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C H A P T E R 4
Trigonometric Functions
Section 4.1
Radian and Degree Measure . . . . . . . . . . . . . . . . 272
Section 4.2
Trigonometric Functions: The Unit Circle
Section 4.3
Right Triangle Trigonometry . . . . . . . . . . . . . . . . 289
Section 4.4
Trigonometric Functions of Any Angle
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
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Practice Test
. . . . . . . . 281
. . . . . . . . . . 300
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
C H A P T E R 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 90. They are supplementary if 180.
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
Vocabulary Check
1. Trigonometry
2. angle
3. standard position
4. coterminal
5. radian
6. complementary
7. supplementary
8. degree
1.
10. angular
The angle shown is
approximately 2 radians.
3. (a) Since
3 7
7
lies in Quadrant IV.
<
< 2,
2
4
4
5 11
11
lies in
<
< 3,
2
4
4
Quadrant II.
(b) Since
272
2.
The angle shown is
approximately 4 radians.
5
5
< < 0,
lies in
2
12
2
Quadrant IV.
4. (a) Since 3
13
13
< < ,
lies in
2
9
9
Quadrant II.
(b) Since © Houghton Mifflin Company. All rights reserved.
9. linear
Section 4.1
5. (a) Since < 1 < 0; 1 lies in Quadrant IV.
2
6. (a) Since < 3.5 <
(b) Since < 2 < ; 2 lies in
2
Quadrant III.
7. (a)
13
4
Radian and Degree Measure
(b) Since
y
8. (a) 3
, 3.5 lies in Quadrant III.
2
< 2.25 < , 2.25 lies in Quadrant II.
2
y
7
4
3π
4
x
x
− 7π
4
(b)
4
3
y
5
(b) 2
y
4π
3
x
x
9. (a)
11
6
− 5π
2
y
10. (a) 4
y
11π
6
4
x
© Houghton Mifflin Company. All rights reserved.
x
(b)
2
3
(b) 3
y
y
2π
3
x
x
11. (a) Coterminal angles for
:
6
−3
(b) Coterminal angles for
13
2 6
6
8
2
2 3
3
11
2 6
6
4
2
2 3
3
2
:
3
273
Chapter 4
Trigonometric Functions
12. (a) Coterminal angles for
7
:
6
13. (a) Coterminal angles for 7
19
2 6
6
9
2 4
4
7
5
2 6
6
9
7
4 4
4
(b) Coterminal angles for
5
:
4
(b) Coterminal angles for 5
13
2 4
4
3
5
2 4
4
15. Complement:
2
3
6
Supplement: 17. Complement:
2
6
3
Supplement: 19. Complement:
2
3
3
14. (a) Coterminal angles for
2
:
15
(b) Coterminal angles for
98
8
2 45
45
32
2
2 15
15
82
8
2 45
45
16. Complement: Not possible;
(a)Supplement: Supplement: 8
:
45
3
is greater than .
4
2
3 4
4
18. Complement: Not possible;
2
is greater than .
3
2
2 3
3
2
Supplement: 2 1.14
20. Complement: None 2 >
Supplement: 1 2.14
7
:
8
7
23
2 8
8
9
7
2 8
8
2
28
2 15
15
5
6
6
1 0.57
2
9
:
4
22.
21.
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.
© Houghton Mifflin Company. All rights reserved.
274
Section 4.1
y
27. (a) 30
Radian and Degree Measure
275
y
(b) 150
150°
30⬚
x
28. (a) 270
x
30. (a) 450
29. (a) 405
y
y
y
−450°
405°
x
x
x
−270°
(b) 120
(b) 600
(b) 780
y
y
y
780°
− 600⬚
x
x
x
−120°
31. (a) Coterminal angles for 52:
52 360 412
114 360 474
52 360 308
114 360 246
(b) Coterminal angles for 36:
© Houghton Mifflin Company. All rights reserved.
32. (a) Coterminal angles for 114:
(b) Coterminal angles for 390:
36 360 324
390 720 330
36 360 396
390 360 30
34. (a) Coterminal angles for 445:
445 720 275
35. Complement: 90 24 66
Supplement: 180 24 156
33. (a) Coterminal angles for 300:
300 360 660
300 360 60
(b) Coterminal angles for 230:
230 360 590
230 360 130
36. Complement: Not possible
Supplement: 180 129 51
445 360 85
(b) Coterminal angles for 740:
740 1080 340
740 720 20
37. Complement: 90 87 3
Supplement: 180 87 93
38. Complement: Not possible
Supplement: 180 167 13
39. (a) 30 30
180 6
(b) 150 150
180 5
6
Chapter 4
Trigonometric Functions
40. (a) 315 315
7
4
2
3
180 (b) 120 120
180 42. (a) 270 270
43. (a)
(b) 144 144
7 7 180
420
3
3 180 2.007 radians
180
0.014 radians
56. 4.2 4.2
58. 0.48 0.48
4
5
15
15 180
450
6
6
48. 83.7 83.7
180 3.776 radians
180
25.714
7
7 4
28 28 180
(b)
336
15
15 46. (a) 13 180
13
39
60
60 51. 0.78 0.78
180 3
720
180
180
540
(b) 3 3 49. 216.35 216.35
180 44. (a) 4 4
7
7 180
210
6
6 47. 115 115
53.
3
180 2
(b) 180 9
(b) 240 240
3 3 180
270
2
2 (b) 45. (a)
41. (a) 20 20
54.
756
180
50. 46.52 46.52
52. 395 395
180
0.812 radians
180
6.894 radians
8 8 180
110.769
13
13 57. 2 2
27.502
180
180
1.461 radians
55. 6.5 6.5
1170
180
114.592
180
59. 64 45 64 64.75
45
60 60. 124 30 124.5
30 61. 85 18 30 85 18
60 3600 85.308
62. 408 16 25 408.274
36 63. 125 36 125 3600
125.01
64. 330 25 330.007
65. 280.6 280 0.660 280 36
66. 115.8 115 48
67. 345.12 345 7 12
68. 310.75 310 45
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276
Section 4.1
69. 0.355 0.355
180
Radian and Degree Measure
70. 0.7865 0.7865
20.34 20 20 24
180
45.0631
45 0.063160 45 3 0.78660 45 3 47
71. s r
72.
6 5
73. 3s r
31 12
65 radians
74.
s r
32 7
7
31
12 212 radians
32
4
3 7 4 7 radians
s r
60 75
4
60
75 5 radians
Because the angle represented is clockwise, this angle is 45 radians.
75. The angles in radians are:
135 © Houghton Mifflin Company. All rights reserved.
0 0
76. The angles in degrees are:
3
4
30 6
45 4
210 7
6
60 3
270 3
2
90 2
330 11
6
180 77. s r
78. 8 15
5
225
4
45
4
4
240
3
60
3
5
300
3
2
120
3
7
315
4
5
150
6
180
s 10
5
radian
r 22 11
79.
8
radians
15
s 160
2 radians
r
80
82. r 9 feet, 60 s r 9
s r
35 14.5
80. r 80 kilometers, s 160 kilometers
30
6
3
3 feet
3
70
2.414 radians
29
81. s r , in radians
s 14180
180
14 43.982 inches
83. s r , in radians
s 27
23 18 meters 56.55 meters
277
278
Chapter 4
Trigonometric Functions
84. r 12 centimeters, s r 12
3
4
85. r s
36
72
feet 22.92 feet
2
s
82
328
miles 34.80 miles
135180
3
34 9 centimeters 28.27 cm
86. r s
3
9
meters 0.72 meter
43 4
87. r 88. r 8
48
s
inches 1.39 inches
330180 11
89. 42 7 33 25 46 32
16 21 1 0.28537 radian
s r 40000.28537 1141.48 miles
s
450
0.07056 radian 4.04
r 6378
91. 90. r 4000 miles
31 46 26 8 57 54 1.0105 rad
4 2 33.02
s r 40001.0105 4042.0 miles
93. 92. r 6378, s 400
94. s 2.5 25
5
radian 23.87
r
6
60 12
s
400
0.0627 rad 3.59
r 6378
s 24
4.8 rad 275.02
r
5
1
95. (a) single axel: 12 revolutions 360 180 540
2 3 radians
1
(b) double axel: 22 revolutions 720 180 900
1
(c) triple axel: 32 revolutions 1260 7 radians
96. Linear speed 97. (a)
s r 6400 12502
436.967 kmmin
t
t
110
Revolutions 2400
40 revsec
Second
60
Angular speed 240 80 radsec
(b) Radius of saw blade Radius in feet Speed 7.5
3.75 in.
2
3.75
0.3125 ft
12
s r
r r angular speed
t
t
t
0.312580 78.54 ftsec
98. (a)
Revolutions 4800
80 revsec
Second
60
Angular speed 280 160 radsec
(b) Radius of saw blade Radius in feet Speed 7.25
3.625 in.
2
3.625
0.3021 ft
12
s r
r r angular speed
t
t
t
0.3021160 151.84 ftsec
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4 5 radians
Section 4.1
99. (a)
Radian and Degree Measure
Revolutions
480
28,800 revhr
Hour
160
Angular speed 2 28,800 57,600 radhr
Radius of wheel Speed 252
5
miles
12 in.ft5280 ftmi 25,344
s r
r r angular speed
t
t
t
5
25,344
57,600 125
35.70 mileshr
11
(b) Let x spin balance machine rate.
70 rangular speed r 2
x
5
revolutions
≤ 500
minute
200 ≤
100. (a)
160 25,344120 x ⇒ x 941.18 revmin
2 200 ≤ angular speed ≤ 2 500
400 radmin ≤ angular speed ≤ 1000 radmin
(b) Speed s r
r r angular speed 6angular speed
t
t
t
6400 ≤ linear speed ≤ 61000
For the outermost track, 6000 cmmin
101. False, 1 radian 57.3, so one radian
180
102. No, 1260 is coterminal with 180.
is much larger than one degree.
© Houghton Mifflin Company. All rights reserved.
103. True:
2 8 3 180
3
4
12
12
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.
1
1
107. A r 2 102
2
2
50
square meters
3
3
106. Let A be the area of a circular sector of radius r
and central angle . Then
A
1
⇒ r 2.
2
r
2
2
108. Because s r, 12
.
15
1
12
1
90 ft 2.
Hence, A r 2 152
2
2
15
279
280
Chapter 4
Trigonometric Functions
1
109. A 2 r 2, s r
8
⇒ A 12 r 2 0.8 0.4r 2
(a) 0.8
s r r 0.8
A
Domain: r > 0
s
Domain: r > 0
0
The area function changes more rapidly for r > 1 because
it is quadratic and the arc length function is linear.
⇒ A 12 10 2 50
(b) r 10
s r 10
12
0
Domain: 0 < < 2
320
Domain: 0 < < 2
A
s
2␲
0
0
110. If a fan of greater diameter is installed, the angular speed does not change.
111. Answers will vary.
112. Answers will vary.
y
113.
y
114.
4
2
3
1
2
x
−4 −3 −2
1
x
−2
2
3
4
5
1
2
3
4
1
2
3
5
6
−2
6
−3
−2
−3
−4
−6
y
115.
y
116.
5
4
4
3
3
2
1
1
−4 −3 −2
1
2
3
x
−5 −4 −3
x
4
−2
−3
y
117.
y
118.
3
4
2
3
1
2
x
−4 −3 −2
1
−2
2
3
1
4
x
−2
1
−2
−4
−3
−5
−4
2
3
© Houghton Mifflin Company. All rights reserved.
−4
−3
Section 4.2
Section 4.2
■
■
■
■
Trigonometric Functions: The Unit Circle
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
1. sin y 2. periodic
15
17
2. x, y 13, 13
12 5
8
17
sin y 5
13
tan y
15
x
8
cos x 12
13
cot x
8
y
15
tan y
513
5
x 1213 12
sec 1
17
x
8
csc 1
1
13
y
513
5
csc 1 17
y
15
sec 1
1
13
x
1213 12
cot x 1213 12
y
513
5
cos x 3. sin y © Houghton Mifflin Company. All rights reserved.
3. odd, even
cos x y
5
x
12
cot x
12
y
5
sec 1 13
x
12
csc 1
13
y
5
4
3
4. x, y , 5
5
3
sin y 5
4
cos x 5
y 35
3
tan x 45 4
1
1
5
csc y
35
3
1
1
5
sec x
45
4
x 45 4
cot y 35 3
12
13
tan 5. t 5
13
2 2
corresponds to
.
,
4
2
2
6. t ⇒
3
1 3
2, 2 281
Chapter 4
Trigonometric Functions
5
⇒
4
7. t 3
1
7
, .
corresponds to 2
2
6
9. t 2
1 3
.
corresponds to ,
3
2 2
10. t 3
corresponds to 0, 1.
2
12. t ⇒ 1, 0
11. t 8. t 2
2
,
2
2
5
1 3
corresponds to , .
3
2
2
4
1 3
corresponds to ,
.
2 2
3
13. t 7
2 2
,
.
corresponds to
2
2
4
14. t 15. t 3
corresponds to 0, 1.
2
16. t 2 corresponds to 1, 0.
17. t 2 2
,
.
corresponds to
4
2
2
sin t y cos t x tan t 19. t 2
2
2
y
1
x
3
7
1
, .
corresponds to 6
2
2
1
2
cos t x tan t 3
tan t 3
y
3
2
cos
1
x
3
2
tan
y 32
3
3
x
12
20. t 3
2
1
2
y
3
x
22. t 2 2
5
,
.
corresponds to 4
2
2
tan t sin
2
2
cos t x 2
2
1 3
corresponds to ,
.
3
2 2
cos t x sin t y 3
1
y
x 3
3
sin t y 1 3
corresponds to ,
.
3
2 2
18. t 2
sin t y 21. t 2
2
y
1
x
5
1 3
.
corresponds to , 3
2
2
sin t y cos t x tan t 3
2
1
2
y
3
x
© Houghton Mifflin Company. All rights reserved.
282
Section 4.2
23. t 5
1 3
.
corresponds to ,
3
2 2
sin t y cos t x tan t 25. t 1
2
2
2
2
3
y
1
4
x
3
2
1
2
y
3
x
sin2 y 0
cos2 x 1
tan2 y
is undefined.
x
2 2
3
corresponds to ,
.
4
2
2
2
2
2
y
1
x
2
csc t 1
2
y
sec t 1
2
x
cot t x
1
y
4
1 3
.
corresponds to ,
3
2 2
30. t 2 corresponds to 1, 0.
cos t x 0
cos t x 28. t tan t 3
corresponds to 0, 1.
2
tan t 2
cos t x y
1
x
sin t y 2
3
x
4
2
2
2 , 2 .
sin t y 2
sin t y 1
31. t 3
y
x
3
tan 2
2
3
y
4
2
cos tan t 3
3
2
1
2
3
corresponds to
4
sin 2 2
7
corresponds to
,
.
4
2
2
tan t 26. t 1
2
3
y
x
3
cos t x © Houghton Mifflin Company. All rights reserved.
tan t sin t y cos t x 3
1
, .
corresponds to
6
2
2
tan t 3
11
1
, .
corresponds to
6
2
2
sin t y 2
y
3
x
cos t x 29. t 24. t 3
sin t y 27. t Trigonometric Functions: The Unit Circle
y 0
0
x 1
283
Chapter 4
Trigonometric Functions
3 1
5
corresponds to , .
6
2 2
32. t sin t y 33. t 1
2
cos t x 3
2
tan t 3
y
1
x
3
3
csc t 1
2
y
sec t 1
2
23
x
3
3
cot t x
3
y
sin t y 1
csc t 1
1
y
cos t x 0
sec t 1
is undefined.
x
cot t x
0
y
tan t 3
corresponds to 0, 1.
2
34. t corresponds to 0, 1.
2
35. t y
is undefined.
x
2
1 3
corresponds to , .
2
2
3
sin
3
y 1
2
sin t y cos
3
x0
2
cos t x tan
3 y 1
⇒ undefined
2
x
0
tan t csc
1
3 1
1
2
y
1
sec
3 1 1
⇒ undefined
2
x
0
cot
0
3 x
0
2
y 1
36. t 2 2
7
corresponds to
,
.
4
2
2
2
7
y
4
2
csc 7
1
2
4
y
2
7
x
4
2
sec 7
1
2
4
x
7
y
1
4
x
cot 7
x
1
4
y
cos tan 2
1
2
y
3
x
37. sin 5 sin 0
sin 3
csc t 23
3
sec 2
cot 3
3
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284
Section 4.2
38. Because 7 6 :
Trigonometric Functions: The Unit Circle
39. cos
cos 7 cos6 cos 1
40. Because
sin
9
2 :
4
4
41. cos 3
13
11
cos cos
6
6
6
2
9
2
sin 2 sin 4
4
4
2
42. Because sin 19
5
4 :
6
6
2
56 12
44. Because cos 8
4
4 :
3
3
46. cos t 4
1
3
2
3
4
(b) sect 3
4
(a) sint sin t 1
1
4
cost cos t
3
(a) sin t sint 1
5
(b) sec t 49. sin t 3
8
4
5
4
5
4
5
(b) cost cos t 4
5
4
5
(b) sint sin t 51. sin
(a) cos t cos t 1
5
1
5
cos t
(a) sin t sin t 1
1
8
sint sint
3
1
3
(b) csc t csc t 3
(a) cos t cost 3
8
(b) csc t 1
3
47. cos t (a) cost cos t 48. sint 45. sin t 8
4
cos 4 3
3
cos
50. cos t 9
4 sin 4 2
43. sin 19
5
sin 4 6
6
sin
© Houghton Mifflin Company. All rights reserved.
2
1
8
cos
3
3
2
7
0.6428
9
4
5
285
286
Chapter 4
52. tan
Trigonometric Functions
2
3.0777
5
53. cos
11
0.8090
5
54. sin
11
0.6428
9
55. csc 1.3 1.0378
56. cot 3.7 1
1.6007
tan 3.7
57. cos1.7 0.1288
58. cos2.5 0.8011
59. csc 0.8 1
1.3940
sin 0.8
60. sec 1.8 1
4.4014
cos 1.8
63. cot 2.5 1
1.3386
tan 2.5
61. sec 22.8 1
1.4486
cos 22.8
62. sin13.4 0.7404
64. tan 1.75 5.5204
65. csc1.5 1
1.0025
sin1.5
66. tan2.25 1.2386
67. sec4.5 1
4.7439
cos4.5
68. csc5.2 1
1.1319
sin5.2
70. (a) sin 0.75 y 0.7
69. (a) sin 5 1
(b) cos 2.5 x 0.8
(b) cos 2 0.4
72. (a) sin t 0.75
71. (a) sin t 0.25
73.
I0.7 5e1.4 sin 0.7
t 4.0 or t 5.4
t 0.25 or 2.89
0.79 amperes
(b) cos t 0.75
(b) cos t 0.25
I 5e2t sin t
t 0.72 or t 5.56
t 1.82 or 4.46
74. At t 1.4,
75. y t 14 cos 6t
I 5e21.4 sin 1.4 0.30 amperes.
(a) y 0 14 cos 0 0.2500 ft
(c) y12 14 cos 3 0.2475 ft
1
76. yt et cos 6t
4
1
1
(a) y0 e0 cos0 foot
4
4
(b)
t
0.50
1.02
1.54
2.07
y
0.15
0.09
0.05 0.03
2.59
0.02
(c) The maximum displacements are decreasing
because of friction, which is modeled by the
et term.
(d)
1 t
e cos 6t 0
4
cos 6t 0
6t 3
,
2 2
t
,
12 4
© Houghton Mifflin Company. All rights reserved.
(b) y14 14 cos 32 0.0177 ft
Section 4.2
43 3
Trigonometric Functions: The Unit Circle
> 0
78. True
79. False. 0 corresponds to 1, 0.
80. True
81. (a) The points have y-axis symmetry.
82. (a) The points x1, y1 and x2, y2 are symmetric
about the origin.
77. False. sin
2
(b) sin t1 sin t1 since they have the same
y-value.
(c) cos t1 cos t1 since the x-values have
opposite signs.
83. cos 1.5 0.0707, 2 cos 0.75 1.4634
(b) Because of the symmetry of the points, you can
make the conjecture that sint1 sin t1.
(c) Because of the symmetry of the points, you can
make the conjecture that cost1 cos t1.
84. sin0.25 sin0.75 0.2474 0.6816 0.9290
Thus, cos 2t 2 cos t.
sin 1 0.8415
Therefore, sin t1 sin t2 sint1 t2.
86. sin y sin 85. cos x cos sec 1
1
sec cos cos y
csc 1
csc y
tan y
tan x
cot x
cot y
(x, y)
θ
x
−θ
(x, −y)
87. ht f tgt is odd.
© Houghton Mifflin Company. All rights reserved.
ht f tgt f tgt ht
287
88. f t sin t and gt tan t
Both f and g are odd functions.
ht f tgt sin t tan t
ht sint tant
sin ttan t
sin t tan t ht
The function ht f tgt is even.
288
Chapter 4
Trigonometric Functions
f x 12 3x 2
89.
6
f
y 12 3x 2
x
1
2 3y
90.
f−1
−9
−9
f−1
9
x 1 14 y3
−6
−6
4x 1 y3
3 4x 1
y 1 y
3
4x 1
f 1x f 1x 3 x 1
2
f x x2 4, x ≥ 2, y ≥ 0
y x2 4
92.
f x 2x
, x > 1
x1
y
2x
, x > 1
x1
7
x y2 4
f−1
x y 4
2
f
x 14 y3 1
2
2x 2 3y
91.
6
1
y 4x3 1
9
2x 3y 2
2
3 x
f x 14 x3 1
2
f
−2
−3
9
−4
2y
x
y1
10
−1
x2 4 y2
4
xy x 2y
x2 4 y, x ≥ 0
x 2y xy
f 1x x2 4, x ≥ 0
x y2 x
x
y, x < 2
2x
x
,x < 2
2x
f 1x 2x
x3
94. f x 5x
5x
x x 6 x 3x 2
2
Asymptotes: x 3, x 2, y 0
Asymptotes: x 3, y 2
y
y
10
8
6
4
8
7
6
5
4
3
x
4 6 8 10
1
−4 −3 −2 −1
x
2
4 5 6 7 8
−6
−8
−2
−3
−4
95. f x x2 3x 10
x 2x 5
2
2x 8
2x 2x 2
x5
,
2x 2
y
4
3
x2
1
Asymptotes: x 2, y 2
2
1
−7 −6 −5
−2 −1
−1
−2
−3
−4
x
1
© Houghton Mifflin Company. All rights reserved.
93. f x Section 4.3
96. f x x3 6x 2 x 1
x
7
15x 4
2x 2 5x 8
2 4 42x 2 5x 8
Slant asymptote: y 289
y
10
8
6
4
2
x
7
2 4
Vertical asymptotes: x 3.608, x 1.108
Right Triangle Trigonometry
−8 −6 −4 −2
x
6 8 10
−8
−10
98. y ln x 4
97. y x2 3x 4
Domain: all real numbers
Domain: all x 0
0 x 3x 4 x 4x 1 ⇒ x 1, 4
ln x4 0 ⇒ x 4 1 ⇒ x ± 1
Intercepts: 0, 4, 1, 0, 4, 0
Intercepts: 1, 0, 1, 0
No asymptotes
Asymptote: x 0
2
100. f x 99. f x 3x1 2
Domain: all real numbers
x7
x7
x 2 4x 4 x 22
Domain: all real numbers x 2
Intercept: 0, 5
Intercepts: 7, 0, 0, Asymptote: y 2
7
4
Asymptotes: x 2, y 0
Section 4.3
© Houghton Mifflin Company. All rights reserved.
■
opp
adj
adj
(f) cot opp
1
csc 1
(d) sec cos sin (g) tan cos (b) csc 1
sin 1
(e) tan cot cos (h) cot sin (j) 1 tan2 sec2 (k) 1 cot2 csc2 nu
θ
You should know the following identities.
(a) sin se
(c) tan te
adj
hyp
hyp
(e) sec adj
(b) cos yp
o
opp
hyp
hyp
(d) csc opp
(a) sin Opposite side
You should know the right triangle definition of trigonometric functions.
H
■
Right Triangle Trigonometry
Adjacent side
1
sec 1
(f) cot tan (c) cos (i) sin2 cos2 1
■
You should know that two acute angles and are complementary if 90, 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.
290
Chapter 4
Trigonometric Functions
Vocabulary Check
1. (a) iii
(b) vi
(c) ii
(d) v
2. hypotenuse, opposite, adjacent
(e) i
(f) iv
3. elevation, depression
2.
1.
13
5
5
θ
3
b
θ
b 132 52 169 25 12
adj 52 32 16 4
sin opp
5
hyp 13
sin opp 3
hyp 5
cos adj
12
hyp 13
cos adj
4
hyp 5
tan opp
5
adj
12
tan opp 3
adj
4
csc csc hyp 5
opp 3
hyp 13
opp
5
sec sec hyp 5
adj
4
hyp 13
adj
12
cot cot adj
4
opp 3
adj
12
opp
5
3.
4.
8
152
opp
8
sin hyp 17
cos tan csc sec cot adj
15
hyp 17
opp
8
adj
15
hyp 17
opp
8
hyp 17
adj
15
adj
15
opp
8
17
C 182 122 468 613
sin 18
313
613
13
cos 12
213
613
13
tan 18 3
12 2
csc sec cot 13
3
13
2
2
3
© Houghton Mifflin Company. All rights reserved.
15
hyp c
θ
12
θ
82
18
Section 4.3
5. opp 102 82 6
sin opp
6
3
hyp 10 5
cos adj
8
4
hyp 10 5
tan Right Triangle Trigonometry
opp 2.52 22 1.5
sin opp 1.5 3
hyp 2.5 5
cos adj
2
4
hyp 2.5 5
opp 6 3
adj
8 4
tan opp 1.5 3
adj
2
4
csc hyp 10 5
opp
6
3
csc hyp 2.5 5
opp 1.5 3
sec hyp 10 5
adj
8
4
sec hyp 2.5 5
adj
2
4
cot adj
8 4
opp 6 3
cot adj
2
4
opp 1.5 3
10
θ
8
291
2.5
θ
2
The function values are the same since the triangles are similar and the corresponding sides are proportional.
adj 7.52 42 © Houghton Mifflin Company. All rights reserved.
6. adj 152 82 161
sin opp
8
hyp 15
cos 15
8
161
7.5
2
θ
sin opp
4
8
hyp 7.5 15
161
adj
hyp
15
cos tan opp
8
8161
161
adj
161
161
161
adj
hyp 2 7.5
15
tan csc hyp 15
opp
8
opp
4
8
8161
adj
161
161
1612
csc hyp 7.5 15
opp
4
8
sec hyp
15
15161
161
adj
161
sec cot 161
adj
opp
8
hyp
7.5
15
15161
adj
161
161
1612
cot 161
161
adj
opp
24
8
θ
The function values are the same because the triangles are similar, and corresponding sides are proportional.
4
292
Chapter 4
Trigonometric Functions
7. adj 32 12 8 22
sin opp 1
hyp 3
θ
adj 62 22 32 42
3
1
sin opp 2 1
hyp 6 3
6
2
θ
cos adj
22
hyp
3
cos adj
42 22
hyp
6
3
tan 2
opp
1
adj
4
22
tan 2
opp
2
1
adj
4
42 22
csc hyp
3
opp
csc hyp 6
3
opp 2
sec hyp
3
32
adj
4
22
sec hyp
6
3
32
adj
4
42 22
cot adj
22
opp
cot adj
42
22
opp
2
The function values are the same since the triangles are similar and the corresponding sides are proportional.
sin 5
opp
1
hyp 5
5
adj
2
25
cos hyp 5
5
θ
1
2
hyp 32 62 35
sin 3
35
5
1
5
5
3
θ
6
2
25
cos 35 5
5
6
tan opp 1
adj
2
tan 3 1
6 2
csc hyp 5
5
opp
1
csc 35
5
3
sec hyp 5
adj
2
sec 35 5
6
2
cot adj
2
2
opp 1
cot 6
2
3
The function values are the same because the triangles are similar, and corresponding sides are proportional.
© Houghton Mifflin Company. All rights reserved.
8. hyp 12 22 5
Section 4.3
9. Given: sin 5 opp
6 hyp
10. hyp 52 12 26
52 adj2 62
6
5
adj 11
adj
5
526
hyp 26
26
opp
5
511
tan adj
11
11
tan opp 1
adj
5
11
adj
cot opp
5
csc hyp 26
26
opp
1
sec hyp 26
adj
5
θ
hyp
6
611
adj
11
11
csc hyp 6
opp 5
4 hyp
1
adj
sin opp 15
cos 4
15
tan 15
4
1
1
4
tan 15
cot 15
1
15
15
csc 4
415
15
15
32
12
hyp
3 opp
1
adj
10
3
hyp 10
cos 10
cot csc 1
3
10
3
2 10
210
3
csc 7
710
210
20
sec 7
3
cot 3
310
210
20
θ
opp
4
sin hyp 17
cos 273
adj
hyp
17
tan opp
4
4273
273
adj
273
sec 1
17
17273
cos 273
273
cot 273
1
tan 4
1
10
sec 10
7
θ
3
14. adj 172 42 273
2
310
sin 10
210
7
θ
13. Given: tan 3 1
5
12. opp 72 32 40 210
opp2 12 42
sin 293
11
sec 11. Given: sec 4 26
θ
26
opp
1
sin hyp 26
26
cos 11
adj
cos hyp
6
© Houghton Mifflin Company. All rights reserved.
Right Triangle Trigonometry
17
θ
273
4
Chapter 4
Trigonometric Functions
15. Given: cot 42
92
9
adj
4 opp
4
hyp
2
tan sec csc 9
8
9
997
97
97
4
9
97
9
97
3
θ
55
55
adj
cos hyp
8
4
497
97
97
cos 3
8
adj 82 32 55
θ
hyp 97
sin 16. sin 97
tan opp
3
355
55
adj
55
csc 1
8
sin 3
sec 1
8
855
cos 55
55
cot 55
1
tan 3
4
(deg)
Function
(rad)
6
Function
Function Value
(deg)
(rad)
Function Value
2
1
2
18. cos
45
4
3
20. sec
45
4
2
22. csc
45
4
2
24. sin
45
4
26. tan
30
6
17. sin
30
19. tan
60
3
21. cot
60
3
3
23. cos
30
6
3
25. cot
45
4
3
2
1
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 33. tan sin cos 34. cot cos sin 35. sin2 cos2 1
36. 1 tan2 sec 2 37. sin90 cos 38. cos90 sin 39. tan90 cot 40. cot90 tan 41. sec90 csc 42. csc90 sec 43. sin 60 3
2
(a) tan 60 , cos 60 1
2
sin 60
3
cos 60
(c) cos 30 sin 60 3
2
(b) sin 30 cos 60 (d) cot 60 1
2
3
cos 60
1
sin 60
3
3
© Houghton Mifflin Company. All rights reserved.
294
Section 4.3
3
1
44. sin 30 , tan 30 2
3
(a) csc 30 Right Triangle Trigonometry
32
4
45. csc 3, sec 1
2
sin 30
(b) cot 60 tan90 60 tan 30 3
3
(a) sin 1
1
csc 3
(b) cos 1
22
sec 3
2
sin 13
cos 22 3
4
(c) cos 30 3
sin 30
12
3
tan 30 33 23
2
(c) tan (d) cot 30 3
33
1
3
tan 30 3
3
(d) sec90º csc 3
46. sec 5, tan 26
(a) cos 1
1
sec 5
(c) cot90 tan 26
(d) sin tan cos 26 6
1
1
(b) cot tan 26
12
47. cos 1
26
5
1
4
(a) sec 1
4
cos (c) cot (b) sin2 cos2 1
sin2 4
1
2
sin 14
154
15
1
15
15
1
4
(d) sin90 cos 15
4
lies in Quadrant I or III.
48. tan 5
(a) cot 15
16
cos sin 1
sin2 © Houghton Mifflin Company. All rights reserved.
5 1
1
tan 5
(b) sec2 1 tan2 ⇒ cos 1
1 tan2 (c) tan 90 cot 1
5
(d) csc 1 cot2 1 251 49. tan cot tan 51. tan cos tan 1
sin 1
cos cos sin 1
1 25
1
26
26
26
26
5
1
50. csc tan sin 52. cot sin sin 1
cos cos sec cos sin cos sin 295
296
Chapter 4
Trigonometric Functions
53. 1 cos 1 cos 1 cos2 54. csc cot csc cot csc2 cot2 sin2 cos2 cos2 1
sin2 sin cos sin2 cos2 cos sin sin cos 56.
1
sin cos 1
sin 1
58. (a) tan 18.5 0.3346
(b) cos 87 0.0523
59. (a) sec 42 12 sec 42.2 (b) cot 71.5 1
1.3499
cos 42.2
1
7 º 1.3432
sin48 60
61. Make sure that your calculator is in radian mode.
(a) cot
1
5.0273
16 tan
16
(b) tan
0.4142
8
63. (a) sin 6
65. (a) sec 2 ⇒ 60 3
(b) cot 1 ⇒ 45 4
(b) sin 2
⇒ 45 60. (a) cos 8 50 25 cos 8 50
25
60 3600
cos 8.840278 0.9881
(b) sec 8 50 25 62. (a) sec1.54 64. (a) cos 1
1.0120
cos 8 50 25 1
32.4765
cos1.54
2
2
⇒ 45 (b) tan 1 ⇒ 45 (b) cos 4
68. (a) cot tan 4
4
66. (a) tan 3 ⇒ 60 23
⇒ 60 3
3
2
1
0.3346
tan 71.5
(b) cos1.25 0.3153
1
⇒ 30 2
6
(b) csc 2 ⇒ 30 67. (a) csc cot 1cot 1 cot 2 csc 2 1
csc sec cos 57. (a) sin 41 0.6561
(b) csc 48º 7 tan cot tan cot tan tan tan 3
1
⇒ 60 2
3
3
3
3
3 ⇒ 60 3
3
(b) sec 2
cos 1
2
2
2
⇒ 45 4
© Houghton Mifflin Company. All rights reserved.
55.
Section 4.3
69. tan 30 cos 30 70. cos 30 y
⇒ y 105 tan 30 105
105
3
3
Right Triangle Trigonometry
353
105
105
105
210
⇒ r
703
r
cos 30 32 3
x
⇒ x 15 cos 30 15
15
3
2
153
2
sin 30 1
15
y
⇒ y 15 sin 30 15
15
2
2
71. cos 60 x
1
⇒ x 16 cos 60 16
8
16
2
sin 60 72. cot 60 3
y
⇒ y 16 sin 60 16
83
16
2
x
1
383
⇒ x 38 cot 60 38 38
3
3
38
38
38
763
⇒ r
r
sin 60 32
3
sin 60 73. tan 45 20
20
⇒ 1
⇒ x 20
x
x
74. tan 45 r 2 202 202 ⇒ r 202
75. tan 45 y
y
⇒ 1
⇒ y 10
10
10
r 2 102 102 ⇒ r 102
25
25
⇒ 1
⇒ x 25
x
x
r 2 252 252 20 20 40 ⇒ r 210
© Houghton Mifflin Company. All rights reserved.
76. tan 45 y
y
⇒ 1
⇒ y 46
46
46
r 2 462 462 96 96 192 ⇒ r 83
78. (a)
77. (a)
h
30
6
16
(b) tan (c) h x
θ
5
6
h
h
6
and tan Thus, .
5
21
5 21
621
25.2 feet
5
75°
(b) sin 75 x
30
(c) x 30 sin 75 28.98 meters
297
298
Chapter 4
79. tan Trigonometric Functions
opp
adj
80. cot 9 c
h
cot 3.5 w
tan 58 100
h
13 c
h
3.5°
13
9°
w 100 tan 58 160 feet
Subtracting,
h
81. (a) tan 50
1 ⇒ 45
50
502 252
ftsec
6
3
rate down the zip line
13
1.295 1.3 miles.
16.3499 6.3138
82. (a) sin 35.4 x
896.5
x 896.5 sin 35.4 519.3 feet
(b) 1693.5 519.3 1174.2 feet above sea level
(c)
50 25
ftsec
6
3
13
cot 3.5 cot 9
h
13
cot 3.5 cot 9
(b) L2 502 502 2 502 ⇒ L 502 feet
(c)
c
not drawn to scale
vertical rate
896.5
minutes to reach top
300
Vertical rate 519.3
896.5300
173.8 feet per minute
83.
(x1, y1)
(x2, y2)
56
56
30°
y1
56
60°
y1 sin 3056 cos 30 1
56 28
2
x1
56
x1 cos 3056 x1, y1 283, 28
sin 60º y2
56
y2 sin 6056 3
2
56 283
cos 60 3
2 56 283
x2
56
x2 cos 6056 x2, y2 28, 283 256 28
1
© Houghton Mifflin Company. All rights reserved.
sin 30 Section 4.3
84. x 9.397, y 3.420
Right Triangle Trigonometry
299
85. True
sin 20 y
0.34
10
cos 20 x
0.94
10
tan 20 y
0.36
x
cot 20 x
2.75
y
sec 20 10
1.06
x
csc 20 10
2.92
y
sin 60 csc 60 sin 60
86. False
1
1
sin 60
87. True
sin 45 cos 45 2
2
2
2
1 cot2 csc2 for all 2 1
88. No. tan2 1 sec2 , so you can find ± sec .
89. (a)
90.
0
20
40
60
80
sin 0
0.3420
0.6428
0.8660
0.9848
cos 1
0.9397
0.7660
0.5000
0.1736
tan 0
0.3640
0.8391
1.7321
5.6713
sin .
cos 0
20
40
60
80
cos sin90 cos 1
0.9397
0.7660
0.5000
0.1736
and 90 are complementary angles.
sin90 1
0.9397
0.7660
0.5000
0.1736
92.
7
−6
93.
1
−3
© Houghton Mifflin Company. All rights reserved.
(c) In each case, tan 91.
−6
−2
10
−4
97.
10
−4
−6
−1
4
−2
6
6
−3
96.
2
−6
6
4
94.
7
3
−1
95.
(b) Sine and tangent are increasing, cosine is
decreasing.
98.
4
−2
10
−4
4
−2
10
−4
Chapter 4
300
Section 4.4
■
Trigonometric Functions
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 x2 y2 0, then:
y
r
sin csc , y 0
r
y
x
r
cos sec , x 0
r
x
y
x
tan , x 0
cot , y 0
x
y
You should know the signs of the trigonometric functions in each quadrant.
■
3
You should know the trigonometric function values of the quadrant angles 0, , , and .
2
2
■
■
■
■
■
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
Vocabulary Check
y
r
2. csc 3.
5. cos 6. cot 7. reference
y
x
4.
r
x
(b) x, y 8, 15
1. (a) x, y 4, 3
r 64 225 17
r 16 9 5
sin y 3
r
5
csc r
5
y 3
sin y
15
r
17
csc r
17
y
15
cos x 4
r
5
sec r
5
x 4
cos x
8
r
17
sec r
17
x
8
tan y 3
x 4
cot x 4
y 3
tan y 15
x
8
cot 8
x
y 15
© Houghton Mifflin Company. All rights reserved.
1.
Section 4.4
2. (a) x 12, y 5
(b) x 1, y 1
r 12 5 13
2
r 12 12 2
2
sin 5
y 5
r
13
13
sin 2
y
1
2
r
2
cos x 12
r
13
cos 2
x
1
2
r
2
tan y 5
5
x
12
12
tan y
1
1
x 1
csc r
13
13
y 5
5
csc 2
r
2
y
1
sec r
13
x 12
sec 2
r
2
x
1
cot 12
12
x
y 5
5
cot x 1
1
y
1
3. (a) x, y 3, 1
(b) x, y 2, 2
r 4 4 22
r 3 1 2
sin y
1
r
2
csc r
2
y
sin y 2
r
2
csc r
2
y
cos x 3
r
2
sec r
23
x
3
cos 2
x
r
2
sec r
2
x
tan y 3
x
3
cot x
3
y
tan y
1
x
cot x
1
y
4. (a) x 3, y 1
r 32 12 10
© Houghton Mifflin Company. All rights reserved.
Trigonometric Functions of Any Angle
(b) x 2, y 4
r 22 42 25
sin 10
y
1
10
r
10
sin y
4
25
r
25
5
cos x
3
310
10
r
10
cos 5
x
2
r
25
5
tan y 1
x 3
tan y 4
2
x
2
csc 10
r
10
y
1
csc 5
r
25
y
4
2
sec 10
r
x
3
sec r
25
5
x
2
cot x 3
3
y 1
cot x
2
1
y 4
2
301
Chapter 4
Trigonometric Functions
5. x, y 7, 24
6. x 8, y 15,
r 49 576 25
r 82 152 17
sin y 24
r
25
csc 25
r
y 24
sin y 15
r
17
cos x
8
r
17
cos 7
x
r
25
sec 25
r
x
7
tan y 15
x
8
csc r
17
y 15
tan y 24
x
7
cot x
7
y 24
sec r
17
x
8
cot x
8
y 15
7. x, y 5, 12
r
52
8. x 24, y 10
12 25 144 169 13
2
r 242 102 26
sin y
12
r
13
sin y 10
5
r
26 13
cos x
5
r
13
cos x 24 12
r
26
13
tan y
12
x
5
tan y
10
5
x 24
12
csc r
13
y
12
csc r
13
y
5
sec r
13
x
5
sec r
13
x
12
cot x
5
y
12
cot x
12
y
5
9. x, y 4, 10
r 16 100 229
10. x 5, y 6
r 52 62 61
sin y 529
r
29
sin y
6
661
61
r
61
cos x
229
r
29
cos 5
x
561
61
r
61
tan y
5
x
2
tan y 6 6
x 5 5
csc 29
r
y
5
csc 61
r
y
6
sec 29
r
x
2
sec 61
r
x
5
cot x
2
y
5
cot x 5
y 6
© Houghton Mifflin Company. All rights reserved.
302
Section 4.4
11. x, y 10, 8
12. x 3, y 9
r 102 82 164 2 41
sin Trigonometric Functions of Any Angle
y
8
441
r
241
41
x
10
541
cos r 241
41
r 32 92 90 310
sin y
9
310
r
310
10
cos 10
x
3
r
310
10
tan y
8
4
x 10
5
tan y 9
3
x
3
csc 41
r
y
4
csc 10
r
y
3
sec r
41
x
5
sec r
10
x
cot x
5
y
4
cot x
1
y
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
> 0 and < 0
x
y
Quadrant IV
15. cot > 0 ⇒ lies in Quadrant I or Quadrant III.
cos > 0 ⇒ lies in Quadrant I or Quadrant IV.
cot > 0 and cos > 0 ⇒ lies in Quadrant I.
16. tan > 0 and csc < 0
y
r
> 0 and < 0
x
y
Quadrant III
17. sin y 3
⇒ x2 25 9 16
r
5
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in Quadrant II ⇒ x 4
18. cos x 4
⇒ y 3
r
5
in Quadrant III ⇒ y 3
sin y 3
r
5
csc r
5
y 3
sin 3
y
r
5
csc 5
3
cos x
4
r
5
sec r
5
x
4
cos x
4
r
5
sec 5
4
tan y
3
x
4
cot x
4
y
3
tan y 3
x 4
cot 4
3
303
Chapter 4
Trigonometric Functions
19. sin < 0 ⇒ y < 0
tan 20. csc y 15
⇒ r 17
x
8
r
4
⇒ x ± 15
y 1
cot < 0 ⇒ x 15
y
15
sin r
17
r
17
csc y
15
sin y 1
r
4
csc 4
x
8
cos r
17
r
17
sec x
8
cos 15
x
r
4
sec tan 15
y
x
15
cot 15
x
8
cot y
15
21. sec r
2
⇒ y2 4 1 3
x 1
22. sin 0 and
3
≤ ≤
⇒ 2
2
cos cos 1
sin ≥ 0 ⇒ y 3
sin 0
cos sin y 3
r
2
csc r
23
y
3
tan cos x
1
r
2
sec r
2
x
csc is undefined.
tan y
3
x
cot 3
x
y
3
sec 1
cot is undefined.
24. tan is undefined and ≤ ≤ 2 ⇒ 23. cot is undefined ⇒ n.
3
≤ ≤
⇒ , y 0, x r
2
2
sin y
0
r
csc r
is undefined.
y
cos x r
1
r
r
sec r
1
x
tan y 0
0
x
x
cot is undefined.
x 1, y 1, r 2
2
1
2
2
cos 2
1
2
2
tan 1
csc 2
sec 2
cot 1
3
.
2
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.
sin 415
15
© Houghton Mifflin Company. All rights reserved.
304
Section 4.4
26.
x, 3 x Quadrant III, x > 0
1
r
x
2
Trigonometric Functions of Any Angle
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.
10x
1
x2 9
3
x 1, y 2, r 5
sin y
x3
r
10
10x3
sin cos x
x
310
r
10
10x3
cos tan y 13 x 1
x
x
3
tan csc r
10x3 10
y
13 x
csc 10
10x 3
sec r
x
cot x
x
3
y 13 x
x
305
10
sec 3
cot 2
25
5
5
5
1
5
5
2
2
1
5
2
5
1
5
2
5
1 1
2 2
4
28. 4x 3y 0 ⇒ y x
3
x, 3 x Quadrant IV, x > 0
4
© Houghton Mifflin Company. All rights reserved.
r
x
2
16 2 5
x x
9
3
sin y 43 x
4
r
53 x
5
csc cos x
x
3
r
53 x 5
sec tan y 43 x
4
x
x
3
cot 29. x, y 1, 0
sec 32. csc 0 r
1
1
x 1
r
1
⇒ undefined
y 0
5
4
5
3
30. tan
3
4
y 1
⇒ undefined
2
x 0
since
corresponds to 0, 1.
2
cot x
1
⇒ undefined
y
0
cot
33. x, y 1, 0
sec 0 35. x, y 1, 0
31. x, y 0, 1
32 xy 10 0
34. csc
r
1
1
x 1
36. csc
2
1
sin
2
1
1
1
3
2
1
1
1
3
1
sin
2
306
Chapter 4
Trigonometric Functions
37. 120
38. 225 180 45
y
y
180 120 60
θ ′ = 60⬚
θ = 225⬚
θ = 120⬚
x
x
θ ′ = 45⬚
40. 330 360 30
39. 135 is coterminal with 225.
225 180 45
y
y
θ = −330⬚
θ ′ = 30⬚
x
θ ′ = 55⬚
x
θ = − 235⬚
5
5 , 2 3
3
3
42. y
y
θ = 5π
3
θ = 3π
4
θ ′ = π4
θ ′ = π3
43. 3 4
4
x
x
5
7
is coterminal with
.
6
6
7
6
6
44. 2
3
3
y
y
x
x
θ′= π
6
45. 208
θ = − 5π
6
θ ′ = π3
θ = − 2π
3
46. 322
y
y
360 322 38
208 180 28
θ = 208°
θ ′ = 28°
θ = 322°
x
x
θ ′ = 38°
© Houghton Mifflin Company. All rights reserved.
41. Section 4.4
47. 292
48. 165 lies in
Quadrant III.
y
360 292 68
θ = −292°
Trigonometric Functions of Any Angle
y
Reference angle:
180 165 15
θ ′ = 68°
x
307
θ ′ = 15⬚
x
θ = −165⬚
49. 11
is coterminal with .
5
5
50. 17 14 3
7
7
7
y
y
5
θ ′ = 3π
7
x
θ′= π
5
θ = 17π
7
x
θ = 11 π
5
51. 1.8 lies in Quadrant III.
52. lies in Quadrant IV.
Reference angle: 1.8 1.342
Reference angle: 4.5 1.358
y
y
θ = 4.5
x
θ ′ = 1.342
x
θ = − 1.8
θ ′= 1.358
54. 300, 360 300 60, Quadrant IV
© Houghton Mifflin Company. All rights reserved.
53. 45, Quadrant III
sin 225 sin 45 2
cos 225 cos 45 sin 300 sin 60 2
2
cos 300 cos 60 2
2
1
2
tan 300 tan 60 3
tan 225 tan 45 1
55. 750 is coterminal with 330, Quadrant IV.
360 330 30
56. 495, 45, Quadrant III
sin495 sin 45 sin750 sin 30 cos750 cos 30 3
1
2
cos495 cos 45 3
tan 495 tan 45 1
2
tan750 tan 30 2
3
3
2
2
2
Chapter 4
57. 5
, Quadrant IV
3
3
3 , 4
4
4
58. 5 3
3
sin
3 2
4
2
53 sin3 23
cos
2
3
4
2
53 cos3 12
tan
3
1
4
2 sin
Trigonometric Functions
cos
tan
53 tan3 3
59. , Quadrant IV
6
6 sin 6 2
sin 1
6 cos 6 cos 3
2
61. sin
cos
3
6 tan 6 3
tan 4
, 3
3
60. , Quadrant II
4
tan
3
4
3
2
4
1
3
2
4
3
3
10
4
is coterminal with .
3
3
62. sin
11
2
sin 4
4
2
cos
2
11
cos 4
4
2
sin
3
10
sin 3
3
2
tan
11
tan 1
4
4
cos
10
1
cos 3
3
2
tan
10
tan 3
3
3
63. 17
7
is coterminal with
.
6
6
sin
17
1
sin
6
6
2
cos
3
17
cos
6
6
2
tan
3
17
tan
6
6
3
sin cos tan 20
, 3
3
64. 7
, Quadrant III
6
6
4
in Quadrant III.
3
3
20
3
3
2
20
1
3
2
20
3
3
© Houghton Mifflin Company. All rights reserved.
308
Section 4.4
sin 65.
3
5
66.
1 32 csc 2 cos2 1 sin2 3
1 5
cos2 1 cos2 10 csc 2 2
csc > 0 in Quadrant II.
10 csc 9
25
16
25
cos > 0 in Quadrant IV.
cos 67.
cot 3
1 cot 2 csc 2 sin2 cos2 1
cos2 Trigonometric Functions of Any Angle
csc 1
sin sin 10
1
1
csc 10
10
4
5
68. cos csc 2
1 cot 2 csc 2 cot 2 csc 2 1
cot 2 22 1
cot 2 3
5
8
cos 1
1
⇒ sec sec cos sec 1
8
58 5
cot < 0 in Quadrant IV.
cot 3
69.
sec 9
4
70.
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tan2 sec 2 1
4
tan 2 9
2
1
65
16
tan > 0 in Quadrant III.
tan 65
4
5
4
⇒ cot 4
5
1 cot2 csc2 1 tan2 sec 2 tan2 tan 1
16
csc2 25
csc ±
41
5
Quadrant IV ⇒ csc 41
5
309
310
Chapter 4
71. sin Trigonometric Functions
2
and cos < 0 ⇒ is in Quadrant II.
5
cos 1 sin2 1 254 21
5
tan sin 25
2
221
cs 215 21
21
csc 1
5
sin 2
sec 1
5
521
cos 21
21
cot 1
21
tan 2
3
72. cos and sin < 0 ⇒ is in Quadrant III.
7
1 499 7 40 27 10
tan sin 2107 210
cs 37
3
cot 1
3
310
tan 210
20
csc 1
7
710
sin 210
20
sec 1
7
cos 3
73. tan 4 and cos < 0 ⇒ is in Quadrant II.
sec 1 tan2 1 16 17
cos 1
1
17
sec 17
17
sin tan cos 4 17
csc 17
1
17
sin 417
4
cot 1
1
tan 4
17
4 1717
74. cot 5 and sin > 0 ⇒ is in Quadrant II.
tan 1
1
cot 5
sec 1 tan2 cos 1 251 5 26
1
5
526
sec 26
26
15526 26 sin tan cos csc 1
26
26
sin 26
26
26
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sin 1 cos2 Section 4.4
1
2
csc 3
cos cos 1 sin2 1 49 5
1
3
sec 4
sin 1 cos2 3
1 169 47
sec 1
3
35
cos 5
5
csc 1
4
47
sin 7
7
tan sin 23
2
25
cos 53 5
5
tan sin 74 7
cos 34
3
cot 1
5
tan 2
cot 1
3
37
tan 7
7
77. sin 10 0.1736
80. csc 320 78. sec 235 1
1.5557
sin 320
311
4
76. sec and cot > 0 ⇒ is in Quadrant III.
3
3
75. csc and tan < 0 ⇒ is in Quadrant IV.
2
sin Trigonometric Functions of Any Angle
1
1.7434
cos 235
79. tan 245 2.1445
82. cot 220 81. cos110 0.3420
1
tan220
1.1918
83. sec280 1
cos280
84. csc 0.33 1
3.0860
sin 0.33
85. tan
29 0.8391
5.7588
86. tan
11
0.8391
9
87. csc 8
9
1
8
sin 9
88. cos
0.9749
15
14 © Houghton Mifflin Company. All rights reserved.
2.9238
89. (a) sin 1
⇒ reference angle is 30 or
2
90. (a) cos 2
2
⇒ reference angle is 45º or
and is in Quadrant I or Quadrant II.
6
and is in Quadrant I or IV.
4
Values in degrees: 30, 150
Values in degrees: 45, 315
Values in radian:
(b) sin 5
,
6 6
1
⇒ reference angle is 30 or
2
and is in Quadrant III or Quadrant IV.
6
Values in degrees: 210, 330
Values in radians:
7 11
,
6 6
Values in radians:
(b) cos 2
2
7
,
4 4
⇒ reference angle is 45 or
and is in Quadrant II or III.
4
Values in degrees: 135, 225
Values in radians:
3 5
,
4 4
Chapter 4
91. (a) csc Trigonometric Functions
23
⇒ reference angle is 60 or
3
92. (a) csc 2 ⇒ sin 1
2
and is in Quadrant I or Quadrant II.
3
Reference angle is 45 or .
4
Values in degrees: 60, 120
Values in degrees: 225, 315
Values in radians:
2
,
3 3
5 7
,
4 4
Values in radians:
(b) cot 1 ⇒ reference angle is 45 or
(b) csc 2 ⇒ sin and is in Quadrant II or Quadrant IV.
4
Reference angle is
Values in degrees: 135, 315
Values in radians:
Values in degrees: 150, 210
Values in radians:
Values in degrees: 120, 240
7
or
4
4
2 4
,
3 3
95. (a) f g sin 30 cos 30 (b) cos 30 sin 30 1 3 1 3
2
2
2
3 1
(d) sin 30 cos 30 (e) 2 sin 30 2
2
23 34
12 23 (b) cos 60 sin 60 2
(f) cos30 cos 30 3
2
1 3 1
2
2
3
4
12 1
2
96. (a) f g sin 60 cos 60 12
5 11
,
6 6
(b) Values in degrees: 45 or 315
Values in radians:
(c) cos 60
2 or 30.
6
Values in degrees: 150, 330
5 7
,
6 6
1
(b) cos ⇒ reference angle is or 60,
2
3
and is in Quadrant II or Quadrant III.
cos 3
sin 94. (a) cot 3 ⇒
Reference angle is
(c) cos 30
2 5
,
6 6
Values in radians:
23
⇒ reference angle is or
3
6
30, and is in Quadrant II or Quadrant III.
Values in radians:
or 30.
6
Values in degrees: 30, 150
3 7
,
4 4
93. (a) sec Values in radians:
1
2
(d) sin 60 cos 60 (e) 2 sin 60 2
1
4
(f) cos60 cos 60 2
2
23 12 1 3 1 3
2
2
2
3
3
3
1
2
3
4
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312
Section 4.4
2
97. (a) f g sin 315 cos 315 2
(b) cos 315 sin 315 (c) cos 315
2 2
2
2 2 2
2
(f) cos315 cos315 2
2
2
2
1
2
© Houghton Mifflin Company. All rights reserved.
(b) cos 300 sin 300 2
(d) sin 300 cos 300 101. (a) f g sin
(b) cos
2 2 2
(f) cos225 cos225 (e) 2 sin 150 2
3 3
2
4
2
2
12 1
(f ) cos150 cos150 3
2
3 1 1 3
2
2
2
1
3
1 3
2
2
2
2 3 1
4
(e) 2 sin 300 2
2 312 4 3
(f) cos300 cos300 3
1
2
7
7
1 3 1 3
cos
6
6
2
2
2
7 3
1
1 3
7
sin
6
6
2
2
2
2 3 43
7
7
1
3
(d) sin
cos
6
6
2
2 (c)
(e) 2 sin 225 2
1 3 1 3
2
2
2
3
4
100. (a) f g sin 300 cos 300 12
2
3 1 1 3
2
2
2
(b) cos 150 sin 150 (d) sin 150 cos 150 2
99. (a) f g sin 150 cos 150 2
2
0
2
2
2
3
2
2
2 2 21
2 2
1
(d) sin 225 cos 225 2 2 2
(c) cos 225
2 2
(b) cos 225 sin 225 (c) cos 300
2 (d) sin 315 cos 315 2
2
cos 76
2
2
(e) 2 sin
3
4
313
2 2 22 12
2
2
(e) 2 sin 315 2
2 0
22 12
98. (a) f g sin 225 cos 225 (c) cos 150
2 Trigonometric Functions of Any Angle
(f) cos
7
1
2 1
6
2
76 cos76 2 3
Chapter 4
Trigonometric Functions
5
5 1 3 1 3
cos
6
6
2
2
2
102. (a) f g sin
(b) cos
5
5 3 1 1 3
sin
6
6
2
2
2
2 3 34
5
5
1 3
3
(d) sin
cos
6
6
2
2 4
(c)
cos 56
2
2
(c)
2
(c)
56 cos56 2 3
1
4
(e) 2 sin
12 3
4
(f) cos
4
3
2
3
3
2
43 cos43 12
5
5 3 1 1 3
cos
3
3
2
2
2
5
5 1
3
1 3
sin
3
3
2
2
2
cos 53 12
2
(d) sin
2
4
3
4
cos
3
3
2
104. (a) f g sin
(b) cos
3 1
4
3
4
1
sin
3
3
2
2
2
cos 43 12
(d) sin
(f) cos
5
1
1
2
6
2
4
4 3 1 1 3
cos
3
3
2
2
2
103. (a) f g sin
(b) cos
(e) 2 sin
2
1
4
5
5
3
cos
3
3
2
(e) 2 sin
12 4 3
105. (a) f g sin 270 cos 270 1 0 1
(f) cos
5
3
2
3
3
2
53 cos53 12
(d) sin 270 cos 270 10 0
(b) cos 270 sin 270 0 1 1
(e) 2 sin 270 21 2
(c) cos 270
02 0
(f) cos270 cos270 0
2
106. (a) f g sin 180 cos 180 0 1 1
(d) sin 180 cos 180 01 0
(b) cos 180 sin 180 1 0 1
(e) 2 sin 180 20 0
(c) cos 180
1 1
(f) cos180 cos180 1
2
2
107. (a) f g sin
(b) cos
(c)
7
7
cos
1 0 1
2
2
7
7
sin
0 1 1
2
2
cos 72
2
02 0
(d) sin
7
7
cos
10 0
2
2
(e) 2 sin
(f) cos
7
21 2
2
72 cos72 0
© Houghton Mifflin Company. All rights reserved.
314
Section 4.4
108. (a) f g sin
(b) cos
(c)
5
5
cos
101
2
2
5
5
sin
0 1 1
2
2
cos 52
2
(f) cos
315
5
5
cos
10 0
2
2
(e) 2 sin
02 0
109. T 49.5 20.5 cos
(d) sin
Trigonometric Functions of Any Angle
5
21 2
2
52 cos52 0
6t 76
(a) January: t 1 ⇒ T 49.5 20.5 cos
61 76 29
(b) July: t 7 ⇒ T 70
(c) December: t 12 ⇒ T 31.75
110. S 23.1 0.442t 4.3 sin
6t, t 1 ↔ Jan. 2004
(a) S1 23.1 0.4421 4.3 sin
6 25.7 thousand
(b) S 14 33.0 thousand
(c) S 5 27.5 thousand
(d) S 6 25.8 thousand
Answers will vary.
111. sin 6
6
⇒ d
d
sin (a) 30
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d
6
6
12 miles
sin 30 12
(b) 90
d
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 90, the tangent
is undefined.
6
6
6 miles
sin 90 1
(c) 120
d
6
6.9 miles
sin 120
113. True. The reference angle for 151 is
180 151 29, and sine is positive
in Quadrants I and II.
114. False. cot
34 1 1 and
4 1
cot 316
Chapter 4
115. (a)
Trigonometric Functions
0
20
40
60
80
sin 0
0.3420
0.6428
0.8660
0.9848
sin180 0
0.3420
0.6428
0.8660
0.9848
(b) It appears that sin sin180 .
116.
Function
sin x
cos x
tan x
Domain
, , All reals except
Range
1, 1
1, 1
, Evenness
No
Yes
No
Oddness
Yes
No
Yes
Period
2
2
Zeros
n
n
2
n
Function
csc x
sec x
Domain
All reals except n
All reals except
Range
, 1
1, , 1
1, , Evenness
No
Yes
No
Oddness
Yes
No
Yes
Period
2
2
Zeros
None
None
n
2
n
2
cot x
n
2
All reals except n
117. 3x 7 14
118. 44 9x 61
3x 21
9x 17
x7
119. x2 2x 5 0
x
2 ± 4 20
1 ± 6
2
x 3.449, x 1.449
17
x 9 1.889
120. 2x2 x 4 0
x
1 ± 1 442 1 ± 33
4
4
x 1.186, 1.686
© Houghton Mifflin Company. All rights reserved.
Patterns and conclusions may vary.
Section 4.5
3
x2
x1
9
121.
5 x4
x
2x
122.
27 x 1x 2
10x x2 4x
x2 x 29 0
x
x2 6x 0
1 ± 1 429 1 ± 117
2
2
xx 6 0
x6
x 5.908, 4.908
123.
Graphs of Sine and Cosine Functions
43x 726
124.
3 x log4 726
(x 0 extraneous)
4500
50
4 e 2x
90 4 e2x
x 3 log4 726 3 ln 726
1.752
ln 4
86 e2x
2x ln 86
x
1
ln 86 2.227
2
125. ln x 6
x e6 0.002479 0.002
126. ln x 10 12 lnx 10 1 ⇒ lnx 10 2 ⇒ x 10 e2 ⇒ x e2 10 2.611
© Houghton Mifflin Company. All rights reserved.
Section 4.5
Graphs of Sine and Cosine Functions
■
■
You should be able to graph y a sinbx c and y a cosbx c.
Amplitude: a
■
Period:
■
Shift: Solve bx c 0 and bx c 2.
■
Key increments:
2
b
1
(period)
4
Vocabulary Check
1. amplitude
2. one cycle
3.
2
b
4. phase shift
317
318
Chapter 4
Trigonometric Functions
1. f x sin x
(a) x-intercepts: 2, 0, , 0, 0, 0, , 0, 2, 0
(b) y-intercept: 0, 0
(c) Increasing on: 2, Decreasing on: 3
3
, , ,
, 2
2
2 2
2
3 3
, , ,
2
2
2 2
(d) Relative maxima: 3
,1 , ,1
2
2
3
Relative minima: , 1 ,
, 1
2
2
2. f x cos x
(a) x-intercepts: 3
3
,0 , ,0 , ,0 ,
,0
2
2
2
2
(b) y-intercept: 0, 1
(c) Increasing on: , 0, , 2
Decreasing on: 2, , 0, (d) Relative maxima: 2, 1, 0, 1, 2, 1
Relative minima: , 1, , 1
Period:
2
2
Amplitude: 3 3
5. y 5
x
cos
2
2
Period:
2
4
12
Amplitude:
7. y 5
5
2
2
2
sin x
3
Period:
4. y 2 cos 3x
Xmin = -4
Xmax = 4
Xscl = Ymin = -3
Ymax = 3
Yscl = 1
6. y 3 sin
Period:
2
2
3
3
2 2
b
3
Amplitude: a 2
Period:
x
3
2
2
6
b
13
Amplitude: a 3 3
8. y 2
2
Amplitude:
Xmin = -2
Xmax = 2
Xscl = 2
Ymin = -4
Ymax = 4
Yscl = 1
3
x
cos
2
2
Period:
2
2
4
b
2
Amplitude: a 3
2
Xmin = -
Xmax = Xscl = 4
Ymin = -3
Ymax = 3
Yscl = 1
Xmin = -6
Xmax = 6
Xscl = Ymin = -4
Ymax = 4
Yscl = 1
© Houghton Mifflin Company. All rights reserved.
3. y 3 sin 2x
Section 4.5
9. y 2 sin x
Period:
10. y cos
2
2
1
Period:
Amplitude: 2 2
12. y 5
x
cos
2
4
Period:
Amplitude: a 5
2
2x
5
11. y 2
2
5
b
25
Amplitude: a 1 1
13. y 2
2
8
b
14
Graphs of Sine and Cosine Functions
1
sin 4x
3
Period:
2 1
4 2
Amplitude:
15. f x sin x
gx sinx 1
2x
cos
4
3
1
1
4
4
Amplitude:
3
x
cos
4
12
Period:
1
1
3
3
2
3
23
Period:
14. y 319
2
2
24
b
12
Amplitude: a 3
4
16. f x cos x, gx cos x g is a horizontal shift of f units to the left.
The graph of g is a horizontal shift to the right
units of the graph of f (a phase shift).
17. f x cos 2x
gx cos 2x
The graph of g is a reflection in the x-axis of the
graph of f.
19. f x cos x
gx 5 cos x
© Houghton Mifflin Company. All rights reserved.
The graph of g has five times the amplitude of f,
and reflected in the x-axis.
21. f x sin 2x
gx 5 sin 2x
18. f x sin 3x, gx sin3x
g is a reflection of f about the y-axis.
(or, about the x-axis)
20. f x sin x, gx 12 sin x
The amplitude of g is one-half that of f. g is a
reflection of f in the x-axis.
22. f x cos 4x, gx 6 cos 4x
g is a vertical shift of f six units downward.
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.
24. The period of g is one-half the period of f.
25. The graph of g is a horizontal shift units to the
right of the graph of f.
26. Shift the graph of f two units upward to obtain the
graph of g.
Chapter 4
320
Trigonometric Functions
27. f x sin x
28. f x sin x
Period: 2
gx sin
Amplitude: 1
x
3
gx 4 sin x
Period: 2
Amplitude: 4 4
x
0
2
3
2
2
sin x
0
1
0
1
0
x
3
0
1
2
3
y
4
1
2
3
2
f
1
−
sin
g
y
3π
2
x
π
2
2
−2
g f
−3
−4
x
6π
–2
30. f x 2 cos 2 x
29. f x cos x
gx cos 4x
Period: 2
Amplitude: 1
gx 4 cos x is a vertical shift of the graph of
f x four units upward.
y
6
x
0
4
2
3
4
2 cos 2x
2
0
2
0
2
cos 4x
1
1
1
1
1
g
4
y
3
2
2
f
−1
2π
g
−2
π
–2
1
x
31. f x sin
2
2
y
4
Period: 4
3
1
Amplitude:
2
1
x
1
sin is the graph of f x
2
2
shifted vertically three units upward.
gx 3 g
2
− 4π
−π
−1
−2
−3
−4
f
2π 3π 4π
x
x
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− 2π
f
x
Section 4.5
32. f x 4 sin x
Graphs of Sine and Cosine Functions
33. f x 2 cos x
gx 4 sin x 2
Period: 2
Amplitude: 2
x
0
1
2
1
3
2
2
f x
0
4
0
4
0
gx
2
2
2
6
2
gx 2 cosx is the graph of f x shifted
units to the left.
y
3
f
y
4
3
2
−1 1
−2
π
f
g
2
x
–3
1
−1
−2
g
35. f x sin x, gx cos x 34. f x cos x
gx cos x 2
sin x cos x x
0
2
3
2
2
cos x
1
0
1
0
1
0
1
0
1
0
cos x 2
Amplitude: 1
2
f=g
−2␲
2␲
−2
g
1
− 2π
© Houghton Mifflin Company. All rights reserved.
2
2
Period: 2
y
f
x
2π
2π
x
−π
π
−1
36. f x sin x, gx cos x x
0
0
sin x
cos x 2
2
1
0
2
2
f=g
−2␲
3
2
2
1
0
2␲
−2
Conjecture: sin x cos x 0
1
0
1
0
2
321
Chapter 4
322
Trigonometric Functions
38. fx cos x, gx cosx 37. f x cos x
gx sin x sin
x cos x
2
2
Thus, f x gx.
x
0
2
3
2
2
cos x
1
0
1
0
1
cosx 1
0
1
0
1
2
f=g
−2␲
2␲
2
f=g
−2
−2␲
2␲
−2
Conjecture: cos x cosx 40. y 39. y 3 sin x
Period: 2
1
cos x
4
Period: 2
Amplitude: 3
3
, 3 , , 0,
, 3 , 2, 0
Key points: 0, 0,
2
2
Amplitude:
y
1
4
y
4
1
3
2
0.5
1
3π
2
−
π
2
π
2
3π
2
π
2
−0.5
π
2π
x
−1
−4
41. y cos
π
2
−
x
2
42. y sin 4x
Period: 4
Period:
Amplitude: 1
Key points: 0, 1, , 0, 2, 1, 3, 0, 4, 1
2 4
2
Amplitude: 1
y
y
2
3
1
2
−1
−2
−3
2π
4π
x
−
3π
8
π
8
−2
3π
8
5π
8
x
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−
x
Section 4.5
; a 1, b 1, c 4
4
43. y sin x Graphs of Sine and Cosine Functions
y
3
Period: 2
2
1
Amplitude: 1
Shift: Set x 0 and x 2
4
4
9
x
x
4
4
3
5
7
9
Key points:
,0 ,
,1 ,
,0 ,
, 1 ,
,0
4
4
4
4
4
44. y sinx −π
x
π
–2
–3
y
Horizontal shift units to the right
2
1
−
π
2
−1
π
2
3π
2
x
−2
45. y 8 cosx y
10
8
Period: 2
Amplitude: 8
2
Key points: , 8, , 0 , 0, 8, , 0 , , 8
2
2
46. y 3 cos x 2
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−π
π
−2
−4
−6
−8
−10
47. y 2 sin
Amplitude: 3
Horizontal shift
− 2π
2π
2 x
3
Amplitude: 2
units to the left
2
Period:
2
3
23
y
4
4
3
−6
6
1
x
−
π
2
−2
π
2
3π
2
−4
−3
−4
Note: 3 cos x 3 sin x
2
x
323
Chapter 4
324
Trigonometric Functions
48. y 10 cos
x
6
49. y 4 5 cos
50. y 2 2 sin
Amplitude: 5
Amplitude: 10
Period:
t
12
2
12
6
Period:
Amplitude: 2
2
24
12
12
Period:
2
3
23
6
20
−18
18
2x
3
−30
30
−6
51. y 2
x
cos 3
2
4
52. y 3 cos6x 2
Amplitude:
3
Period:
−2
−20
−12
2
4
12
6
53. y 2 sin4x Amplitude: 3
Amplitude: 2
2 Period:
6
3
Period:
2
4
6
2
−
− 4π
π
2
π
2
5π
−␲
␲
−4
−6
−2
54. y 4 sin
3x 3 2
55. y cos 2x 1
2
Amplitude: 1
Period: 1
Amplitude: 4
Period: 3
56. y 3 cos
x
2
3
2
Amplitude: 3
Period: 4
3
1
8
−6
12
−3
6
3
−1
−7
−8
57. y 5 sin 2x 10
58. y 5 cos 2 x 6
1
59. y 100
sin 120 t
Amplitude: 5
Amplitude: 5
Amplitude:
Period: Period: Period:
20
1
60
0.02
12
−
−␲
1
100
␲
180
␲
180
␲
−4
−␲
␲
0
−0.02
© Houghton Mifflin Company. All rights reserved.
−12
Section 4.5
60. y 1
cos50 t
100
Amplitude:
Period:
Graphs of Sine and Cosine Functions
61. f x a cos x d
1
100
Amplitude:
62. f x a cos x d
1
8 0
4
2
Amplitude:
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,
1
25
0.04
−0.06
0.06
63. f x a cos x d
Reflected in the x-axis:
a 1
4 1 cos 0 d
d 3
f x 4 cos x 4
y 3 cos x
65. f x a sinbx c
64. y a cos x d
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:
66. y a sinbx c
Amplitude: a 3
1
2
Since the graph is reflected
about the x-axis, we have
a 3.
Period: 2
Reflected in x-axis, a 1
2
2
⇒ b2
b
Phase shift: c 0
Period:
d 4
y 4 1
cos x
2
Thus, f x 3 sin 2x.
67. f x a sinbx c
Amplitude: 2 ⇒ a 2
Amplitude: a 1
Period: 4
Period: 2 ⇒ b 1
2
1
4 ⇒ b b
2
Phase shift: bx c 0 when x 1
Phase shift: c 0
© Houghton Mifflin Company. All rights reserved.
2 4
1
2
4 4 cos x.
−0.04
y 2 sin
2
Thus, f x sin x 68. y a sinbx c
y2 Period: 4
2
4 ⇒ b
b
2
.
4
⇒ c
4
.
4
69. y1 sin x
Amplitude: 2 ⇒ a 2
c
1 ⇒ c b
2
x y 2 sin
2
2
4 c 0
x
Phase shift:
325
1
2
In the interval 2, 2
, sin x 12 when
x
5 7 11
, , ,
.
6
6 6 6
2
y1
−2␲
2␲
y2
−2
Chapter 4
326
Trigonometric Functions
71. v 0.85 sin
70. y1 cos x
y2 1
(a)
y1 y2 when x , .
t
3
2
4␲
0
2
y1
−2␲
−2
2␲
y2
(b) Time for one cycle one period −2
(c) Cycles per min 2
6 sec
3
60
10 cycles per min
6
(d) The period would change.
72. S 74.50 43.75 cos
(a)
t
6
73. h 25 sin
(a)
120
1
t 75 30
15
60
0
12
135
0
0
(b) Minimum: 30 25 5 feet
(b) Maximum sales: December t 12
Maximum: 30 25 55 feet
Minimum sales: June t 6
74. P 100 20 cos
8
t
3
Period:
130
3
2
83 4
4
1 heartbeat
⇒ heartbeatssecond 80 heartbeatsminute
34
3
1.5
60
75. C 30.3 21.6 sin
(a) Period:
2t
365 10.9
2
2
365 days
b
2365
(c)
60
This is to be expected: 365 days 1 year
(b) The constant 30.3 gallons is the average daily
fuel consumption.
0
365
0
Consumption exceeds 40 gallonsday when
124 ≤ x ≤ 252. (Graph C together with y 40.)
(Beginning of May through part of September)
© Houghton Mifflin Company. All rights reserved.
0
Section 4.5
76. (a) Yes, y is a function of t because for each
value of t there corresponds one and only
one value of y.
Graphs of Sine and Cosine Functions
327
(c) One model is y 0.35 sin 4 t 2.
(d)
4
(b) The period is approximately
20.375 0.125 0.5 seconds.
0
The amplitude is approximately
1
2 2.35 1.65 0.35 centimeters.
y
77. (a)
0.9
0
y
(c)
2.0
2.0
1.5
1.5
1.0
1.0
0.5
−10
x
10 20 30 40 50 60 70
− 0.5
−10
x
10 20 30 40 50 60 70
− 0.5
(b) y 0.506 sin0.209x 1.336 0.526
The model is a good fit.
(d) The period is
2
30.06.
0.209
(e) June 29, 2007 is day 545. Using the model,
y 0.2709 or 27.09%.
78. (a) At 19.73 sin0.472t 1.74 70.2
(b)
(d) Nantucket: 58
Athens: 70.2
80
The constant term (d) gives the average daily high
temperature.
0
12
0
(e) Period for Nt The model somewhat fits the data.
(c)
© Houghton Mifflin Company. All rights reserved.
Period for At 95
2
11
211
2
13
0.472
You would expect the period to be 12 (1 year).
0
12
45
(f) Athens has greater variability. This is given by the
amplitude.
The model is a good fit.
79. True. The period is
20
2
.
310
3
80. False. The amplitude is
1
that of y cos x.
2
81. True
82. Answers will vary.
83. The graph passes through 0, 0 and has period .
84. The amplitude is 4 and the period 2. Since
0, 4 is on the graph, matches (a).
Matches (e).
85. The period is 4 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).
Chapter 4
328
Trigonometric Functions
87. (a) hx cos2 x is even.
h(x) = cos2 x
2
2
−␲
(c) hx sin x cos x is odd.
(b) hx sin2 x is even.
␲
h(x) = sin2 x
−␲
2
−␲
␲
−2
h(x) = sin x cos x
␲
−2
−2
88. (a) In Exercise 87, f x cos x is even and we saw that hx cos 2 x is even.
Therefore, for f x even and hx f x
2, we make the conjecture that hx is even.
(b) In Exercise 87, gx sin x is odd and we saw that hx sin 2x 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
sin x
x
0.8415
0.9983
1.0
1.0
(b)
1.1
−1
1
0.8
0
0.001
0.01
0.1
1
sin x
x
Undef.
1.0
1.0
0.9983
0.8415
As x → 0, f x sin x
approaches 1.
x
(c) As x approaches 0,
90. (a)
x
1
0.1
0.01
0.001
1 cos x
x
0.4597
0.05
0.005
0.0005
x
0
1 cos x
x
Undef. 0.0005 0.005
0.001
0.01
0.1
1
0.05
0.4597
(b)
0.5
−1
1
−0.5
As x → 0,
1 cos x
approaches 0.
x
(c) As x approaches 0,
91. (a)
(c) Next term for sine approximation: 2
−2 π
2π
x7
7!
Next term for cosine approximation: x6
6!
−2
2
(b)
2
2
−2 π
−2 π
2π
−2 π
2π
2π
−2
−2
sin x
approaches 1.
x
−2
1 cos x
approaches 0.
x
© Houghton Mifflin Company. All rights reserved.
x
Section 4.6
92. (a) sin1 0.8415
1
0.4794
(b) sin
2
0.3827
(c) sin
8
(d) cos1 0.5403
0.7071
(e) cos 4
1
(f ) cos 0.8776
2
Graphs of Other Trigonometric Functions
y
93.
6
5
4
3
2
1
In all cases, the approximations are very accurate.
−4 −3 −2 −1
−1
Slope (−1, 4)
4
(0, 1)
x
1
2
3
4
71
3
20
95. 8.5 8.5
y
94.
(2, 7)
7
487.014
180
3
2
1
−4 −3 −2 −1
−1
−2
x
1
2
3
4
(3, −2)
−3
−4
m
2 4 6 3
31
4
2
96. 0.48 0.48
Section 4.6
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■
■
■
27.502
180
97. Answers will vary. (Make a Decision)
Graphs of Other Trigonometric Functions
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
329
Chapter 4
330
Trigonometric Functions
1. f x tan x
(a) x-intercepts: 2, 0, , 0, 0, 0, , 0, 2, 0
(c) Increasing on:
(b) y-intercept: 0, 0
2, 32, 32, 2 , 2 , 2 , 2 , 32, 32, 2
(d) No relative extrema
Never decreasing
(e) Vertical asymptotes: x 3 3
, , ,
2
2 2 2
2. f x cot x
(a) x-intercepts:
32, 0, 2 , 0, 2 , 0, 32, 0
(b) No y-intercepts
(d) No relative extrema
(c) Decreasing on: 2, , , 0, 0, , , 2
Never increasing
(e) Vertical asymptotes: x 2, , 0, , 2
3. f x sec x
4. f x csc x
(a) No x-intercepts
(a) No x-intercepts
(b) y-intercept: 0, 1
(b) No y-intercepts
(c) Increasing on intervals:
(c) Increasing on intervals:
2, 32, 32, , 0, 2 , 2 , 32, , , 2 , 2 , , , 32
Decreasing on intervals:
Decreasing on intervals:
, 2 , 2 , 0, , 32, 32, 2
2, 32, 2 , 0, 0, 2 , 32, 2
(d) Relative minima: Relative maxima: , 1, , 1
1
6. y tan x
4
Period: Period: 4
x
y
1
2
0
4
0
1
2
and x 2
2
Asymptotes: x ±
2
y
y
4
3
3
2
2
1
1
−
π
2
π
2
π
(e) Vertical asymptotes: x ± , ± 2
1
tan x
2
Two consecutive asymptotes: x 3
Relative maxima: , 1 ,
, 1
2
2
3 3
(e) Vertical asymptotes: x , , ,
2
2 2 2
5. y 3
,1 , ,1
2
2
x
π
−1
−2
−3
π
x
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(d) Relative minima: 2, 1, 0, 1, 2, 1,
Section 4.6
8. y 3 tan 4x
7. y 2 tan 2x
Period:
2
Period:
Two consecutive asymptotes:
2x 2x ⇒ x
2
4
y
6
x
8
0
8
y
2
0
2
4x − 3π
4
−π
4
π
4
3π
4
−8
⇒ x
2
8
1
10. y 4 sec x
3
y
3
Period: 2
2
Period: 2
2
1
One cycle: 0 to 2
1
−π
x
π
11. y sec x 3
− 2π
1
x
−2
−1
1
2
−3
One cycle: 0 to 4
2 4
2
π
2
6
4
−π
4
x
16
0
16
y
0.828
0
0.828
14. y csc
10
8
6
4
2
Period:
x
−8
−10
π
4
x
y
x ,x
8
8
y
− 3π − 2π − π
2π
2
−5
2
Period:
4
12
Period:
π
Asymptotes:
−4
x
first.
2
−π
12. y 2 sec 4x 2
y
Period: 2
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−4
x
y
x
2
x
π
2
−6
Graph y 12 cos x first.
Graph y 3 sin
4
−4
9. y 12 sec x
Shift graph of sec x
down three units.
331
y
Asymptotes:
4x ⇒ x
2
8
⇒ x
2
4
13. y 3 csc
Graphs of Other Trigonometric Functions
π
2π 3π 4π
x
3
x
y
3
2
6
13
− 4π
Asymptotes:
−π
−1
−2
x 0, x 3
−3
x
2
4
y
1.155
1.155
1.155
π
4π
x
Chapter 4
15. y Trigonometric Functions
1
x
cot
2
2
16. y 3 cot x
y
6
Period:
2
12
Period:
−π
Two consecutive
asymptotes:
Period:
x
2
3
2
y
1
2
0
4
4
2
−6
−2
2
4
6
x
x ⇒ x2
4
2
y
1
2
0
2
19. y 2
3
1
sec2x 2
2
1
1
4
x
−1
4
0
1
4
0
−3
1
2
y
8
6
2
Asymptotes: x ±
1
x
−π
2
−1
−2
Period: 2
3
Asymptotes: x ±
4
1
20. y secx y
2
Period:
2
y
3
Asymptotes:
1
1
x ,x
2
2
x
⇒ x 2
4
2
2
1
Period: 1
4
Two consecutive
asymptotes:
y
x
−1
1
18. y tan x
2
x
1
−2
1
2
6
0
−3
−3
y
1
1
−6
x
4
x
2
−4
x
0 ⇒ x0
2
x
⇒ x 2
2
17. y 2 tan
3
1
Asymptotes:
x 0, x 1
x
π
y
π
2
2
−2π
−2
−4
−6
−8
21. y csc x
y
6
Graph y sin x first.
4
Period: 2
2
Asymptotes: Set x 0 and x 2
x
x −2π
−π
π
2π
x
2π
x
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332
Section 4.6
22. y csc2x Period:
23. y 2 cot x 2
2
π
2
π
2
−1
−2
−3
−4
−5
π
25. y 2 csc 3x Period: y
−
−π
0 ⇒ x
2
2
x
3
⇒ x
2
2
x
3
4
5
4
y
2
0
2
2
sin3x
2
Period:
3
−4
−6
26. y csc4x y
1
sin4x 10
−␲
π
2
−2
−3
−4
−5
π
2
␲
−␲
2
␲
2
x
3π
2
−3
−10
3
10
−␲
␲
−␲
2
␲
2
−3
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−10
27. y 2 sec 4x
y
π
−2
3
5
4
3
2
1
3π
2
y
x
x
1
24. y cotx 4
−
Two consecutive
asymptotes:
y
−
2
28. y 2
cos 4x
1
1
sec x 4
4 cos x
29. y 2
4
−␲
2
−2
2
1
x sec
3
2
2
1
x 3 cos
2
2
Period: 4
␲
2
−2
2
2
−4
−6
−2
4
6
2
−2
−␲
2
333
Period: 5
4
3
−π
Graphs of Other Trigonometric Functions
␲
2
−2
2
−4
−6
6
−2
x
Chapter 4
334
30. y Trigonometric Functions
1
csc2x 2
31. tan x 1
3
x
3
−␲
−␲
␲
−3
␲
−3
32. cot x 3
33. sec x 2
The solutions appear to be:
x±
7 5 11
, , ,
6
6 6 6
(or in decimal form: 3.665, 0.524, 2.618, 5.760)
x
34. csc x 2
The solutions appear to be:
7 3 5
, , ,
4
4 4 4
2 4
,±
3
3
35. The graph of f x sec x has y-axis symmetry.
Thus, the function is even.
7 5 3
, , ,
4
4 4 4
(or in decimal form: 5.498, 3.927, 0.785, 2.356)
36. f x tan x
37. The function
tanx tan x
Thus, the function is odd and the graph of y tan x
is symmetric with the origin.
f x csc 2x has origin symmetry. Thus, the function is odd.
39. y1 sin x csc x and y2 1
38. f x cot 2 x
f x cot2x
2
cos2x
cos2x
f x
sin2x
sin2x
−3
3
Odd
−2
Not equivalent because y1 is not defined at 0.
sin x csc x sin x
40. y1 sin x sec x, y2 tan x
41. y1 sin x 1,
1
sin x 0
cos x
1
and y2 cot x tan
x
sin x
4
Equivalent
−2␲
2␲
−4
cos x
sin x
−2␲
2␲
−4
It appears that y1 y2.
sin x sec x sin x
cot x 4
1
sin x
tan x
cos x cos x
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1
sin 2x
Section 4.6
42. y1 sec 2 x 1, y2 tan2 x
Graphs of Other Trigonometric Functions
335
43. f x x cos x
As x → 0, f x → 0.
3
Odd function
−
3␲
2
3␲
2
f
−1
32 0
Matches graph (d).
It appears that y1 y2.
1 tan2 x sec2 x
tan2 x sec2 x 1
44. f x x sin x
45. gx x sin x
Matches graph (a) as x → 0,
f x → 0, and f x ≥ 0 for all x.
46. gx x cos x
As x → 0, gx → 0.
Even function
Odd function
Matches graph (c) as
x → 0, gx → 0.
g2 0
Matches graph (b).
47. f x sin x cos x , gx 0
2
48. f x sin x cos x f x gx
f=g
−2␲
2␲
1
49. f x sin2 x, gx 1 cos 2x
2
50. f x cos2
2
gx © Houghton Mifflin Company. All rights reserved.
f=g
−2␲
2␲
cos2
x→
2 sin x.
2
x
2
2
f=g
1
1 cos x
2
−2␲
−2
2
−3
6
f x → 0 as x →
, f x → 0
−2
54. gx ex 2 sin x
2
2
2
Damping factor: ex 4
1
Damping factor: y ex 2
2
2
h → 0 as x →
3
−3
53. h x ex 4 cos x
2␲
x 1
1 cos x.
2
2
Damping factor: e2x
−3
2␲
−3
52. f x e2x sin x
3
Damping factor: ex
f=g
−2␲
It appears that f x gx.
That is, that
−2
51. f x ex cos x
3
It appears that f x gx.
That is,
sin x cos x −2
f x gx
gx 2 sin x
2
The graph is the line y 0.
2
−3
3
−2
ex 2 ≤ gx ≤ ex 2
2
x → ± , gx → 0
−8
8
2
−1
Chapter 4
336
55. (a) As x →
Trigonometric Functions
, f x → .
2
56. f x sec x
(b) As x →
, f x → .
2
(a) As x →
, f x → .
2
(c) As x → , f x → .
2
(b) As x →
, f x → .
2
(d) As x → , f x → .
2
(c) As x → , f x → .
2
(d) As x → , f x → .
2
58. f x csc x
57. f x cot x
(a) As x → 0 , f x →
(a) As x → 0 , f x →
.
.
(b) As x → 0 , f x → .
(b) As x → 0 , f x → .
(c) As x → , f x →
(c) As x → , f x → .
.
(d) As x → , f x →
(d) As x → , f x → .
.
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
t
15.69 sin
6
6
Lt 39.36 15.70 cos
t
t
14.16 sin
6
6
(b) From the graph, it appears that the greatest difference
between high and low temperatures occurs in summer.
The smallest difference occurs in winter.
90
H
L
0
(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.
12
0
Period of cos
t 2
:
12
6 6
Period of sin
t 2
:
12
6 6
Period of Ht : 12
Period of Lt : 12
61. tan x d
5
d
5
5 cot x
tan x
62. cos x 36
0
−36
␲
d
36
d
36
36 sec x
cos x
72
−␲
2
36
␲
2
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(a)
Section 4.6
63. (a)
Graphs of Other Trigonometric Functions
337
(b) The displacement function is not periodic, but damped.
It approaches 0 as t increases.
0.6
4␲
0
−0.6
64. (a)
1700
850 revmin
2
(e) Straight line lengths change faster.
(b) The direction of the saw is reversed.
2 cot , 0 < < 2
(c) L 60
(d)
(f)
0.3
0.6
0.9
1.2
1.5
L
217.9
195.9
189.6
188.5
500
␲
2
0
306.2
0
3
and x is a vertical
2
2
2
asymptote for the tangent function.
65. True. 67. f x 2 sin x, gx (a)
66. True. For x → , 2x → 0.
1
csc x
2
(b) f x > gx for
3
f
5
< x <
6
6
(c) As x → , 2 sin x → 0 and
1
csc x → , since gx is the
2
reciprocal of f x.
g
␲
0
−1
68. (a)
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−3
4
1
1
1
sin x sin 3 x sin 5 x sin 7 x
3
5
7
(b) y3 2
3
2
−2
−3
3
2
−2
−3
3
(c) y4 y3 −2
69. (a)
x
1
0.1
0.01
0.001
tan x
x
1.5574
1.0033
1.0
1.0
4 1
sin 9 x
9
(b)
2
−1.1
1.1
0
x
0
0.001
0.01
0.1
1
tan x
x
Undef.
1.0
1.0
1.0033
1.5574
As x → 0, f x tan x
→ 1.
x
(c) The ratio approaches 1 as x approaches 0.
Chapter 4
338
70. (a)
Trigonometric Functions
x
1
0.1
0.01
0.001
tan 3x
3x
0.0475
1.0311
1.0003
1.0
(b)
1.1
−1
1
−1.1
x
0
0.001
0.01
0.1
tan 3x
3x
Undef.
1.0
1.0003
1.0311 0.0475
1
As x → 0, f x tan 3x
→ 1.
3x
(c) The ratio approaches 1 as x approaches 0.
and graph is increasing on , .
2
4 4
Matches (a).
71. Period is
73.
y = tan(x)
y= x+
and
, 1 is on the graph.
2
8
Matches (b).
74.
2 x 3 16 x 5
+
3!
5!
72. Period is
y=1+
x 2 5x 4
+
2!
4!
4
y = sec(x)
2
−3
3
−3
3
0
−2
2x3 16x5
The graphs of y1 tan x and y2 x 3!
5!
are similar on the interval , .
2 2
The graphs of y1 sec x and y2 1 are similar on the interval , .
2 2
76. 77 1
1
75. Distributive Property
x 2 5x 4
2!
4!
77. Additive Identity Property
Multiplicative Inverse Property
78. a b 10 a b 10
80. Not one-to-one
79. Not one-to-one
81. y 3x 14, x ≥
14
3,
y ≥ 0
x 3y 14, y ≥
14
3,
x ≥ 0
x 2 3y 14
y 13 x 2 14
1
f 1x 3 x 2 14, x ≥ 0
82. One-to-one
y x 513
x y 513
x3 y 5
f 1x x 3 5
© Houghton Mifflin Company. All rights reserved.
Associative Property of Addition
Section 4.7
Section 4.7
■
■
Inverse Trigonometric Functions
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 <
≤ y ≤
2
2
< y <
2
2
You should know the inverse properties of the inverse trigonometric functions.
sinarcsin x x, 1 ≤ x ≤ 1 and arcsinsin y y, ≤ y ≤
2
2
cosarccos x x, 1 ≤ x ≤ 1 and arccoscos y y, 0 ≤ y ≤ tanarctan x x and arctantan y y, ■
< y <
2
2
You should be able to use the triangle technique to convert trigonometric functions or inverse trigonometric
functions into algebraic expressions.
Vocabulary Check
2. y arccos x, 0 ≤ y ≤ 1. y sin1 x, 1 ≤ x ≤ 1
3. y tan1 x, < x <
© Houghton Mifflin Company. All rights reserved.
1. (a) arcsin
1 2
6
2. (a) y arccos
, 2
< y <
2
(b) arcsin 0 0
1
1
⇒ cos y for 0 ≤ y ≤ ⇒ y 2
2
3
(b) y arccos 0 ⇒ cos y 0 for 0 ≤ y ≤ ⇒ y 3. (a) arcsin 1 because sin 1 and ≤
≤ .
2
2
2
2
2
(b) arccos 1 0 because cos 0 1 and 0 ≤ 1 ≤ .
4. (a) arctan 1 because tan 1 and <
< .
4
4
2
4
2
(b) arctan 0 0 because tan 0 0 and < 0 < .
2
2
2
339
Chapter 4
340
5. (a) arctan
Trigonometric Functions
3
6
3
(b) arctan1 4
(b) y arctan 3 ⇒ tan y 3 ⇒ y 2
8. (a) y arccos 1
2
(b) y arcsin
2
⇒ cos y ⇒ sin y 2
3
9. (a) y sin1
2
2
(b) arcsin 7. (a) y arctan 3 ⇒ tan y 3 for ⇒ sin y 2
2
3
2
3
2
4
6. (a) arccos 2
4
< y <
⇒ y
2
2
3
3
1
2
for 0 ≤ y ≤ ⇒ y 2
3
for for ≤ y ≤
⇒ y
2
2
4
≤ y ≤
⇒ y
2
2
3
3 3 ⇒ tan y 3 3 ⇒ y 6
(b) y tan1
10. (a)
x
1.0
0.8
0.6
0.4
0.2
y
1.5708
0.9273
0.6435
0.4115
0.2014
y
(b)
2
1
x
0
0.2
0.4
0.6
0.8
1
y
0
0.2014
0.4115
0.6435
0.9273
1.5708
(c)
x
–1
1
–1
–2
2
−1
1
(d) 0, 0, symmetric to the origin
11. y arccos x
(a)
x
1
0.8
0.6
0.4
0.2
y
3.1416
2.4981
2.2143
1.9823
1.7722
(b)
y
4
3
x
0
0.2
0.4
0.6
0.8
1.0
y
1.5708
1.3694
1.1593
0.9273
0.6435
0
1
x
(c)
4
–2
−1
1
0
2 , 1, 0, no symmetry
(d) Intercepts: 0,
–1
1
2
© Houghton Mifflin Company. All rights reserved.
−2
Section 4.7
12. (a)
Inverse Trigonometric Functions
y
(b)
x
10
8
6
4
2
y
1.4711
1.4464
1.4056
1.3258
1.1071
341
2
1
x
y
0
2
4
6
8
10
0
1.1071
1.3258
1.4056
1.4464
1.4711
(c)
x
–10
–5
5
10
–2
2
−10
10
−2
(d) Horizontal asymptotes: y ±
2
13. y arctan x ↔ tan y x
14. y arccos x
x 1 ⇒ y , cos 1
3, 3 , 33, 6 , 1, 4 x
y
cos 6 23 3
⇒ x
,
6
2
15. cos10.75 0.72
16. sin1 0.56 0.59
17. arcsin0.75 0.85
18. arccos0.7 2.35
19. arctan6 1.41
20. tan1 5.9 1.40
21. tan x
8
22. cos x
arctan
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1
2
2
1
⇒ y
, cos
2
3
3
2
23. sin x
8
8
x2
5
24. tan 5
x+2
tan x
x
⇒ arcsin
2
2
4 x 2
2
x
4 x 2
⇒ arccos
⇒ arctan
θ
4
x+1
x1
arctan
10
θ
10
26. Let y be the third side. Then
y2 22 x 2 4 x 2 ⇒ y 4 x2
cos 4
x
x1
10
θ
25. Let y be the third side. Then
sin x
arccos
θ
x2
arcsin
5
4
x
y2 32 x 2 9 x 2 ⇒ y 9 x2.
sin 4 x 2
2
x
4 x 2
cos tan x
x
⇒ arcsin
3
3
9 x 2
3
x
9 x 2
⇒ arccos
⇒ arctan
9 x 2
3
x
9 x 2
342
Chapter 4
Trigonometric Functions
27. Let y be the hypotenuse. Then y 2 x 12 22 x 2 2x 5 ⇒ y x 2 2x 5.
sin cos tan x 2
x1
x1
⇒ arcsin
2
x 2x 5
2x 5
2
⇒ arccos
x 2 2x 5
2
x 2 2x 5
x1
x1
⇒ arctan
2
2
28. Let y be the hypotenuse. Then y 2 x 22 32 x 2 4x 13 ⇒ y x 2 4x 13.
sin cos tan x2
x 2 4x 13
3
x 2 4x 13
⇒ arcsin
x2
x 2 4x 13
⇒ arccos
3
x 2 4x 13
x2
x2
⇒ arctan
3
3
31. cosarccos0.3 0.3
30. tanarctan 35 35
29. sinarcsin 0.7 0.7
33. arcsinsin 3 arcsin0 0
32. sinarcsin0.1 0.1
Note: 3 is not in the range of the arcsine function.
7
arccos0 2
2
35. arctan tan
3
11
arctan 6
3
5
sin1 1 2
2
5
sin1 1 4
2
2
7
arcsin 4
2
4
37. sin1 sin
3
cos1 0 2
2
39. sin1 tan
3
cos11 4
41. tanarcsin 0 tan 0 0
36. arcsin sin
38. cos1 cos
40. cos1 tan
4 42. cosarctan1 cos 2
2
4 22
44. sinarctan1 sin 6 arccos 12 23
46. arccos sin 6
43. sinarctan 1 sin
4 © Houghton Mifflin Company. All rights reserved.
34. arccos cos
2
2
6 arcsin 23 3
45. arcsin cos 4
47. Let y arctan . Then
3
4
tan y , 0 < y < , and
3
2
4
sin y .
5
5
4
y
3
Section 4.7
3
48. Let u arcsin ,
5
sin y 3
u
24
7
, and cos y .
25
25
25
24
4
3
sec u
5
hyp 5
adj
4
12
,
5
50. Let u arctan tan u 5
343
24
. Then
25
49. Let y arcsin
3
sin u , 0 < u < .
5
2
sec arcsin
Inverse Trigonometric Functions
y
7
5. Then,
3
51. Let y arctan 34
3
tan y , < y < 0 and sec y .
5
2
5
12 , < u < 0.
5
2
y
5
x
y
5
u
13
−3
34
−12
csc arctan 12
5
csc u opp 12
hyp
13
4 ,
52. Let u arcsin 3. Then,
3
53. Let y arccos 3 sin u , < u < 0.
4
2
2
5
2 cos y ,
< y < and sin y .
3 2
3
© Houghton Mifflin Company. All rights reserved.
y
7
u
5
4
3
−3
y
−2
3
4 tan u 7 tan arcsin 3
5
54. Let u arctan ,
8
5
tan u , 0 < u < .
8
2
37
7
55. Let y arctan x. Then,
89
u
5
8
cot arctan
cot u 8
5
x
8
5
x2 + 1
1
tan y x and cot y .
x
x
y
1
344
Chapter 4
Trigonometric Functions
x
56. Let u arctan x, tan u x .
1
57. Let y arccosx 2, cos y x 2.
Opposite side: 1 x 22
sin y x2 + 1
1 x 22
1
x
1
x2 4x 3
1 − (x + 2)2
u
1
y
opp
x
sinarctan x sin u 2
hyp
x 1
58. Let u arcsinx 1, sin u x 1 x+2
x1
.
1
x
x
59. Let y arccos . Then cos y , and
5
5
tan y 1
25 x 2
x
x −1
u
5
2x − x 2
.
25 − x 2
y
hyp
1
secarcsin x 1 sec u 2x x2
adj
4
4
60. Let u arctan , tan u .
x
x
x
61. Let y arctan
csc y x 2 + 16
x
7
7 x 2
x
. Then tan y x
7
and
.
4
7 + x2
u
x
x
cot arctan
4
adj
x
cot u x
opp 4
62. Let u arcsin
xh
xh
, sin u .
r
r
7
63. Let y arctan
sin y r
x−h
14
y
u
x
r 2 − (x − h) 2
r 2 x h 2
xh
cos u r
r
14
14
. Thus, y arcsin
.
2
196 x
196 x 2
196 + x 2
cos arcsin
14
14
. Then tan y and
x
x
© Houghton Mifflin Company. All rights reserved.
y
Section 4.7
64. If arcsin
36 x 2
6
u, then sin u 36 x 2
6
.
65. Let y arccos
cos y 6
Inverse Trigonometric Functions
3
y arcsin
u
x
36 x2
arcsin
6
66. If arccos
arccos
3
. Then,
2x 10
x2 2x 10
and sin y 36 − x 2
x2
345
x 1
3
x 12 9
x 12 9
x 1
x 12 9
. Thus,
arcsin
x 1
x2 2x 10
.
x
6
x2
x2
u, 2 < x < 4 then cos u .
2
2
67. y 2 arccos x
Domain: 1 ≤ x ≤ 1
Range: 0 ≤ y ≤ 2
Vertical stretch of f x arccos x
2
4x − x 2
2␲
u
x−2
−2
arccos
4x x
x2
arctan
, since 2 < x < 4
2
x2
68. y arcsin
x
2
69. The graph of f x arcsinx 2 is a horizontal
translation of the graph of y arcsin x by two units.
Domain: 2 ≤ x ≤ 2
␲
2
Range: ≤ y ≤
2
2
© Houghton Mifflin Company. All rights reserved.
2
0
2
␲
2
0
−3
4
3
−␲
2
−␲
2
70. gt arccost 2
71. f x arctan 2x
Domain: 3 ≤ t ≤ 1
Domain: all real numbers
This is the graph of y arccos t shifted two units
to the left.
Range: ␲
< y <
2
2
␲
2
−3
−4
3
0
0
−␲
2
Chapter 4
346
Trigonometric Functions
72. f x arccos
x
4
␲
Domain: 4, 4
Range: 0, −5
5
0
74. f t 4 cos t 3 sin t
73. f t 3 cos 2t 3 sin 2t
32 32 sin 2t arctan
3
3
42 32 sin t arctan
32 sin2t arctan 1
32 sin 2t 4
5 sin t arctan
4
3
4
3
The graph suggests that
6
A cos t B sin t A2 B2 sin t arctan
−2␲
2␲
is true.
A
B
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
10
10
⇒ arcsin
s
s
(b) s 52: arcsin
s 26: arcsin
0.1935, 11.1
10
52 0.3948, 22.6
10
26 83. (a) tan 82. (a)
11 ft
s
750
arctan
34 ft
11
(b) tan ⇒ 0.5743 or 32.9
17
(c) tan0.5743 h
⇒ h 20 tan0.5743
20
12.94 feet
750s (b) When s 400,
arctan
0.49 radian, 28.
400
750 When s 1600, 1.13 radians, 65.
© Houghton Mifflin Company. All rights reserved.
81. (a) sin Section 4.7
84. arctan
(a)
Inverse Trigonometric Functions
3x
,x > 0
x2 4
(b) is maximum when x 2 feet.
1.5
0
(c) The graph has a horizontal asymptote at 0.
As the camera moves further from the picture, the
angle subtended by the camera approaches 0.
6
−0.5
6
x
85. (a) tan 86. (a) tan arctan
6x arctan
(b) When x 10,
arctan
87. False. arcsin
x
20
(b) When x 5,
arctan
6
0.54 radian, 31.
10
5
14.0, (0.24 rad).
20
When x 12,
When x 3,
arctan
x
20
63 1.11 radians, 63.
arctan
1 2
6
88. False. tan x 89. y arccot x if and only if cot y x,
< x < and 0 < y < .
12
31.0, 0.54 rad.
20
sin x
cos x
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
is , 1 1, and the range is
0, 2 2, .
y
π
y
© Houghton Mifflin Company. All rights reserved.
π
2
−2
π
x
−1
1
π
2
2
−2
91. y arccsc x if and only if csc y x.
1
y
Domain: , 1 1, Range:
x
−1
π
2
2 , 0 0, 2 −2
x
−1
1
−π
2
2
2
347
348
Chapter 4
Trigonometric Functions
92. (a) y arccot x if and only if x cot y, < x <
Thus,
and 0
< y < .
1
1
tan y and y arctan
.
x
x
arctan1x, x < 0
Hence, graph y 2,
x 0.
arctan1x,
x > 0
(b) y arcsec x if and only if x sec y, x ≤ 1 or x ≥ 1, and 0 ≤ y <
1
1
1
cos y and y arccos
. Hence graph y arccos , x , 1 1, .
x
x
x
(c) y arccsc x if and only if x csc y, x ≤ 1 or x ≥ 1, and Thus,
≤ y < 0 or 0 < y ≤ .
2
2
1
1
1
sin y and y arcsin
. Hence, graph y arcsin
, x , 1 1, .
x
x
x
93. y arcsec 2 ⇒ sec y 2 and 0 ≤ y <
94. y arcsec 1 ⇒ sec y 1 and 0 ≤ y <
< y ≤ ⇒ y
2 2
4
< y ≤ ⇒ y0
2 2
95. y arccot 3 ⇒ cot y 3 and 0 < y < ⇒ y 96. y arccsc 2 ⇒ csc y 2 and 5
6
≤ y < 00 < y ≤
⇒ y
2
2
6
97. Let y arcsinx. Then,
sin y x
y arctanx
98.
tan y x, sin y x
< y <
2
2
tan y x
siny x
y arcsin x
tany x, y arcsin x.
Therefore, arcsinx arcsin x.
< y <
2
2
arctantany arctan x
y arctan x
y arctan x
99. arcsin x arccos x 2
Let arcsin x ⇒ sin x.
Let arccos x ⇒ cos x.
Hence, sin a cos ⇒ a and are complementary angles ⇒ a ⇒ arcsin x arccos x .
2
2
© Houghton Mifflin Company. All rights reserved.
Thus,
or < y < .
2
2
Section 4.7
Inverse Trigonometric Functions
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
0 0.785
4
4
(c) a 0, b 3
Area arctan 3 arctan1
1.25 0 1.25
103.
42
2
1
2
2
102.
23 3
6
3
105. sin 104.
5
6
6
5
© Houghton Mifflin Company. All rights reserved.
23
33
sin cos 23
3
2
5
2
25
5
5
5
2
θ
5
1
5
2
sec 5
6
11
611
11
cot 1
2
11
5
3
4
108. sec 3, 0 < <
4
3
7
4
θ
sin 22
3
cos 1
3
3
37
7
7
csc 4
3
csc sec 4
47
7
7
cot 7
3
2
3
7
tan cot 55
55
5
52
4
210 225 22
csc Adjacent side: 16 9 7
cos 5
11
csc 107. sin θ
6
5
511
11
11
cot 6
11
tan sec 2
3
106. tan 2, 0 < <
Adjacent side: 62 52 11
cos 4 2.03
1.25 1.25
4
(d) a 1, b 3
Area arctan 3 arctan 0
101.
1.571
4
4
2
θ
1
tan 22
3
22
1
22
32
4
2
4
2 2
349
Chapter 4
350
Section 4.8
■
■
■
Trigonometric Functions
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
⇒ a b tan A 10 tan 30 5.77
b
cos A b
b
10
⇒ c
11.55
c
cos A cos 30
2. Given: B 60, c 15
A 90 60 30
sin B b
153
⇒ b c sin B 15 sin 60 12.99
c
2
cos B 15
a
⇒ a c cos B 15 cos 60 7.50
c
2
tan B b
b
14
⇒ a
4.82
a
tan B tan 71
b
14
b
⇒ c
14.81
sin B c
sin B sin 71
A 90 71 19
5. Given: a 6, b 12
c2
a2
tan A b2
⇒ c 36 144 13.42
6
1
1
a
⇒ A arctan 26.57
b 12 2
2
B 90 26.57 63.43
4. Given: A 7.4, a 20.5
B 90 7.4 82.6
tan A a
a
20.5
⇒ b
157.84
b
tan A tan 7.4
sin A a
a
20.5
⇒ c
159.17
c
sin A sin 7.4
6. Given: a 25, c 45
b c2 a2 1400 1014 37.42
sin A a
25
⇒ A arcsin
33.75
c
45
cos B a
25
⇒ B arccos
56.25
c
45
© Houghton Mifflin Company. All rights reserved.
3. Given: B 71, b 14
Section 4.8
7. Given: b 16, c 54
a
c2
cos A b2
Applications and Models
8. Given: b 1.32, c 18.9
2660 51.58
a c2 b2 355.4676 18.85
16
16
b
⇒ A arccos
72.76
c
54
54
B 90 72.76 17.24
9. A 12 15, c 430.5
cos A 1.32
1.32
b
⇒ A arccos
86.00
c
18.9
18.9
sin B b
1.32
⇒ B arcsin
4.00
c
18.9
B
c = 430.5
a
B 90 12 15 77 45
12°15′
b
C
A
a
sin 12 15 430.5
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
a
145.5
⇒ c
346.88
c
cos B cos65 12
tan B b
⇒ b a tan B 145.5 tan65 12 314.89
a
© Houghton Mifflin Company. All rights reserved.
11. tan h
b2
12. tan h
b2
1
h b tan 2
1
h b tan 2
1
h 8 tan 52 5.12 in.
2
1
h 12 tan 18 1.95 meters
2
13. tan h
b2
14. tan h
b2
1
h b tan 2
1
h b tan 2
1
h 18.5 tan 41.6 8.21 ft
2
1
h 3.26 tan 72.94 5.31 cm
2
(b) tan 15. (a)
60 ft
θ
L
351
L
60
L
60
tan 60 cot (c)
10
20
30
40
50
L
340
165
104
72
50
(d) No, the shadow lengths do not increase in
equal increments. The cotangent function
is not linear.
Chapter 4
352
Trigonometric Functions
(b) tan 16. (a)
L
850 ft
850
tan 850 cot θ
L
(c)
850
L
10
20
30
40
50
L
4821
2335
1472
1013
713
17. sin 80 (d) No, the cotangent function is not a
linear function.
h
20
h
58.2
18. tan13 h 20 sin 80
20
h 58.2 tan13 13.44 meters
h
19.70 feet
h
80°
13°
58.2 meters
19. sin 50 h
100
20. sin 31.5 x
4000
x 4000 sin 31.5
h 100 sin 50
76.6 feet
100
31.5°
x
4000
2089.99 feet
h
50°
h
a
100
tan 39.75 47° 40′
35°
as
100
a s 100 tan 39.75
50 ft
s 100 tan 39.75 a
(b) Let the height of the church x and the height
of the church and steeple y. Then:
x
y
tan 35 and tan 47 40 50
50
⇒ a 100 tan 28
100 tan 39.75 100 tan 28
30 feet
s
x 50 tan 35 and y 50 tan 47 40
h y x 50tan 47 40 tan 35
39.75°
a
(c) h 19.9 feet
100
28°
© Houghton Mifflin Company. All rights reserved.
22. tan 28 21. (a)
Section 4.8
23. (a) l 2 1002 20 3 h2
24. (a) sin 60 l 1002 17 h2
h
θ
60°
h
l
d
(b) tan 20 ft
θ
3 ft
100 ft
(c) If 35, then l tan 25 ≤ tan ≤ tan 30
tan 25 ≤
100
122.077. Then
cos 35
h
≤ tan 30
d
h
h
≥ d ≥
tan 25
tan 30
17 h2 100 2 l 2
17 h2 l 2 1002 4902.906
55.72 ≥ d ≥ 45.00
17 h 70.02
d is in the interval 45.0, 55.72.
h 53.02 feet
53 feet,
h 153 25.98
d
d
d
25 ≤ ≤ 30
(c)
100
100
(b) cos l ⇒ arccos l
1
inch.
4
75
95
27. sin 26. (a)
arctan
h
⇒ h 30 sin 60
30
θ
30
25. tan 353
153 25.98 ft
h 2 34h 10,289
3 ft
Applications and Models
15
38.29
19
1
12 2 ft
4000
⇒ 14.03
16,500
90 75.97
θ
1
17 3 ft
16,500
1
12
(b) tan 21
173
© Houghton Mifflin Company. All rights reserved.
75
α
θ
4000
1
(c) arctan
θ
95
28. tan 250
2.55280
arctan
122
35.8
1713
29. Since the airplane speed is
250
1.09
2.55280
275 sec60 min 16,500 min,
ft
sec
ft
after one minute its distance travelled is 16,500 feet.
2.5 miles
θ
250 feet
not drawn to scale
sin 18 a
16,500
a 16,500 sin 18
5099 ft
16500
18°
a
354
Chapter 4
30. sin 18 s
Trigonometric Functions
h
275s
31. sin 9.5 275(s)
h
h
275 sin 18
18°
x
4
4
x
9.5°
x 4 sin 9.5
0.66 mile
10,000
If h 10,000, s 117.7 seconds.
275 sin 18
If h 16,000, s 32. sin25.2 16,000
188.3 seconds.
275 sin 18
1808
L
L
1808
⇒ L
4246.3 feet
sin25.2
1808
25.2°
33. 90 29 61
N
206 120 nautical miles
sin 61 a
⇒ a 120 sin 61 104.95 nautical miles
120
b
61°
W
a
b
cos 61 ⇒ b 120 cos 61 58.18 nautical miles
120
E
120
S
34. c 6001.5 900
y 900 sin 38 554.1 miles
35. 32, 68
c
52°
38°
x
y
Note: ABC forms a right triangle.
N
(a) 90 32 58
B
φ
C
γ
β
α
50
β
Bearing from A to C: N 58 E
d
(b) 32
90 22
C 54
tan C d
d
⇒ tan 54 ⇒ d 68.82 m
50
50
W
θ
A
S
E
© Houghton Mifflin Company. All rights reserved.
x 900 cos 38 709.2 miles
Section 4.8
36. tan 14 tan 34 d
⇒ x d cot 14
x
37. tan 45 3
⇒ 56.31
30 2
N
30 d cot 14
d
Port
d cot 34 30 d cot 14
d
355
Bearing: N 56.31 W
d
d
d
⇒
y
30 x 30 d cot 14
cot 34 Applications and Models
45
30
θ
W
E
Ship
30
5.46 kilometers
cot 34 cot 14
S
85
⇒ 27.98
160
38. tan 39. tan 6.5 Bearing: S 27.98 W
tan 4 N
350
⇒ D 5005.23 ft
D
Distance between ships: D d 1933.3 ft
Plane
85
W
350
⇒ d 3071.91 ft
d
E
6.5°
θ
160
4°
350
θ
S2
S1
d
D
Airport
S
40.
41. tan 57 28°
10 km
55°
P1
T1
55°
d
28°
a
tan 16 x 556
T2
D
© Houghton Mifflin Company. All rights reserved.
cot 55 d
⇒ d 7 kilometers
10
cot 16 D d 18.8 7 11.8 kilometers
h
h
10°
2.5°
x
not drawn to scale
h
h
h 18 tan 10
18 tan 2.5 tan 10
tan 10
h tan 10 h tan 2.5 18tan 10tan 2.5
h
x
550
60
55
6
⇒ a 3.23 miles 17,054 feet
h
h
42. tan 2.5 , tan 10 x
x 18
h
h
,x
18
tan 2.5
tan 10
a
16°
a cot 57 556
a
a cot 16 a cot 57 Distance between towns:
P2
57°
H
a
tan 16 a cot 57 556
D
cot 28 ⇒ D 18.8 kilometers
10
x
a
⇒ x a cot 57
x
18tan 10tan 2.5
1.04 miles 5515 feet
tan 10 tan 2.5
x − 18
not drawn to scale
Chapter 4
356
Trigonometric Functions
44. (a) Using the two triangles with acute angle ,
43. (a)
3
3
and sin .
L1
L2
cos 50 + 6 = 56
Hence, L L 1 L 2 3 sec 3 csc .
θ
40
3
β
56
arctan
54.5
40
L1
β
L2
Similarly for the back row,
arctan
56
20.5.
150
(b)
3
20
(b) For 45, you need to be 56 feet away since
56
45.
56
␲
2
0
0
(c) The least value is 0.785. 3
5
3
45. L1: 3x 2y 5 ⇒ y x ⇒ m1 2
2
2
L2 x 5y 4 ⇒ m2 1 32
52
5
1 132
12
tan arctan 5 78.7
m2 m1
1 m2m1
arctan
arctan
48. tan 47. The diagonal of the base has a length of
a2 a2 2a. Now, we have:
1
tan 2a
2
49. cos 30 1
2
35.3
2
θ
2a
a
50.
b 30°
r
y
3r
c
35
17.5
2
sin 15º a
c
c
15°
a c sin 15 17.5 sin 15 4.53
2
x
y 2b
a2
2
a
θ
r
2
23r 2
arctan323 74.7
1 152
1
5
a
b
r
2
m2 m1
1 m2m1
a
b cos 30r
b
1
5
arctan 2 54.7º
a
arctan
46. L1 2x y 8 ⇒ m1 2
L2: x y 1 ⇒ y x 1 ⇒ m2 1
tan 4
3r
Distance 2a 9.06 centimeters
a
© Houghton Mifflin Company. All rights reserved.
arctan
Section 4.8
51. d 0 when t 0, a 8, period 2
Applications and Models
52. Displacement at t 0 is 0 ⇒ d a sin t.
Use d a sin t since d 0 when t 0.
Amplitude: a 3
2
2 ⇒ Period:
Thus, d 8 sin t.
2
6 ⇒ 3
d 3 sin
53. d 3 when t 0, a 3, period 1.5
t
3
54. Displacement at t 0 is 2 ⇒ d a cos t
Use d a cos t since d 3 when t 0.
Amplitude: a 2
2
4
1.5 ⇒ 3
2
Period:
10 ⇒ 5
Thus, d 3 cos
43 t.
357
d 2 cos
t
5
55. d 4 cos 8t
(a) Maximum displacement amplitude 4
(b) Frequency 8
2 2
(c) d 4 cos8 5 4
(d) 8t 1
⇒ t
2
16
4 cycles per unit of time
56. d 1
cos 20 t
2
57. d (a) Maximum displacement: a (b) Frequency:
20
10
2
2
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(c) When t 5, d 1
1
2
2
1
1
cos20 5
.
2
2
(d) Least positive value for t for which d 0:
1
cos 20 t 0
2
cos 20 t 0
20 t 2
t
2
1
1
20 40
1
sin 140t
16
(a) Maximum displacement amplitude (b) Frequency 140
2
2
70 cycles per unit of time
(c) d 0
(d) 140t ⇒ t 1
140
1
16
Chapter 4
358
58. d Trigonometric Functions
1
sin 792 t
64
(a) Maximum displacement: a 792
(b) Frequency:
396
2
2
1
1
64
64
(d) Least positive value for t for which d 0:
1
sin 792 t 0
64
sin 792 t 0
1
(c) When t 5, d sin792 5
0.
64
792 t t
59. d a sin t
Period 60. At t 0, buoy is at its high point ⇒ d a cos t.
1
2
frequency
Distance from high to low 2 a 3.5
7
a 4
2
1
264
Returns to high point every 10 seconds:
2264 528
Period d
61. y 1
cos 16t, t > 0
4
(a)
1
792 792
(b) Period:
0.5
2
10 ⇒ 5
7
t
cos
4
5
2 seconds
16
8
(c)
␲
0
1
cos 16t 0 when
4
16t ⇒ t
seconds.
2
32
62. (a)
0.1
0.2
0.3
0.4
L1 L 2
L1
L2
2
sin 0.1
3
cos 0.1
23.0
2
sin 0.2
3
cos 0.2
13.1
2
sin 0.3
3
cos 0.3
9.9
2
sin 0.4
3
cos 0.4
8.4
(c) L L1 L2 2
3
sin cos (b)
0.5
2
sin 0.5
3
cos 0.5
7.6
0.6
2
sin 0.6
3
cos 0.6
7.2
0.7
2
sin 0.7
3
cos 0.7
7.0
0.8
2
sin 0.8
3
cos 0.8
7.1
The minimum length of the elevator is 7.0 meters.
(d)
12
0
0
␲
2
From the graph, it appears
that the minimum length is
7.0 meters, which agrees
with the estimate of part (b).
© Houghton Mifflin Company. All rights reserved.
−0.5
Section 4.8
Applications and Models
359
(c) A 12 b1 b2 h
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
12 8 8 16 cos 8 sin 641 cos sin (d) Maximum area is approximately 83.1 square feet
for 60.
100
0
90
0
Maximum 83.1 square feet
15
2
12 ⇒ b b
6
12
9
6
Shift: d 14.3 6.3 8
3
t
2
4
6
S
Average sales
(in millions of dollars)
1
(b) a 14.30 1.70 6.3
2
S
Average sales
(in millions of dollars)
64. (a)
15
12
9
6
3
t
S d a cos bt
8 10 12
Month (1 ↔ January)
2
12
6
This corresponds to the 12 months in a year.
Since the sales of outerwear is seasonal, this
is reasonable.
(c) Period:
S 8 6.3 cos
2
4
6
8 10 12
Month (1 ↔ January)
t
6
The model is a good fit.
© Houghton Mifflin Company. All rights reserved.
(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.
65. St 18.09 1.41 sin
(a)
6t 4.60
22
0
(b) The period is
12
2
12 months, which is 1 year.
6
(c) The amplitude is 1.41. This gives the maximum change
in time from the average time 18.09 of sunset.
14
66. False
It means 24 degrees east of north.
67. False. The other acute angle is 90 48.1 41.9.
Then
tan41.9 opp
adj
a
⇒ a 22.56
22.56
tan41.9.
Chapter 4
360
Trigonometric Functions
y 2 4x 1
69.
68. False
y 0 12x 13 70.
2y x 13
4x y 6 0
3x 6y 1 0
71. Slope 62
4
2 3
5
72. m 4
y 2 x 3
5
13 23
1
4
12 14 34
3
y
2
4
1
x
3
3
4
3y 2 4x 1
5y 10 4x 12
4x 3y 1 0
4x 5y 22 0
74. Domain: , 73. Domain: , 75. Domain: , 76. Domain: 7 x ≥ 0
or x ≤ 7
Review Exercises for Chapter 4
2. 250 or 4.4 radians
y
3. (a)
(b) Quadrant III
(c)
4π
3
y
4
10
2 3
3
(b) Quadrant III
(c) x
− 5π
6
7. Complement:
3
2
8
8
Supplement: 9. Complement:
7
8
8
3 2
10
5
Supplement: (b) Quadrant IV
(c)
11 π
6
4
2
2 3
3
x
5. (a)
y
4. (a)
3 7
10
10
5
7
2 6
6
17
5
2 6
6
11
2 6
6
x
y
6. (a)
11
23
2 6
6
(b) Quadrant I
(c) − 7π
4
x
8. Complement:
5
2
12
12
Supplement: 10. Complement:
11
12
12
2 17
2
21
42
Supplement: 2 19
21
21
7
2 4
4
7
15
2 4
4
© Houghton Mifflin Company. All rights reserved.
1. 40 or 0.7 radian
Review Exercises for Chapter 4
11. (a)
y
12. (a)
y
210°
45°
x
x
(b) Quadrant I
(b) Quadrant III
(c) 45 360 405
(c) 210 360 570
210 360 150
45 360 315
13. (a)
361
y
14. (a)
y
−405°
x
x
− 135°
(b) Quadrant III
(b) Quadrant IV
(c) 135 360 225
(c) 405 720 315
135 360 495
405 360 45
15. Complement of 5: 90 5 85
16. Complement of 84: 90 84 6
Supplement of 5: 180 5 175
18. Complement of 136: not possible
17. Complement: not possible
© Houghton Mifflin Company. All rights reserved.
Supplement: 180 171 9
19. 135 16 45 135 21. 5 22 53 5 Supplement of 84: 180 84 96
Supplement of 136: 180 136 44
16
45 135.279
60 3600
53 22
5.381
60 3600
20. 234 40 234 22. 280 8 50 280 40
234.011
3600
8
50
60 3600
280 0.13 0.01 280.147
23. 135.29 135 0.2960 135 17 24
24. 25.8 25 48
25. 85.36 85 0.3660 85 21 36
26. 327.93 327 55 48
27. 415 415 rad 83
rad 7.243 rad
180
36
28. 355 355 rad
180
71
rad 6.196 rad
36
362
Chapter 4
Trigonometric Functions
29. 72 72 31.
5 5 180
128.571
7
7 33. 3.5 3.5
35.
rad
2
rad 1.257 rad
180
5
30. 94 94 32. 200.535
180
rad 47
rad 1.641 rad
180
90
3
3 180
108
5
5 34. 1.55 1.55
88.808
180
36. s r
⇒ s r
s 245
49
4.083 radians
r
60
12
25 12
25
2.083
12
38. s r
15
37. s r
s 20138
5 15.71 cm
3
180
s 48.171 m
t
40. (a) 28 miles per hour 13 73
Circumference of wheel: C 2
1
1
3
minutes 60 seconds.
500
500
25
Linear speed Number of revolutions per minute:
s
12
100 cmsec
t
325
2464
7392
336.1 revmin
73
7
(b)
41. t 2
2
7
corresponds to
,
.
4
2
2
42. cos
7392
14,724
2 2112 rad/min
t
7
7
2
3 2
3
, sin
4
2
4
2
x, y 43. cos
sin
3
5
6
2
285280
2464 ft/min
60
44. t 2 2
2
,
2
4
1 3
.
corresponds to , 3
2
2
5 1
6
2
x, y 45. t 3 1
2
,
2
2
1 3
corresponds to , .
3
2
2
46. t 3 1
7
corresponds to , .
6
2 2
© Houghton Mifflin Company. All rights reserved.
39. In one revolution, the arc length traveled is
s 2 r 2 6 12 cm. The time
required for one revolution is
Review Exercises for Chapter 4
47. t 2 2
5
,
.
corresponds to 4
2
2
cot
7
3
6
cos
3
7
6
2
sec
7
2
23
6
3
3
tan
1 cot
4
4
tan
3
7
1
6
3
3
csc
7
2
6
sec
2 csc
4
4
sin 2 y 0
x
cot 2 , undefined
y
cos 2 x 1
sec 2 y
0
x
1
1
x
1
csc 2 , undefined
y
3 1
11
, .
corresponds to
6
2 2
53. t 2
cos
4
2
4
50. sin
52. t corresponds to 1, 0.
51. t 2 corresponds to 1, 0.
tan 2 cos x 1
tan 5
6
5
6
5
6
5
x
3
6
y
5
1
23
6
x
3
5
1
2
6
y
11
6
11
6
11
x
3
6
y
cot 11
1 23
6
x
3
sec 11
1
2
6
y
csc cos tan 3
yx 3
cos 2
tan 3
cot sec csc 55. t sin x
corresponds to 0, 1.
2
2 y 1
cot 2 x 0
sec 2 yx, undefined
csc sin cos tan 2 xy 0
2 1x, undefined
2 1y 1
y
0
x
3
5
1
, .
corresponds to 6
2
2
54. t 11
6
y 21
x
cot , undefined
y
1
sec 1
x
1
csc , undefined
y
sin y 0
sin 3
5
1
corresponds to , .
6
2
2
7
1
6
2
49. sin
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48. t 363
y 21
x 23
yx 3
3
Chapter 4
56. t Trigonometric Functions
2
2
corresponds to
,
.
4
2
2
4 y 22
cot 4 xy 1
4 x sec sin cos 4 1x 2
2
2
4 yx 1
4 1y 2
tan 57. sin
114 sin34 59. sin csc 2
58. cos 4 cos 0 1
2
17
7
1
sin
6
6
2
61. sin t 60. cos 3
5
62. cos t (a) sint sin t (b) csct 63. sint 3
5
1
5
sint
3
67. cos
5
13
(b) sect 2
3
64. cost 23 23
(b) csc t (a) cost cos t 1
3
sin t 2
5
8
(b) sect 1
0.8935
tan 2.3
66. sec 4.5 5 1
3
2
68. tan 69. The opposite side is 92 42 81 16 65.
sin opp 65
hyp
9
csc 1
9
965
sin 65
65
cos adj
4
hyp 9
sec 1
9
cos 4
tan opp 65
adj
4
cot 1
4
465
tan 65
65
5
13
1
13
cost
5
(a) cos t cost (a) sin t sint 65. cot 2.3 13
5 1
cos
3
3
2
5
8
1
8
cost 5
1
4.7439
cos 4.5
11
0.5774
6
© Houghton Mifflin Company. All rights reserved.
364
Review Exercises for Chapter 4
70. sin 15
541
41
341
cos 12
441
41
341
71. The hypotenuse is 122 102 244 261.
sin opp
10
5
561
hyp 261 61
61
cos adj
12
6
661
hyp 261 61
61
tan opp 10 5
adj
12 6
3 41
15
15 5
12 4
tan θ
41
csc 41
541
sec 41
41
4
441
csc 61
1
sin 5
cot 4
5
sec 61
1
cos 6
cot 1
6
tan 5
72. sin cos tan 12
5
2
1
10 5
csc 5
46 26
10
5
sec 2
46
6
cot 12
73. csc tan 5
26
sin 1
79. tan 62 125
1
sin cos cos sec 56
12
12
26
6
10
2
θ
96 = 4 6
74.
tan cot tan 1
1 tan2 sec2 cot cot © Houghton Mifflin Company. All rights reserved.
76. (a) csc52 12 (b) sec54 7 75. (a) cos 84 0.1045
(b) sin 6 0.1045
1
1
1.2656
sin52 12 sin 52.2
1
1
1.7061
cos54 7 cos54.1167
77. (a) cos
0.7071
4
78. (a) tan
(b) sec
1.4142
4
(b) cot
320 0.5095
320 tan3120
1.9626
(b) sin 30 80. (a)
152 ft
30⬚
365
h
(c) h 152
h
⇒ h 152 sin 30
152
12 76 feet
x 125 tan 62
235 feet
366
Chapter 4
Trigonometric Functions
81. x 12, y 16, r 144 256 400 20
y 4
r
5
csc r
5
y 4
cos x 3
r
5
sec r
5
x 3
tan y 4
x 3
cot x 3
y 4
82. x 2, y 10, r 4 100 104 226
83. x 7, y 2, r 49 4 53
sin y
10
526
r
26
226
sin y
2
253
r
53
53
cos 26
x
2
r
26
226
cos x
7
753
r
53
53
tan y 10
5
x
2
tan y
2
x
7
csc 26
r
y
5
csc sec r
26
x
sec x 1
y 5
cot cot 84. x 3, y 4, r 32 42 5
y
4
sin r
5
85. x 53
2
53
7
7
2
2
5
and y , r 3
8
23 58
2
sin y
58
15481
48124
r
481
cos x
3
r
5
cos tan y
4
x
3
x
23
16481
48124
r
481
tan csc x
5
y
4
y 58 15
x 23 16
csc sec r
5
x 3
481
r
y
15
sec cot r
3
y
4
481
r
x
16
cot x 16
y 15
2
481
24
© Houghton Mifflin Company. All rights reserved.
sin Review Exercises for Chapter 4
86. x 10
2
,y ,r
3
3
103 23
2
2
226
3
sin y
23
1
26
r
26
2 263
26
csc r
26
y
cos x
103
5
526
r
26
2263 26
sec 26
r
x
5
tan y
23
1
x 103 5
cot x
5
y
6
87. sec , tan < 0 ⇒ is in Quadrant IV.
5
y
θ
r 6, x 5, y 36 25 11
sin 11
y
r
6
csc cos x 5
r
6
sec tan 11
y
x
5
cot © Houghton Mifflin Company. All rights reserved.
88. tan 5
611
11
x
−
6
11
6
5
511
11
y
12
⇒ r 13, sin > 0 ⇒ y 12, x 5
x
5
sin y 12
r
13
csc r
13
y 12
cos x
5
r
13
sec r
13
13
x 5
5
cot x
5
y
12
3
89. sin , cos < 0 ⇒ is in Quadrant II.
8
y
y 3, r 8, x 55
y 3
r
8
csc cos 55
x
r
8
sec tan y
3
355
x
55
55
cot sin 8
3
8
3
−
8
855
55
55
55
3
θ
x
55
367
368
Chapter 4
90. cos Trigonometric Functions
x 2
⇒ y ± 21, sin > 0 ⇒ y 21
r
5
sin y 21
r
5
sec r
5
5
x 2
2
tan 21
y
x
2
cot x
2
221
y 21
21
csc r
5
521
y 21
21
y
91. Reference angle:
92. 635 is coterminal with 275.
264 180 84
Reference angle:
y
360 275 85
θ = 264⬚
x
θ = 635°
θ ′ = 84⬚
x
θ ′ = 85⬚
y
93. Coterminal angle:
6 4
5
5
y
Reference angle:
θ′= π
5
Reference angle:
x
θ = − 6π
5
4 5
5
95. 240 is in Quadrant III with reference angle 60.
sin 240 sin 60 3
cos 240 cos 60 tan 240 17
5
is coterminal with
.
3
3
5 3
3
θ = 17π
3
x
θ′= π
3
96. 315 is in Quadrant IV with reference angle 45.
sin 315 sin 45 2
1
2
cos 315 cos 45 32
3
12
2
2
2
2
tan 315 tan 45 1
97. 210 is coterminal with 150 in Quadrant II with
reference angle 30.
98. 315 is coterminal with 45 in Quadrant I.
sin315 1
sin210 sin30 2
cos210 cos30 2 3
2
3
12
1
tan210 3
3
32
cos315 2
2
2
tan315 1
2
© Houghton Mifflin Company. All rights reserved.
2 94.
Review Exercises for Chapter 4
99. 94 is coterminal with 74 in Quadrant IV
with reference angle 4.
100. 116 is in Quadrant IV.
116 12
2
9
sin
sin 4
4
2
sin
2
9
cos cos
4
4
2
cos
9
1
4
22
tan
tan 22
101. sin4 sin0 0
102. sin
cos4 1
73 sin3 cos
tan4 0
tan
104. csc 105 1
1.0353
sin 105
3
2
116 33
3
103. tan 33 0.6494
2
73 21
12
1
3.2361
5
cos125
9 0.3420
106. sin 108. f x 2 cos x
y
4
Amplitude: 3
116 73 3
105. sec
107. f x 3 sin x
369
y
3
Amplitude: 2
3
2
2
1
x
π
− 2π
2π
−π
π
−1
2π
x
−2
−3
−4
109. f x 14 cos x
Amplitude: 14
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110. f x 72 sin x
y
1
Amplitude:
7
2
1
2
y
4
3
2
1
− 2π
x
2π
−
x
π
−1
111. Period:
2
2
Amplitude: 5
2π
1
2
112. Period:
−4
2
4
12
Amplitude:
3
2
113. Period:
2
2
Amplitude: 3.4
114. Period:
2
4
2
Amplitude: 4
370
Chapter 4
Trigonometric Functions
115. y 3 cos 2x
3
Amplitude: 3
Period:
2
1
2
Period:
116. y 2 sin x
y
1
1
2
2
Period:
–3
x
12
0
1
2
y
2
0
2
y
6
4
Period:
2
2
5
25
–2
x
118. f x 8 cos 4
y
6π
x
1
–1
Reflected in x-axis
−3
Amplitude: 5
1
1
−2
2x
117. f x 5 sin
5
2
2
Amplitude: 2 2
x
−1
y
3
8
2
8
14
4π
8π
x
x
Amplitude: 8
–2
–4
–6
Reflected in y-axis
–8
–6
Amplitude:
Period:
2
8
14
121. f x 2
0
2
4
y
8
0
8
0
8
1
x
120. f x sin
2
4
y
5
4
3
2
1
5
2
4
Amplitude:
− 4π
4π
8π
1
1
2
x
x
−6
Period: 8
122. f x 3 cosx 4
–1
–2
–3
–4
4
3
2
1
Shift:
π
x
Amplitude: 3
1
π
This is the graph of
y 3 cos x shifted
to the left units.
x 0 and x 2
x 3
6
y
Period: 2
3
Period: 2
2
−1
y
5
Amplitude:
2
−2
−1
2
−3
−4
−5
5
sinx 2
x
y
x
f x
3
0
2
–3
–4
0
2
3
0
3
x
© Houghton Mifflin Company. All rights reserved.
5
x
119. f x cos
2
4
x
Review Exercises for Chapter 4
123. f x 2 cos
x
2
124. f x y
4
1
sin x 3
2
Amplitude:
2
x
−4 −3 −2 −1
−1
1
2
3
4
−2
−4
2x 4 1
2
4
Period:
−4
π
x
2π 3π
y
2
8
6
4
2
−3
−4
π
2
−3
−2
4
π
4
−2
Period: 2
2
2
−1
Amplitude: 2
3
127. f x 2 cos x 1
126. f x 4 2 cos4x y
−π
x
−1
Vertical shift
downward three units
−3
125. f x 3 cos
y
−2
3
371
−π
2
−π
4
−2
2
128. f x a cosbx c
129. f x 4 cos 2x x
Amplitude: 3
Period: ⇒ f x 3 cos2x
130. f x a cosbx c
Amplitude:
1
2
Period: 2 ⇒ f x © Houghton Mifflin Company. All rights reserved.
131. S 48.4 6.1 cos
132. S 56.25 9.50 sin
t
6
56
Maximum sales: t 6
(June)
1
cos x
2
t
6
133. f x tan
70
Maximum sales: t 3
(March)
Minimum sales: t 9
(September)
0
12
45
0
Minimum sales: t 12
(December)
12
40
x
4
y
6
Period:
4
4
4
2
x
−6
Asymptotes:
x 2, x 2
−4
−2
−2
−4
Reflected in x-axis
x
1
0
1
y
1
0
1
−6
2
4
6
Chapter 4
372
Trigonometric Functions
135. f x 134. f x 4 tan x
1
tan x 4
2
136. f x 2 2 tan
x
3
y
y
y
6
4
4
3
2
2
6
4
2
1
x
1
− 3π
2
−1
π
2
x
3π
2
−3π −2π
−2
Period:
138. f x y
6
2
12
1
x
cot
2
2
1
2 tan
4
−4π
x
2
y
2
Two consecutive
asymptotes:
x
−6
−4
x
2
3π 4π
−4
−3
137. f x 3 cot
π
−2π
2π
4π
x
2
1
x
0 ⇒ x0
2
x
−2
−1
1
2
x
⇒ x 2
2
139. f x 1
cot x 2
2
140. f x 4 cot x y
5
4
3
Period: 4
4
tan x 4
y
x
x
−2 π
0 ⇒ x
2
2
− π −1
π
2π
x
4
− 2π
−π
1
sec x
4
142. f x Period: 2
x
1
1
csc x 2
2 sin x
y
2
y
1
2
1
−π
2π
−2
2π
3
⇒ x
2
2
141. f x −1
π
π
x
−2
2π
3π
x
© Houghton Mifflin Company. All rights reserved.
Two consecutive
asymptotes:
Review Exercises for Chapter 4
143. f x 1
csc 2x
4
144. f x 1
1
sec 2 x 2
2 cos 2 x
Period: y
y
3
2
x
−1
1
x
π
145. f x sec x 4
1
−1
146. f x y
3
1
1
csc2x 2
2 sin2x y
2
Secant function shifted
to right
4
1
−π
π
2π
x
−π
−π
2
π
2
x
π
−1
147. f x 1
x
tan
4
2
148. f x tan x 1
−π
−2π
© Houghton Mifflin Company. All rights reserved.
−1
4
tan2x 150. f x 2 cot4x 10
2
tan4x 4
−π
2
π
−10
π
2
−4
151. f x 2 secx 2
cosx 152. f x 2 cscx 10
8
2π
−10
2π
−4
149. f x 4 cot2x − 2π
4
π
−π
4
−2π
2π
−8
2
sinx 373
374
Chapter 4
Trigonometric Functions
1
2
sin3x 2
153. f x csc 3x 3
4
sin2x 4
154. f x 3 csc 2x 4
8
−π
π
−π
π
−8
−4
156. f x ex cos x
155. f x ex sin 2x
Damping factor: y ex
Damping factor: ex
5
20
−4
−4
4
4
−5
−20
157. f x 2x cos x
158. f x x sin x
Damping factor: gx 2x
6
As x → , f oscillates
between x and x.
14
−9
9
−6
−10
10
−14
As x → , f oscillates between 2x and 2x.
12 6 because sin 6 12.
160. (a) arcsin (b) arcsin 4 does not exist because the domain of
arcsin is 1, 1.
(b) arcsin 3
2
3 because
3 23.
sin 22 4 because cos 4 22.
3
5
3
5
(b) arccos
because cos
.
2 6
6
2
161. (a) arccos
162. (a) arctan 3 (b) arctan1 3
4
163. arccos0.42 1.14
164. arcsin 0.63 0.68
165. sin10.94 1.22
166. cos10.12 1.69
167. arctan12 1.49
168. arctan 21 1.52
169. tan10.81 0.68
170. tan1 6.4 1.42
171. sin x3
x3
⇒ arcsin
16
16
172. tan x 1 ⇒ arctan x 1
20
20
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because sin 1.
2
2
159. (a) arcsin1 Review Exercises for Chapter 4
x
x
174. Let u arccos , cos u .
2
2
173. Let y arcsinx 1. Then,
sin y x 1 sec y x1
and
1
tan arccos
x
tan u
2
1
x2 2x
x2 2x
x2 2x
1
4 x2
2
x
sin y .
y
u
x
x2
x2
and
. Then cos y 2
4x
4 x2
4 x22 x22
4 x2
176. Let u arcsin 10x, sin u 10x.
cscarcsin 10x csc u
.
16 8x2
4
x2
24 2x2
.
4 x2
177. sin1 10 4 − x2
u
x2
1 − 100x 2
a
3.5
3.5
a
1° 10'
not drawn to scale
12
100
arctan
sin
12
100
0.1194 or 6.84
h
4
h
4
θ
h 4 sin0.1194 0.48 miles or 2534 feet (answer depends on angle accuracy)
179. tan 14 tan 58 1
(4 − x 2 ) 2 − (x 2) 2
© Houghton Mifflin Company. All rights reserved.
1
hyp
opp 10x
y
7
a 3.5 sin1 10 3.5 sin
0.0713 or 71 meters
6
178. tan 4 − x2
x−1
1 − (x − 1) 2
175. Let y arccos
375
y
⇒ y 37,000 tan 14 9225.1 feet
37,000
32°
xy
⇒ x y 37,000 tan 58 59,212.4 feet
37,000
x 59,212.4 9225.1 49,987.2 feet
The towns are approximately 50,000 feet apart or 9.47 miles.
76°
58°
37,000
14°
x
y
10x
376
Chapter 4
Trigonometric Functions
d1
⇒ d1 483
650
180. sin 48 d2
⇒ d2 734
cos 25 810
d3
cos 48 650
⇒ d3 435
d4
⇒ d4 342
sin 25 810
tan N
d1 d2 1217
48°
W
d3
A
d3 d4 93
B
25°
48° 65° 810
650
D
θ
d1
d2
E
d4
C
S
93
⇒ 4.4
1217
sec 4.4 D
⇒ D 1217 sec 4.4 1221
1217
The distance is 1221 miles and the bearing is N 85.6 E.
182. Use a cosine model with amplitude
181. Use cosine model with amplitude 3 feet.
Period: 15 seconds
Period: 3 seconds
2
t
y 3 cos
15
y 0.75 cos
183. False. y sin is a function, but it is not
one-to-one.
(a)
23 t
184. False. The sine and cosine functions are useful for
modeling simple harmonic motion.
0.672s 2
3000
s
10
20
30
40
50
60
1.28
5.12
11.40
19.72
29.25
38.88
(b) increases at an increasing rate. The function is not linear.
186. (a)
(b) Next term:
10
−␲
␲
−10
x9
9
arctan x x 10
x3 x 5 x7 x9
3
5
7
9
The accuracy of the approximation
increases as more terms are added.
−␲
␲
−10
© Houghton Mifflin Company. All rights reserved.
185. tan 1.5
0.75.
2
Practice Test for Chapter 4
Chapter 4
377
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 23, use the trigonometric identities to
find tan .
6. Find given sin 0.9063.
7. Solve for x in the figure below.
8. Find the magnitude of the reference angle for
65.
35
20°
x
10. Find sec given that lies in Quadrant III and
tan 6.
9. Evaluate csc 3.92.
x
11. Graph y 3 sin .
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.
17. Evaluate arcsin 1.
18. Evaluate arctan3.
© Houghton Mifflin Company. All rights reserved.
19. Evaluate sin arccos
.
4
4
.
35
20. Write an algebraic expression for cos arcsin
For Exercises 21–23, solve the right triangle.
21. A 40, c 12
22. B 6.84, a 21.3
B
c
A
b
a
C
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.
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?
23. a 5, b 9
x
.
4