9 opp
adj
a
22.56
90 48.1 41.9.
⇒ a 22.56 tan41.9.

360 Chapter 4 Trigonometric Functions
68. False 69. y 2 4x 1
70.
4x y 6 0
y 0 1 2x 1 3
2y x 1 3
3x 6y 1 0
71.
Slope 6 2
2 3 4 5
y 2 4 x 3
5
72.
m
13 23
12 14 1
34 4 3
y 2 3 4 3 x 1 4
5y 10 4x 12
3y 2 4x 1
4x 5y 22 0
4x 3y 1 0
73. Domain: , 74. Domain: , 75. Domain: , 76. Domain:
or x ≤ 7
7 x ≥ 0
Review Exercises for Chapter 4
1. 40 or 0.7 radian
y
3. (a) (b) Quadrant III
4 π
3
x
(c)
4
3 2 10
3
4
3 2 2
3
2. 250 or 4.4 radians
y
4. (a) (b) Quadrant IV
11π
6
x
(c)
11
6 2 23
6
11
6 2
6
y
5. (a) (b) Quadrant III
7. Complement:
Supplement:
9. Complement:
Supplement:
−
5 π
6
x
8 7
8
(c)
2 8 3
8
2 3
10 5
3
10 7
10
5
6 2 7
6
5
6 2 17
6
y
6. (a) (b) Quadrant I
−
7 π
4
8. Complement:
Supplement:
10. Complement:
Supplement:
x
(c)
2 12 5
12
2
12 11
12
2 2
21 17
42
21 19
21
7
4 2
4
7
4 2 15
4
© Houghton Mifflin Company. All rights reserved.

Review Exercises for Chapter 4 361
11. (a)
y
12. (a)
y
45°
x
210°
x
(b) Quadrant I
(c) 45 360 405
45 360 315
(b) Quadrant III
(c) 210 360 570
210 360 150
13. (a)
y
14. (a)
y
x
−405°
x
−135°
(b) Quadrant III
(c) 135 360 225
135 360 495
(b) Quadrant IV
(c) 405 720 315
405 360 45
15. Complement of
5: 90 5 85
16. Complement of
Supplement of 5: 180 5 175
84:
90 84 6
Supplement of 84: 180 84 96
17. Complement: not possible
Supplement: 180 171 9
18. Complement of 136: not possible
Supplement of 136: 180 136 44
© Houghton Mifflin Company. All rights reserved.
19. 135 16 45
135 16
60 3600 45 135.279 20. 234 40 234
3600 40 234.011
21. 5 22 53 5 22
60 3600 53 5.381
23. 135.29 135 0.2960 135 17 24
25. 85.36 85 0.3660 85 21 36
27. 415 415
rad
22.
280 8 50 280 8
60 50
3600
24. 25.8 25 48
280 0.13 0.01 280.147
26. 327.93 327 55 48
rad
180 83
36 rad 7.243 rad 28. 355 355
180
71 rad 6.196 rad
36

362 Chapter 4 Trigonometric Functions
rad
rad
29. 72 72 2 rad 1.257 rad 30. 94 94
180 5
180 47 rad 1.641 rad
90
31.
7 5
7 180
128.571
5
32. 3
5 3
5 180
108
33. 3.5 3.5
180
200.535
34. 1.55 1.55
180
88.808
35.
s r
25 12
25
12 2.083
s r 245
49
36. s r ⇒
4.083 radians
60 12
37.
s r
s 20138
180
s 48.171 m
38. s r 15
cm
3 5 15.71
39. In one revolution, the arc length traveled is
s 2r 26 12 cm. The time
required for one revolution is
t 1
500 minutes 1
500 60 3 25 seconds.
Linear speed s t 12
325 100 cmsec
41. t 7 corresponds to
2
4
2 , 2 2 .
43.
cos 5
6 3 2
sin 5
6 1 2
x, y 3
2 , 1 2
45. t 2 corresponds to 1 3
2 , 3 2 .
40. (a) 28 miles per hour 285280 2464 ft/min
60
42.
(b)
Circumference of wheel:
Number of revolutions per minute:
2464
73
7
t 7392
7392 336.1 revmin
7
2 14,724
7
cos 3
4 2 3
, sin
2 4 2
2
x, y 2
2 , 2
2
44. t 4 corresponds to 1 3
2 , 3 2 .
46. t 7 corresponds to 3
2 , 1 6
2 .
C 21 3 7 3
2112 rad/min
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Review Exercises for Chapter 4 363
47. t 5 corresponds to 2
2 , 2
4
2 .
48. t 5 corresponds to 3
6
2 , 1 2 .
49.
7
sin
6 1 2
cot 7
6 3
50.
sin
4 2
2 cos 4
cos 7
6 3 2
sec 7
6 2 3 23 3
tan
4 1 cot 4
7
tan
6 1
3 3
3
csc 7
6 2
sec 2 csc
4 4
51. t 2 corresponds to 1, 0.
52. t corresponds to 1, 0.
sin 2 y 0
cot 2 x y ,
undefined
sin y 0
cot x y ,
undefined
cos 2 x 1 sec 2 1 x 1
tan 2 y csc 2 1 undefined
x 0 y ,
cos x 1 sec 1 x 1
tan y csc 1 undefined
x 0 y ,
© Houghton Mifflin Company. All rights reserved.
53. t 11 corresponds to
6
sin 11
6 y 1 2
cos 11
6 x 3
2
tan 11
6 y x 3
3
cot 11
6 x y 3
sec 11
6 1 x 23
3
csc 11
6 1 y 2
3
2 , 1 2 .
55. t corresponds to 0, 1.
2
2 y 1
sin
cos 2 x 0
tan 2 y x ,
undefined
cot
2 x y 0
sec
2 1 x ,
undefined
csc
2 1 y 1
54. t 5 corresponds to
6
sin 5
6 y 1 2
cos 5
6 x 3 2
tan 5
6 y x 3
3
cot 5
6 x y 3
sec 5
6 1 x 23 3
csc 5
6 1 y 2
3 2 , 1 2 .

364 Chapter 4 Trigonometric Functions
56. t corresponds to
4
sin 4 y 2 2
cos 4 x 2
2
tan 4 y x 1
2
2 , 2 2 .
cot
4 x y 1
sec
4 1 x 2
csc
4 1 y 2
57. sin
11
4 sin 3
4 2
2
59. sin 17
6 sin 7
6 1 2
58. cos 4 cos 0 1
60. cos 13 5
3 cos
3 1 2
61.
sin t 3 5
62.
cos t 5 13
(a)
sint sin t 3 5
(a)
cost cos t 5 13
(b) csct 1
sint 5 3
(b) sect 1
cost 13
5
63.
sint 2 3
64.
cost 5 8
(a)
sin t sint 2 3 2 3
(a)
cos t cost 5 8
(b) csc t 1
sin t 3 2
(b) sect 1
cost 8 5
65. cot 2.3 1
tan 2.3 0.8935
67. cos 5
3 1 2
69. The opposite side is 9 2 4 2 81 16 65.
sin opp
hyp 65
9
cos adj
hyp 4 9
tan opp
adj 65
4
csc 1 9
sin 65 965
65
sec 1
cos
9 4
cot 1 4
tan 65 465
65
66. sec 4.5 1
cos 4.5 4.7439
68. tan 11
6 0.5774
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Review Exercises for Chapter 4 365
70.
sin 15
341 541
41
cos 12
341 441
41
tan 15
12 5 4
csc 41
541 41
5
sec 41
441 41
4
cot 4 5
θ
3 41
12
15
71. The hypotenuse is 12 2 10 2 244 261.
sin opp
hyp 10
261 5
61 561
61
cos adj
hyp 12
261 6
61 661
61
tan opp
adj 10
12 5 6
csc 1 61
sin 5
sec 1 61
cos 6
cot 1
tan
6 5
72.
sin 2 10 1 5
csc 5
73. csc tan 1 sin
1 sec
sin cos cos
cos 46
10 26
5
sec 5
26 56
12
tan 2
46 6
12
cot 12
6 26
10
θ
96 = 4 6
2
74.
cot tan
cot
1 tan
1 tan
cot
2 sec 2
75. (a)
cos 84 0.1045
(b) sin 6 0.1045
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76. (a)
csc52 12
1
sin52 12
(b) sec54 7 1
1
cos54 7 cos54.1167 1.7061
1
sin 52.2 1.2656
77. (a) cos
4 0.7071 78. (a) tan
3
79. tan 62
20 0.5095
125
(b) sec
4 1.4142
x 125 tan 62
(b) cot
3
20 1
tan320
235 feet
80. (a) (b)
30
152 ft
h
(c)
sin 30
h
152
h 152
1
2 76
1.9626
⇒ h 152 sin 30
feet

366 Chapter 4 Trigonometric Functions
81.
x 12, y 16, r 144 256 400 20
sin y r 4 5
csc r y 5 4
cos x r 3 5
sec r x 5 3
tan y x 4 3
cot x y 3 4
82.
x 2, y 10, r 4 100 104 226
83.
x 7, y 2, r 49 4 53
sin y r 10
226 526
26
sin y r 2
53 253
53
cos x r 2
226 26
26
cos x r 7
53 753 53
tan y x 10 2 5
tan y x 2 7
csc r y 26
5
csc 53
2
sec r x 26
sec 53
7
cot x y 1 5
cot 7 2
84.
cot r y 3 4
x 3, y 4, r 3 2 4 2 5
85.
x 2 3 and y 5 8 ,
r
2
3 2
5
8 2 481
24
sin y r 4 5
sin y r 58
48124 15481
481
cos x r 3 5
cos x r 23
48124 16481
481
tan y x 4 3
csc x y 5 4
sec r x 5 3
tan y x 58
23 15
16
csc r y 481
15
sec r x 481
16
cot x y 16
15
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Review Exercises for Chapter 4 367
86.
x 10 3 , y 2 3 ,
sin y r 23
2263 1
26 26
26
cos x r 103
2263 5
26 526
26
tan y x 23
103 1 5
r 10 3 2 2 3 2 226
3
csc r y 26
sec r x 26 5
cot x y 5
87. sec 6 is in Quadrant IV.
5 , tan < 0 ⇒
r 6, x 5, y 36 25 11
θ
y
5
x
sin y r 11 6
csc 611
11
6
− 11
cos x r 5 6
tan y x 11 5
sec 6 5
cot 511
11
88.
tan y x 12 5 ⇒ r 13,
sin > 0 ⇒ y 12, x 5
sin y r 12
13
cos x r 5 13
csc r y 13
12
sec r x 13
5 13 5
cot x y 5 12
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89. sin 3 is in Quadrant II.
8 , cos < 0 ⇒
y 3, r 8, x 55
sin y r 3 8
cos x r 55 8
tan y x 3
55 355 55
csc 8 3
sec 8
55 855 55
cot 55
3
3
− 55
y
8 θ
x

368 Chapter 4 Trigonometric Functions
90.
cos x r 2
5
sin y r 21
5
tan y x 21 2
⇒ y ±21,
csc r y 5
21 521
21
sin > 0 ⇒ y 21
sec r x 5
2 5 2
cot x y 2
21 221 21
91. Reference angle:
y
92. 635 is coterminal with 275.
264 180 84
Reference angle:
y
θ = 264
360 275 85
θ ′ = 84
x
θ = 635°
x
θ ′ = 85
93. Coterminal angle:
2 6
Reference angle:
4
5 4
5
5 5
θ ′ = π 5
y
θ =
−
6π
5
x
17
Reference angle:
2 5
3 3
5
94. is coterminal with
3
3 .
θ =
17π
3
y
θ ′ = π 3
x
95. 240 is in Quadrant III with reference angle 60. 96. 315 is in Quadrant IV with reference angle 45.
sin 240 sin 60 3
2
cos 240 cos 60 1 2
tan 240 32
12
tan210
3
12
32 1 3 3 3
sin 315 sin 45 2
2
cos 315 cos 45 2
2
tan 315 tan 45 1
97. 210 is coterminal with 150 in Quadrant II with 98. 315 is coterminal with 45 in Quadrant I.
reference angle 30.
sin210 sin30 1 sin315 2
2
2
cos210 cos30 3
cos315 2
2
2
tan315 1
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Review Exercises for Chapter 4 369
99. 94 is coterminal with 74 in Quadrant IV
with reference angle 4.
sin 9
cos 9
4 sin
4 cos
4 2 2
4 2
2
tan 9
4 22
22
1
100. 116 is in Quadrant IV.
sin
11
6 1 2
cos
11
6 3
2
tan
11
6 3 3
101.
sin4 sin0 0
cos4 1
tan4 0
102.
sin
7
3 sin
cos
7
3 1 2
3 3
2
103. tan 33 0.6494
tan
7
3 3
104. csc 105 1
sin 105 1.0353
105. sec 12
5 1
cos125 3.2361
106. sin
9 0.3420
107.
f x 3 sin x
y
108.
f x 2 cos x
y
Amplitude: 3
4
3
2
Amplitude: 2
3
2
1
π
2π
x
−2π
−π
−1
π
2π
x
−2
−4
−3
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109.
y
f x 1 4 cos x 110. f x 7 2 sin x
Amplitude: 1 1
4
Amplitude: 7 2
111. Period:
2
Amplitude: 5
2
−2π
1
2
1
−
2
−1
112. Period:
2π
2
Amplitude: 3 2
x
12 4
2
113. Period:
2
Amplitude: 3.4
114. Period:
y
4
3
2
1
x
π 2π
−4
2
Amplitude: 4
2 4

370 Chapter 4 Trigonometric Functions
115.
y 3 cos 2x
y
116.
y 2 sin x
y
Amplitude: 3
Period: 2 1
2
1
−
2
3
1
−2
1
2
1
x
Period: 2 2
Amplitude: 2 2
Reflected in x-axis
3
1
–1
–2
1
x
−3
1 2
1
2
x 0
y 2 0 2
–3
117. fx 5 sin 2x
5
Amplitude: 5
Period:
2
25 5
y
6
4
2
–2
–6
6π
x
118.
f x 8 cos x 4
2
Period:
14 8
Amplitude: 8
Reflected in y-axis
2
4
x 0
y 8 0 8 0 8
4
2
8
–4
–6
–8
y
4π 8π
x
119.
f x 5 2 cos x 4
5
Amplitude:
Period: 2
14 8
5
4
3
2
2 1
−4π
y
−3
−4
−5
4π
8π
x
120.
f x 1 2 sin
Amplitude: 1 2
Period: 8
x
4
−6
−2
1
−
1
2
−1
y
2 6
x
121.
f x 5 2 sinx
Amplitude:
Period:
Shift:
2
5
2
x 0 and x 2
x x 3
y
4
3
1
–1
–2
–3
–4
π
x
122.
f x 3 cosx
Period: 2
Amplitude: 3
This is the graph of
y 3 cos x shifted
to the left units.
x
f x
2
0
3 0 3 0 3
2
4
3
2
1
–3
–4
y
π
x
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Review Exercises for Chapter 4 371
123.
f x 2 cos
x
2
y
4
3
2
124.
f x 1 2 sin x 3
Amplitude: 1 2
−2
−1
y
−1
1
2
x
−4 −3 −2 −1
−1
1 2 3 4
−2
−3
−4
x
Period: 2
Vertical shift
downward three units
−2
−3
−4
125.
fx 3 cos
x
2
4
y
4
3
2
−π π
−2
−3
2π 3π
x
126.
f x 4 2 cos4x
Amplitude: 2
Period:
2
y
8
6
4
2
−4
− π 2
− π 4
−2
π
4
π
2
x
127. fx 2 cos x
128.
4
fx a cosbx c
Amplitude: 3
Period: ⇒ f x 3 cos2x
129. fx 4 cos 2x
2
130.
fx a cosbx c 131.
Amplitude:
1
2
Period: 2 ⇒ f x 1 2 cos x
S 48.4 6.1 cos
Maximum sales:
(June)
Minimum sales:
(December)
t
6
t 6
t 12
56
0
40
12
© Houghton Mifflin Company. All rights reserved.
132.
S 56.25 9.50 sin
Maximum sales:
(March)
Minimum sales:
(September)
t 3
t 9
t
6
70
0
45
12
133.
fx tan
Period:
4 4
Asymptotes:
−4
x 2, x 2
Reflected in
x
y
x
4
x-axis
1 0 1
1 0 1
6
4
2
−6 −2 2 4 6
−2
−4
−6
y
x

372 Chapter 4 Trigonometric Functions
134.
f x 4 tan x135.
y
6
4
2
x
1
−
3π
2
4
3
2
1
−1
−2
−3
−4
y
π
2
3π
2
f x 1 4 tan x
136.
2
x
f x 2 2 tan x 3
−3π −2π
6
4
2
−4
−6
y
π
3π
4π
x
137.
f x 3 cot x 2
Period:
12 2
y
6
4
2
138.
f x 1 2 cot
y
x
2 1
2 tan
x
2
Two consecutive
asymptotes:
−4π
−2π
2π
4π
x
2
1
x
2 0 ⇒ x 0
−2 −1
1 2
x
x
2 ⇒ x 2
139.
141.
f x 1 2 cot x
Period:
Two consecutive
asymptotes:
x
2 0 ⇒ x 2
x
2 ⇒ x 3
2
f x 1 4 sec x 142. f x 1 2 csc x 1
2 sin x
y
Period:
y
2
2
1
x
1
−1
−2
2
2π
2
x
5
4
3
−2π
−π
−1
y
π
2π
x
140.
f x 4 cot x
−2π
−π
−2
−π
4
y
π
π
2π
2π
3π
4 4
tan x
x
4
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Review Exercises for Chapter 4 373
143.
f x 1 4 csc 2x 144.
Period:
y
f x 1 2 sec 2x
y
1
2 cos 2x
3
2
1
−1
1
x
π
x
−1
145.
f x sec x
Secant function shifted
to right
y
146.
4
3
4 −π
π 2π
2
1
x
f x 1 2 csc2x
y
1
2 sin2x
−π
− π 2
π
2
π
x
−1
147.
fx 1 4 tan
x
2
148.
fx tan x
4
1
4
− π
π
−2π
2π
−1
−4
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149.
151.
fx 4 cot2x
− π
10
−10
f x 2 secx
− 2π
10
π
2π
4
tan2x
2
cosx
150.
152.
2
fx 2 cot4x
tan4x
− π
2
−2π
4
− 4
π
2
fx 2 cscx
8
2π
2
sinx
−10
− 8

374 Chapter 4 Trigonometric Functions
153.
fx csc 3x
2 1
sin3x 2
154.
fx 3 csc 2x
4 3
sin2x 4
4
8
−π
π
− π
π
−4
− 8
155.
f x e x sin 2x 156.
f x e x cos x
Damping factor:
y e x
Damping factor:
e x
20
5
−4
4
−4
4
−20
−5
157.
f x 2x cos x 158.
f x x sin x
6
Damping factor:
14
gx 2x
As x →, f oscillates
between x and x.
−9
9
−10
10
−6
−14
As x →, f oscillates between 2x and 2x.
159. (a) arcsin1 because sin
2
2 1.
(b) arcsin 4 does not exist because the domain of
arcsin is 1, 1.
161. (a) arccos 2 because cos
4 2
2 4
2 .
(b) arccos 3 because cos 5
2 5
6
6 3 2 .
160. (a) arcsin 1 because sin
2 6
6 1 2 .
(b)
162. (a)
arcsin 3 because
2 3
sin
3 3 2 .
arctan3
3
(b) arctan1
4
163. arccos0.42 1.14 164. arcsin 0.63 0.68
165. sin 1 0.94 1.22
166. cos 1 0.12 1.69 167. arctan12 1.49
168. arctan 21 1.52
169. tan 1 0.81 0.68
170. tan 1 6.4 1.42
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171. sin x 3
16
⇒
arcsin
x 3
16
172. tan x 1
20
⇒
arctan
x 1
20

Review Exercises for Chapter 4 375
173. Let y arcsinx 1. Then,
174. Let u arccos x cos u x 2 , 2 .
sin y x 1 x 1 and
1
tan arccos
2 x tan u
1
sec y
4
x 2 x2
2x
x2 1
2x
x 2 2x . x − 1
y
u
1 − (x − 1) 2
x 2
175. Let y arccos Then cos y
x2
and
4 x 2. 176. Let u arcsin 10x, sin u 10x.
4 x 2
sin y 4 cscarcsin 10x csc u
x2 2 x 2 2
.
4 x 2
16 8x2
4 x 2
24 2x2
4 x 2 .
y
4 − x 2
x 2
(4 − x 2 ) 2 − (x 2 ) 2
2
x 4 − x 2
hyp
opp 1
10x
1
u
x
1 − 100x 2
10x
177.
sin1 10 a
3.5
a 3.5 sin1 10 3.5 sin
7
6 0.0713
or 71 meters
3.5
a
1° 10'
not drawn to scale
© Houghton Mifflin Company. All rights reserved.
178.
179.
tan 12
100
arctan
12
sin h 4
tan 14
y
37,000
0.1194 or 6.84
100
h 4 sin0.1194 0.48 miles or 2534 feet (answer depends on angle
⇒ y 37,000 tan 14 9225.1 feet
tan 58 x y
37,000 ⇒ x y 37,000 tan 58 59,212.4 feet 37,000
14°
x 59,212.4 9225.1 49,987.2 feet
The towns are approximately 50,000 feet apart or 9.47 miles.
h
4
θ
x
accuracy)
32°
76°
58°
y

376 Chapter 4 Trigonometric Functions
180.
sin 48 d 1
650 ⇒ d 1 483
cos 25 d d 1 d 2 1217
2
810 ⇒ d 2 734 W 48°
48°
cos 48 d 3
650 ⇒ d 3 435
sin 25 d 4
810 ⇒ d 4 342
tan 93
1217 ⇒
sec 4.4
D
1217
4.4
⇒ D 1217 sec 4.4 1221
The distance is 1221 miles and the bearing is N 85.6 E.
d 3 d 4 93
N
B
65° 25°
d 810 d 4
3 650
D C
A
θ
d1 d2
S
E
181. Use cosine model with amplitude 3 feet.
Period: 15 seconds
y 3 cos
2
15 t
183. False. y sin is a function, but it is not
one-to-one.
182. Use a cosine model with amplitude
Period: 3 seconds
y 0.75 cos
2
3 t
1.5
2 0.75.
184. False. The sine and cosine functions are useful for
modeling simple harmonic motion.
185.
tan 0.672s 2
3000
(a)
s 10 20 30 40 50 60
1.28 5.12 11.40 19.72 29.25 38.88
(b)
186. (a)
−
increases at an increasing rate. The function is not linear.
10
−10
(b) Next term:
x 9
9
arctan x x x3
3 x5
5 x7
7 x9
9
The accuracy of the approximation
increases as more terms are added.
−
10
−10
© Houghton Mifflin Company. All rights reserved.

Practice Test for Chapter 4 377
Chapter 4
Practice Test
1. Express 350 in radian measure.
2. Express 59 in degree measure.
3. Convert 135 14 12 to decimal form.
4. Convert 22.569 to D M S form.
5. If cos 2 3 ,6. use the trigonometric identities to
Find given sin 0.9063.
find tan .
7. Solve for x in the figure below.
8. Find the magnitude of the reference angle for
65.
35
20°
x
9. Evaluate csc 3.92. 10. Find sec given that lies in Quadrant III and
tan 6.
11. Graph y 3 sin x 2 . 12. Graph y 2 cosx .
13. Graph y tan 2x. 14. Graph y csc x
15. Graph y 2x sin x, using a graphing calculator. 16. Graph y 3x cos x, using a graphing calculator.
4 .
17. Evaluate arcsin 1. 18. Evaluate arctan3.
19. Evaluate sin arccos
4
35 . 20. Write an algebraic expression for cos arcsin x 4 .
© Houghton Mifflin Company. All rights reserved.
For Exercises 21–23, solve the right triangle.
21. A 40, c 12 22.
B 6.84, a 21.3 23. a 5, b 9
24. A 20-foot ladder leans against the side of a barn. Find the height of the top
of the ladder if the angle of elevation of the ladder is 67.
A
c
b
B
a
C
25. An observer in a lighthouse 250 feet above sea level spots a ship off the shore.
If the angle of depression to the ship is 5, how far out is the ship?