Chapter 5 Analytic Trigonometry

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1
Precalculus with Limits, Answers to Section 5.1
Chapter 5
sec Section 5.1 (page 379)
1. sin x 2. cos u
6. sec2 u
10. tan u
3
cos x 4. csc u
8. csc u
1
2
1
2
3
cos x 2
3
tan x 3
tan x 23
csc x 3
csc x 2
sec x 2
sec x cot x 3
3
2
3. sin 2
2
cos 2
tan 1
sec 2
csc 2
cot 1
Copyright © Houghton Mifflin Company. All rights reserved.
3. cot u
7. cos u
2. sin x 2
5
13
12
cos x 13
5
tan x 12
13
sec x 12
13
csc x 5
12
cot x 5
5. sin x 7. sin 5
3
2
cos 3
tan 5
2
csc x 35
5
25
cot 5
1
9. sin x 3
22
cos x 3
2
tan x 4
csc x 3
32
sec x 4
csc Vocabulary Check (page 379)
1. tan u
5. cot2 u
9. cos u
5
3
5
sec x 4
4
cot x 3
3
2
3
23
3
3
5
4
cos 5
3
tan 4
5
sec 4
5
csc 3
4
cot 3
tan x 15
sec x 4
415
csc x 15
15
cot x 15
1
12. sin 5
26
cos 5
6
tan 12
56
sec 12
csc 5
25
5
5
cos 5
11. sin tan 2
4. sin csc 5
2
sec 5
1
cot 2
13. sin 1
10
15
4
1
cos x 4
cot x 22
cot x 3
6. sin 10. sin x cot 26
14. sin 1
cos 0
cos 0
tan is undefined.
tan is undefined.
cot 0
csc 1
csc 1
sec is undefined.
sec is undefined.
cot 0
10
310
cos 10
1
tan 3
19. e
23. f
cot 3
26. d
27. csc 28. sin 29. cos2 30. 1
31. cos x
csc 10
sec 3
5
4
cos x 5
3
tan x 4
8. sin x 10
3
15. d
16. a
17. b
18. f
20. c
21. b
22. c
24. a
25. e
32. cot 33. sin x
34. cos 2 x
35. 1
36. sin2 37. tan x
38. sin x
39. 1 sin y
40. sec t
41. sec 42. 2 sec 43. cos u sin u
2
44. csc sec 45.
47. sin 2 x tan 2 x
48. 1
50. cos x 2
51.
sin 2
sec 4
x
46. cos2 x
49. sec x 1
x
52. sin 4 x
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2
Precalculus with Limits, Answers to Section 5.1
(Continued)
53.
55.
57.
59.
62.
65.
sin 2 x cos 2 x
54.
56.
58.
61.
64.
66.
cot2 xcsc x 1
1 2 sin x cos x
60. 9 cos 2 x
4 cot2 x
63. 2 sec x
2 cot 2 x
1 cos y
67. 3sec x tan x
sec2 x tan2 x
tan2 xsec x 1
1
2 csc 2 x
cot x
5sec x tan x
x
1.2
1.4
y1
5.3319
11.6814
y2
5.3319
11.6814
12
68. tan 4 x csc x 1
␲
2
0
69.
x
0.2
0.4
0.6
0.8
1.0
y1
0.1987
0.3894
0.5646
0.7174
0.8415
y2
0.1987
0.3894
0.5646
0.7174
0.8415
x
1.2
1.4
y1
0.9320
0.9854
y2
0.9320
0.9854
1
y1 y2
72.
x
0.2
0.4
0.6
0.8
1.0
y1
0.0428
0.2107
0.6871
2.1841
8.3087
y2
0.0428
0.2107
0.6871
2.1841
8.3087
x
1.2
1.4
y1
50.3869
1163.6143
y2
50.3869
1163.6143
1
1200
␲
2
0
0
y1 y2
␲
2
0
0
Copyright © Houghton Mifflin Company. All rights reserved.
70.
y1 y2
x
0.2
0.4
0.6
0.8
1.0
y1
0.0403
0.1646
0.3863
0.7386
1.3105
73. csc x
74. cot x
75. tan x
y2
0.0403
0.1646
0.3863
0.7386
1.3105
76. sec 77. 3 sin 78. 8 sin 79. 3 tan 80. 2 tan 81. 5 sec x
1.2
1.4
82. 10 sec 83. 3 cos 3; sin 0; cos 1
y1
2.3973
5.7135
84. 6 cos 3; sin ±
y2
2.3973
5.7135
3
85. 4 sin 22; sin 6
2
; cos 2
2
; cos 86. 10 sin 53; sin 87. 0 ≤ ≤ ␲
2
0
0
89. 0 ≤ <
y1 y2
71.
3
,
< < 2
2 2
lncsc t sec t
1
2
3
2
2
2
; cos 88.
1
2
3
≤ ≤
2
2
90. 0 < < 91. ln cot x
92. ln tan x
94. 0
x
0.2
0.4
0.6
0.8
1.0
93.
y1
1.2230
1.5085
1.8958
2.4650
3.4082
95. (a) csc 2 132 cot 2 132 1.8107 0.8107 1
y2
1.2230
1.5085
1.8958
2.4650
3.4082
(b) csc 2
2
2
cot2
1.6360 0.6360 1
7
7
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3
Precalculus with Limits, Answers to Section 5.1
(Continued)
112. Not an identity because
96. (a) tan 3462 1 sec 3462 1.0622
(b) tan 3.12 1 sec 3.12 1.0017
97. (a) cos90 80 sin 80 0.9848
1
1
sin 113–114. Answers will vary.
115. x 25
116. 4z 12z 9
(b) cos
0.8 sin 0.8 0.7174
2
117.
6x 8
x 5x 8
118.
32x 1
x4
98. (a) sin250 sin 250 0.9397
119.
120.
(b) sin 12 sin12 0.4794
5x2 8x 28
x2 4x 4
xx2 5x 1
x2 25
121.
x2
122.
99. tan y
y
2
3
100. Answers will vary.
1
101. True. For example, sinx sin x.
x
102. False. A cofunction identity can be used to transform a
tangent function so that it can be represented by a cotangent function.
103. 1, 1
104. 1, 1
105.
, 0
1
110. Not an identity because
sin k
tan k
cos k
5
1
cos 5 cos Copyright © Houghton Mifflin Company. All rights reserved.
111. An identity because sin 1
sin 1
x
1
3
−1
−2
−3
−2
123.
124.
108. Not an identity because cot ± csc2 1
109. Not an identity because
−1
−1
106. 0, 107. Not an identity because cos ± 1 sin2 −3
3
y
y
4
5
3
4
2
3
1
π
2
−2
3π 2π
2
1
x
−π
−1
−3
−2
−4
−3
π
2π
x
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4
Precalculus with Limits, Answers to Section 5.2
Section 5.2 (page 387)
44. (a)
(b)
1
Vocabulary Check (page 387)
1. identity
4. cot u
7. csc u
2. conditional equation
5. cos2 u
6. sin u
8. sec u
1–38. Answers will vary.
39. (a)
(b)
3. tan u
−2␲
2␲
−1
Identity
(c) Answers will vary.
45. (a)
(b)
3
5
−5
−2␲
5
y2
y1
2␲
−3
−5
Not an identity
Identity
(c) Answers will vary.
46. (a)
(c) Answers will vary.
40. (a)
(b)
(b)
3
3
−2␲
−2␲
2␲
2␲
−5
Not an identity
−1
Identity
(c) Answers will vary.
41. (a)
(b)
5
y2
y1
−2␲
2␲
−1
Copyright © Houghton Mifflin Company. All rights reserved.
Not an identity
(c) Answers will vary.
42. (a)
(b)
5
−␲
␲
−5
Not an identity
(c) Answers will vary.
43. (a)
(b)
5
−2␲
2␲
−1
Identity
(c) Answers will vary.
(c) Answers will vary.
47–50. Answers will vary.
52. 1
53. 2
54. 2
56. (a) Answers will vary.
(b)
10
20
51. 1
55. Answers will vary.
30
40
50
4.20
s
28.36
13.74
8.66
5.96
60
70
80
90
s
2.89
1.82
0.88
0
(c) Greatest: 10; Least: 90 (d) Noon
57. False. An identity is an equation that is true for all real
values of .
58. True. An identity is an equation that is true for all real
values in the domain of the variable.
59. The equation is not an identity because sin ±1 cos2 .
7
Possible answer:
4
60. The equation is not an identity because tan ± sec 2 1.
3
Possible answer:
4
61. 2 3 26 i
62. 21 20i 63. 8 4i
5 ± 53
64. 9 46i
65. 3 ± 21 66.
2
67. 1 ± 5
68. 141 ± 7 333202CB05_AN.qxd
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Page 5
5
Precalculus with Limits, Answers to Section 5.3
43. 2 6n, 2 6n
Section 5.3 (page 396)
44. 2 8n, 2 8n
45. 2.678, 5.820
Vocabulary Check (page 396)
1. general
2. quadratic
1–6. Answers will vary.
8.
10.
7
11
2n,
2n
6
6
2
n
3
2
12.
n,
n
3
3
3. extraneous
2
4
7.
2 n,
2n
3
3
2
2n ,
2n
3
3
9.
5
n,
n
6
6
11.
3
13. n,
2n
2
5
2
14.
n,
n, n,
n
6
6
3
3
15.
2
n,
n
3
3
n 3 n
17.
,
8
2 8
2
19.
2
n,
n
3
3
16.
n 2 n
18.
,
9
3 9
3
n , n
3 4
3
, ,
2
2
23. 0, ,
Copyright © Houghton Mifflin Company. All rights reserved.
25.
5 7 11
,
,
,
6 6 6
6
5
,
,
3 3
24.
3 2 4
, , ,
2 2 3 3
26.
5
,
3 3
30.
7 3 11
, ,
6 2 6
5 7 11
, , ,
6 6 6 6
32.
5
,
3 3
33.
2
34.
5
,
4 4
7
4n,
4n
2
2
41. 1 4n
51. 0, 2.678, 3.142, 5.820
52. 0.515, 2.726, 3.657, 5.868
53. 0.983, 1.768, 4.124, 4.910
54. 0.524, 0.730, 2.412, 2.618
55. 0.3398, 0.8481, 2.2935, 2.8018
56. 0.5880, 2.0344, 3.7296, 5.1760
57. 1.9357, 2.7767, 5.0773, 5.9183
58. 1.7794, 4.5038
59.
5
, , arctan 5, arctan 5 4 4
5
60. arctan2 , arctan2 2, ,
4 4
5
,
3 3
62.
5
,
6 6
(b)
3
2␲
0.7854
4
5
3.9270
4
36.
−3
Maximum: 0.7854, 1.4142
Minimum: 3.9270, 1.4142
64. (a)
2
5
n,
n
3
6
n 5 n
38.
,
12
2 12
2
40.
8
10
4n,
4n
3
3
42.
3
4
n
(b)
3
2␲
0
31.
39.
50. 4.917
0
5
29. , ,
3 3
n
37.
12
3
49. 0.860, 3.426
22. 0, 28. 0
5
n,
n
6
6
48. 0.524, 2.618
63. (a)
27. No solution
35.
47. 1.047, 5.236
61.
n 2
4
20.
,
2n,
2n
4
2 3
3
21. 0,
46. 0.785, 2.356, 3.665, 3.927, 5.498, 5.760
−3
0.5236
6
1.5708
2
5
2.618
6
3
4.7124
2
Maximum: 0.5236, 1.5
Maximum: 2.6180, 1.5
Minimum: 1.5708, 1
65. 1
Minimum: 4.7124, 3
66. 0.739
67. (a) All real numbers x except x 0
(b) y-axis symmetry; Horizontal asymptote: y 1
(c) Oscillates
(d) Infinitely many solutions
(e) Yes, 0.6366
68. (a) All real numbers x except x 0
(b) y-axis symmetry
(c) y approaches 1.
(d) Four solutions: ± , ± 2
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6
Precalculus with Limits, Answers to Section 5.3
(Continued)
78. (a)
69. 0.04 second, 0.43 second, 0.83 second
70.
(b)
1.96 seconds
4
10
3
4
0
6
f
10
0
g
−4
For 3.5 ≤ x ≤ 6, the approximation appears to be good.
−4
(c) 3.46, 8.81
3.46 is close to the zero of f in the interval 0, 6
.
71. February, March, and April
72. January, October, November, December
73. 36.9, 53.1
74. 1.9
75. (a) Between t 8 seconds and t 24 seconds
(b) 5 times: t 16, 48, 80, 112, 144 seconds
79. True. The first equation has a smaller period than the second equation, so it will have more solutions in the interval
0, 2.
80. False. There is no value of x for which sin x 3.4.
81. 1
82. 3
83. C 24
84. C 19
6
a 54.8
a 15.4
4
b 50.1
c 5.0
76. (a)
Unemployment rate
y
8
2
85. sin 390 t
2
4
6
1
2
86. sin600 cos 390 (b) 1
tan 390 (c) The constant term, 5.45
3
cos600 2
3
87. sin1845 (e) 2010
Copyright © Houghton Mifflin Company. All rights reserved.
(b) 0.6 < x < 1.1
2
−2
A 1.12
␲
2
cos1845 1
2
tan600 3
3
(d) 13.37 years
0
2
8 10 12 14
Year (0 ↔ 1990)
77. (a)
3
2
2
2
88. sin1410 2
cos1410 tan1845 1
tan1410 89. 1.36
91. Answers will vary.
90. 30 feet
1
2
3
2
3
3
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7
Precalculus with Limits, Answers to Section 5.4
Section 5.4 (page 404)
11 2
3 1
12
4
11. sin
Vocabulary Check (page 404)
cos
tan u tan v
1 tan u tan v
2
11
3 1
12
4
tan
11
2 3
12
tan u tan v
1 tan u tan v
12. sin
1. sin u cos v cos u sin v
2. cos u cos v sin u sin v
3.
4. sin u cos v cos u sin v
5. cos u cos v sin u sin v
1. (a)
2. (a)
3. (a)
2 6
4
(b)
6 2
(b)
4
2 6
(b)
4
4. (a) 6 2
(b)
4
1 2
2
2 3
2
2 1
2
2 1
2
5. (a)
1
2
(b)
3 1
2
6. (a)
2 6
4
(b)
2 3
2
7. sin 105 cos 105 2
4
3 1
2
4
1 3 tan 105 2 3
8. sin 165 Copyright © Houghton Mifflin Company. All rights reserved.
6.
2
4
cos 165 3 1
2
4
3 1
tan 165 2 3
9. sin 195 2
4
cos 195 1 3 2
4
3 1
tan 195 2 3
10. sin 255 cos 255 2
4
2
4
3 1
1 3 tan 255 2 3
7 2
3 1
12
4
cos
7 2
1 3 12
4
tan
7
2 3
12
2
17
3 1
12
4
13. sin
cos
17 2
1 3 12
4
tan
17
2 3
12
12 2
2
14. sin 12 cos 1 3 4
3 1
4
12 2 3
tan 15. sin 285 cos 285 2
2
4
3 1
4
3 1
tan 285 2 3 16. sin105 cos105 2
4
2
4
3 1
1 3 tan105 2 3
17. sin165 2
cos165 4
3 1
2
4
1 3 tan165 2 3
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Page 8
8
Precalculus with Limits, Answers to Section 5.4
(Continued)
2
18. sin 15 4
3 1
2
cos 15 4
1 3 cos
2
13
1 3 12
4
tan
13
2 3
12
7
2 3
12
2
13
3 1
12
4
cos x sin x Copyright © Houghton Mifflin Company. All rights reserved.
24. sin 190
30. cos3x 2y
31.
3
2
3
36.
3
34.
37.
12
cos
35
3
2
2
2
63
65
40.
33
65
42.
65
33
43.
65
56
45.
3
5
46.
4
5
39.
16
65
63
16
16
44. 63
41.
4
(b) 2 cos 4
(b) 13 cos3 1.1760
4
89. 2 cos 90. 52
52
sin cos 2
2
(b) 2 cos 2 92. (a) All real numbers x except x 0
(b)
h
0.01
0.02
0.05
f h 0.504 0.509 0.521
gh 0.504 0.509 0.521
h
0.1
(b) 5 cos 2 0.6435 88. (a) 2 sin 2 25. tan 239
29. tan 3x
56
38.
65
87. (a) 13 sin 3 0.3948
28.
35. 1
5 7
,
4 4
sin x cos cos x sin cos x
2
2
2
86. (a) 5 sin2 0.9273
5 2
3 1
12
4
33.
71.
27. sin 1.8
2
67. cos 5
70. ,
3 3
cos x cos sin x sin sin x
2
2
2
85. (a) 2 sin 26. tan 80
32.
66. cos x
69.
2
81– 84. Answers will vary.
tan
2
55–64. Answers will vary.
80. True.
5
3 2
12
23. cos 40
53. 0
79. False.
5 2
22. sin
1 3 12
4
cos
117
125
78. False. cosu ± v cos u cos v sin u sin v
13
tan 2 3
12
x x1 x2
x2 1
50.
77. False. sinu ± v sin u cos v ± cos u sin v
2
13
cos 3 1
12
4
2 x 2 1 x2
4x2 1
5
3
76. Answers will vary.
21. sin 52.
49.
7
5
72. 0, , ,
73. ,
74. 0, 3
3
4 4
5
75. (a) y sin 2t 0.6435
12
5
1
(b)
feet
(c) cycle per second
12
2
7
cos 1 3 12
4
tan 51. 1
68. tan 2
7
3 1 12
4
117
44
65. sin x
13 2
19. sin
1 3 12
4
48.
54.
tan 15 2 3
20. sin 44
47. 117
0.2
0.5
f h 0.542 0.583 0.691
gh 0.542 0.583 0.691
4
91. Proof
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Page 9
9
Precalculus with Limits, Answers to Section 5.4
(Continued)
(c)
sin 2 2
−3
97. f 1x (d) As h → 0, the function f h approaches 0.5 and the
function gh approaches 0.5.
94. 15
x 15
5
98. f 1x 8x 7
99. Because f is not one-to-one, f 1 does not exist.
3
−2␲
2␲
−3
Copyright © Houghton Mifflin Company. All rights reserved.
96. (a) and (b) Answers will vary.
3
−2
93. 15
95.
sin 2 1
4
4
100. f 1x x 2 16
101. 4x 3
102. 3x 2
103. 6x 3
104. x 2 10x
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Page 10
10
Precalculus with Limits, Answers to Section 5.5
1
29. 83 4 cos 2x cos 4x
Section 5.5 (page 415)
30.
Vocabulary Check (page 415)
1. 2 sin u cos u
2. cos2 u
3. cos2 u sin2 u 2 cos2 u 1 1 2 sin2 u
1 cos u
4. tan2 u
5. ±
2
sin u
1 cos u
6.
sin u
1 cos u
7. 12cosu v cosu v
u 2 v cosu 2 v
uv
uv
10. 2 sin
sin
2 2 1.
6.
17
1
2.
4
17
17
15
7.
17
8
15
3.
17
8.
32.
1
128 3
33.
1
16 1
cos 2x cos 4x cos 2x cos 4x
34.
1
16 1
cos 2x cos 4x cos 2x cos 4 x
35.
417
17
36.
37.
1
4
38.
Copyright © Houghton Mifflin Company. All rights reserved.
41. sin 75 7 3 11
, , ,
2 6 2 6
11.
12.
3 3 5 7
, , , , ,
2 2 4 4 4 4
13. 0,
14.
7 11
, ,
2 6 6
15.
5 13 17
, ,
,
12 12 12 12
2 4
,
3 3
5 7 3 11
, , , , ,
2 6 6 6 2 6
3
17. 0, , ,
2
2
5 3 7
3
18. 0, , , , , , ,
4 2 4
4 2 4
19. 3 sin 2x
20. 3 cos 2x
22. cos 2x
23. sin 2u 21. 4 cos 2x
24
25
7
25
24
tan 2u 7
25. sin 2u 24
25
cos 2u 8
5.
15
5
9. 0, , ,
3
3
10.
5 3
16. , , ,
6 2 6 2
45
9
1
cos 2u 9
24. sin 2u tan 2u 45
cos 2u 15
17
tan 2u 24
7
8
tan 2u 15
42
28. sin 2u 3
7
cos 2u 9
42
tan 2u 7
17
17
17
4
40. 4
1
2
2 3
42. sin 165 122 3
cos 75 122 3
cos 165 122 3
tan 75 2 3
tan 165 3 2
2 2
1
cos 112 30 22 2
43. sin 112 30 1
2
tan 112 30 1 2
44. sin 67 30 1
1
2 2 45. sin 2 2
2
8
2
cos
1
2 2
8
2
tan 67 30 1 2
tan
2 1
8
46. sin
1
2 3
12 2
47. sin
3 1
2 2
8
2
cos
1
2 3
12 2
cos
3 1
2 2
8
2
tan
2 3
12
tan
3
2 1
8
cos 67 30 48. sin
1
2 2
2
7 1
2 3
12
2
49. sin
u 526
2
26
cos
7
1
2 3
12
2
cos
u 26
2
26
tan
7
2 3
12
tan
u
5
2
8
26. sin 2u 17
7
cos 2u 25
421
27. sin 2u 25
17
cos 2u 25
421
tan 2u 17
4 cos 4x cos 8x
39. 17
8
4.
17
15
8
cos 2x 28 cos 4x 16 cos 2x cos 4x cos 8x
1
31. 81 cos 4x
8. 12sinu v sin u v
9. 2 sin
1
128 35 48
50. sin
u 5
2
5
51. sin
u
2
89 1788
89
89 1788
cos
u 25
2
5
cos
u
2
tan
u 1
2 2
tan
u 8 89
2
5
89
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Page 11
11
Precalculus with Limits, Answers to Section 5.5
(Continued)
83.
u
1
10 3
cos 2
2
5
52. sin
10 310
5
u 1
2 2
53. sin
10
cos
u 310
2
10
3 1
84.
2
10
u
2
10
2
2␲
0
u 314
54. sin 2
14
u 70
cos 2
14
u 35
tan 2
5
55. sin 3x
−2
5 3 7
3
88. 0, , , , , , ,
4 2 4
4 2 4
57. tan 4x
58. sin
x1
2
5
60. 0, ,
3 3
59. 2
2
−2
1
2␲
0
5
,
6 6
2␲
0
−2
1
2
2␲
0
−1
144
169
91.
92.
93.
95–110. Answers will vary.
3
111.
−2␲
Copyright © Houghton Mifflin Company. All rights reserved.
−2
63. 3 sin
sin 0
2
64. 2 sin
7
sin
6
2
66. 3sin 60 sin 30
67. 12 sin 10 sin 2
68. 32 cos cos 5
69. 52cos 8 cos 2
70. 12 cos 2 cos 6
73.
1
2 sin
94.
36
65
112.
2␲
3
−2␲
2␲
−3
113.
−3
114.
3
−2␲
71.
4
13
−3
65. 5cos 60 cos 90
1
2 cos
2␲
0
−2
25
169
3
2␲
2␲
0
3
62. 0, ,
2 2
0
3 3 5 7
90. 0, , , , , , ,
2
2 4 4 4 4
2
−2
5
, ,
3
3
2␲
0
89.
61.
2
85. 2
86. 2
3
5 3 7
87. 0, , , , , , ,
4 2 4
4 2 4
u
tan 3
2
u
tan 3 10
2
56. cos 2x
3 1
2y cos 2x
72.
1
2 sin
2 x sin 2y
75. 2 cos 4 sin 76. 2 sin 2 cos 77. 2 cos 4x cos 2x
78. 2 sin 3x cos 2 x
79. 2 cos sin 80. 2 cos cos 2 cos 2
82. 2 sin x cos
0
2
␲
−3
−3
115.
116.
2 sin 2
81. 2 sin sin
−␲
2␲
y
74. 12cos 2 cos 2 121 cos 2
3
y
2
2
1
1
π
2π
x
π
−1
−1
−2
−2
2π
x
333202CB05_AN.qxd
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09:41 AM
Page 12
12
Precalculus with Limits, Answers to Section 5.5
(Continued)
128. (a)
117. 2x1 x2
118. 2x 2 1
120. (a) A 100 sin
cos
2
2
119. 23.85
2
−2␲
−2
(b) gx sin 2x
(b) A 50 sin The area is maximum when 2.
129. (a)
(c) Answers will vary.
y
6
121. (a) (b) 0.4482
(−1, 4)
(c) 760 miles per hour; 3420 miles per hour
(d) 2 sin1
1
1
2
3
4
5
−2
(c) Midpoint: 2, 3
(b) Distance 210
130. (a)
y
12
2 sinu cosu
(6, 10)
10
8
2sin u cos u
6
4
2 sin u cos u.
x
−3 − 2 − 1
−1
sin 2u sin2u
u
2
(5, 2)
2
1
M
123. False. For u < 0,
124. False. sin
5
3
122. x 2r1 cos 2
1 cos u
u
when is in the second
2
2
quadrant.
125. (a)
2␲
(b) 4
−6 −4
x
2
−2
4
6
8 10
(−4, − 3)
7
(c) Midpoint: 1, 2 (b) Distance 269
131. (a)
y
3
( 43 , 52)
2
2␲
0
Copyright © Houghton Mifflin Company. All rights reserved.
0
Maximum: , 3
126. (a)
1
(0, 12 )
7 3 11
(b) , , ,
2 6 2 6
2
1
132. (a)
127. (a)
1
3
Minima:
2 , 3 , 2 , 1
Maxima:
6 , 2 , 1
4 3
7 3
cos 4x
(c) 1 2 sin 2 x cos 2 x
11 3
,
6 2
−2
(b) 2 cos 4 x 2 cos 2 x 1
(d) 1 12 sin 2 2 x
(e) No. There is often more than one way to rewrite a
trigonometric expression.
(c) Midpoint:
23, 32 y
−3
2
2
(b) Distance 313
2␲
0
x
−1
( 13 , 23)
x
−1
1
−1
(−1, − 32 ) − 2
1
(b) Distance 6233
1
5
(c) Midpoint: 3, 12 133. (a) Complement: 35; supplement: 125
(b) No complement; supplement: 18
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09:41 AM
Page 13
13
Precalculus with Limits, Answers to Section 5.5
(Continued)
136. (a) Complement: 0.62; supplement: 2.19
134. (a) No complement; supplement: 71
(b) No complement; supplement: 0.38
(b) Complement: 12; supplement: 102
17
4
135. (a) Complement:
; supplement:
9
18
Copyright © Houghton Mifflin Company. All rights reserved.
(b) Complement:
11
; supplement:
20
20
137. September: $235,000; October: $272,600
138. 15.7 gallons
139. 127 feet
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09:41 AM
Page 14
14
Precalculus with Limits, Answers to Review Exercises
Review Exercises (page 420)
1. sec x
4. cot x
2. csc x
5. cot x
3. cos x
6. sec x
213
8. sin 13
313
cos 13
13
csc 2
3
cot 2
3
7. tan x 4
5
csc x 3
5
sec x 4
4
cot x 3
9. cos x 51. sin 285 2
1
10. cos 9
2
tan x 1
csc x 2
sec x 2
cot x 1
4
2
4
3 1
3 1
tan 285 2 3
52. sin 345 cos 345 2
4
1 3 2
4
1 3 tan 345 2 3
53. sin
25 2
3 1
12
4
95
csc 20
cos
25 2
3 1
12
4
sec 9
tan
25
2 3
12
tan 45
cot 5
54. sin
20
11. sin2 x
12. sec csc 13. 1
14. cos2 x
15. cot 16. tan u sec u
17. cot2 x
18. 1
cos
19. sec x 2 sin x
20. 2 tan2 x 2 sec x tan x 1
21. 2 tan2 22. 1 sin x
23–32. Answers will vary.
Copyright © Houghton Mifflin Company. All rights reserved.
cos 285 2
2
19
3 1
12
4
19 2
3 1
12
4
19
2 3
12
55. sin 15
56. cos 165
58. tan47
3
59. 52 5 47 tan
57. tan 35
33.
2
2n,
2n
3
3
34.
5
2n,
2n
3
3
60.
960 5077
1121
61.
1
57 36
52
35.
n
6
36.
5
2n,
2n
3
3
62.
127 15
52
63.
1
57 36
52
37.
2
n,
n
3
3
38.
5
n,
n
6
6
64.
960 5077
1121
65–70. Answers will vary.
40.
5
, ,
6 2 6
71.
7
,
4 4
72.
73.
11
,
6 6
74. 0, 39. 0,
2 4
,
3 3
41. 0, , 2
42. 0
3 9 11
43. , , ,
8 8 8 8
2
4 5
44. 0, , , , ,
3 3
3 3
3 5 7 9 11 13 15
45. 0, , , , , ,
,
,
8 8 8 8 8 8
8
8
46. No solution
47. 0, 48.
24
75. sin 2u 25
3
,
2 2
50.
3 7
, , arctan5 , arctan5 2
4 4
cos 2u 35
tan 2u 43
78.
2
−2␲
4
2␲
−2
49. arctan4 , arctan4 2, arctan 3,
arctan 3
4
76. sin 2u 5
7
cos 2u 25
tan 2u 24
7
77.
3
2
−2
1 cos 4x
1 cos 4x
3 4 cos 2x cos 4x
81.
41 cos 2x
79.
2
−1
1 cos 6x
2
1 cos 2x
82.
2
80.
333202CB05_AN.qxd
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09:41 AM
Page 15
15
Precalculus with Limits, Answers to Review Exercises
(Continued)
83. sin75 12
cos75 1
2
102. (a) V sin
2 3
1
(b) V 2 sin cubic meters
2 3
Volume is maximum when 2.
tan75 2 3
103.
cos 15 1
2
−2
1
19
2 3
12
2
1
1
104. y 210 sin8t arctan 3 1
105. 210 feet
19 1
2 3
cos
12
2
17
1
2 3
12
2
17
1
2 3
12
2
87. sin
cos
17
tan 2 3
12
u
88. sin 2
u
cos 2
89 8
289
89 8
289
108. False. Using the sum and difference formula,
sinx y sin x cos y cos x sin y.
u 10
2
10
109. True. 4 sinx cosx 4sin x cos x
u 310
10
2
4 sin x cos x
22 sin x cos x
u 1
tan 2 3
2 sin 2x
110. True by the product-to-sum formula
178889 89
111. Reciprocal identities:
178
178889 89
Copyright © Houghton Mifflin Company. All rights reserved.
cos
u
90. sin 2
u 70
2
14
cos
u 35
tan 2
5
91. cos 5x
u
2
21
7
12
6
125 93.
1
sin
2
3
1
95. 2cos 2 cos 8
96. 2sin 5 sin 97. 2 cos 3 sin 5
cos
2
2
100. 2 cos x sin
4
1
1
1
, cos , tan ,
csc sec cot csc 1
1
1
, sec , cot sin cos tan 6
99. 2 sin x sin
Quotient identities: tan sin cos , cot cos sin Pythagorean identities: sin2 cos2 1,
1 tan2 sec2 , 1 cot2 csc2 112. No. For an equation to be an identity, the equation must
1
be true for all real numbers x. sin 2 has an infinite
number of solutions but is not an identity.
113. 1 ≤ sin x ≤ 1 for all x
94. 3cos30 cos 60
98. 2 cos
15
35
u
7
tan 2 35
5
92. tan 3x
sin 178
u 89 8
tan 2
5
u 314
89. sin 2
14
106. 4 cycles per second
107. False. If 2 < < , then cos2 > 0. The sign of
cos2 depends on the quadrant in which 2 lies.
19
2 3
tan
12
cos 2␲
0
2 3
tan 15 2 3
86. sin 2
2 3
84. sin 15 1
2
85. sin
cos cubic meters
2
2
6
101. 15 or
12
114. (a) 49.91479, 59.86118
(b) 54.735616
115. y1 y2 1
116. y1 1 y2
117. 1.8431, 2.1758, 3.9903, 8.8935, 9.8820
118. 3.1395, 2.0000, 0.4378, 2.0000
333202CB05_AN.qxd
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09:41 AM
Page 16
16
Precalculus with Limits, Answers to Chapter Test
Chapter Test
(page 423)
15. 2sin 6 sin 2
313
2. 1
3. 1
4. csc sec 13
213
cos 13
13
csc 3
13
sec 2
2
cot 3
3
5. 0, < ≤ ,
< < 2
2
2
3
6.
7–12. Answers will vary.
1. sin −2␲
16. 2 cos
17. 0,
7
sin
2
2
7
3
, ,
4
4
18.
5 3
, , ,
6 2 6 2
19.
5 7 11
, , ,
6 6 6 6
20.
5 3
, ,
6 6 2
21. 2.938, 2.663, 1.170
25. t 0.26 minute
0.58 minute
−3
y1 y2
13.
1 10 15 cos 2x 6 cos 4x cos 6x
16
1 cos 2x
Copyright © Houghton Mifflin Company. All rights reserved.
14. tan 2
0.89 minute
2 6
4
23. sin 2u 45 , tan 2u 43, cos 2u 35
24. Day 123 to day 223
2␲
22.
1.20 minutes
1.52 minutes
1.83 minutes
333202CB05_AN.qxd
1/1/70
09:41 AM
Page 17
17
Precalculus with Limits, Answers to Problem Solving
Problem Solving (page 427)
1. (a) cos ± 1 sin tan ±
1 sin2 1 sin2 cot ±
sin 1
sec ±
1 sin2 1
csc sin 2 –3. Answers will vary.
sin2
sin 2 1 cos 0.6 W cos 8. (a) F sin 12
(b) 550
tan
(b) sin ± 1 1 cos2 tan ±
cos 1
csc ±
1 cos2 1
sec cos cos cot ±
1 cos2 cos2
(c) Maximum: 0
Minimum: 90
0
90
0
9. (a)
20
4. (a) p1t sin524 t
p2t 12 sin1048 t
0
0
p3t sin1572 t
1
3
10. (a) High tides: 6:12 A.M., 6:36 P.M.
Low tides: 12:00 A.M., 12:24 P.M.
(b) The water depth never falls below 7 feet.
(c) 70
p5t 15 sin2620 t
p6t 16 sin3144 t
1.4
1.4
p1(t)
−0.006
p2(t)
0.006
−0.006
0.006
− 1.4
− 1.4
1.4
1.4
p3(t)
−0.006
−0.006
− 1.4
p6(t)
Copyright © Houghton Mifflin Company. All rights reserved.
5
4
2
(b)
≤ x ≤
≤ x ≤
6
6
3
3
3
(c)
< x < ,
< x < 2
2
2
5
(d) 0 ≤ x ≤ ,
≤ x ≤ 2
4 4
11. (a)
12. (a) n 0.006
(b) 76.5
sin u cos v cos w sin u sin v sin w
1
1
p3: 786
p6: 1572
1
p5: 1310
cos u sin v cos w cos u cos v sin w
1
p is periodic with period 262
.
(b) tanu v w
(c) 0, 0, 0.00096, 0, 0.00191, 0
0.00285, 0, 0.00382, 0
(d) Maximum: 0.00024, 1.19524
Minimum: 0.00357, 1.19525
6. y 1 cos cos 2
2
7. sin 2
1
cot 3
2
2
13. (a) sinu v w
− 1.4
5. u v w
24
0
0.006
− 1.4
1.4
1
(b) p1: 262
1
p2: 524
0
p5(t)
0.006
−0.006
365
(b) t 91, t 274; Spring
Equinox and Fall Equinox
(c) Seward; The amplitudes:
6.4 and 1.9
(d) 365.2 days
1 2
64 v0
sin tan u tan v tan w tan u tan v tan w
1 tan u tan v tan u tan w tan v tan w
14. (a) cos 3 cos 4 sin2 cos (b) cos 4 cos4 6 sin2 cos2 sin4 2
15. (a)
(b) 233.3 times per second
15
1 cos 2
0
1
0