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d.Precalculus with Limits, Answers to Section 5.1 1
Chapter 5Section 5.1 (page 379)
Vocabulary Check (page 379)1. 2. 3. 4.5. 6. 7. 8.9. 10.
1. 2. sin
csc
cot
3. 4. sin
cos
sec
cot
5. 6.
cos
tan
csc
sec
7. 8. sin
tan
csc
sec
cot
9. 10.
11. sin 12.
cos
csc
sec
cot
13. 14. sin
cos
is undefined. is undefined.
csc
sec is undefined.
is undefined. cot
15. d 16. a 17. b 18. f
19. e 20. c 21. b 22. c
23. f 24. a 25. e
26. d 27. 28.
29. 30. 1 31.
32. 33. 34.
35. 1 36. 37.
38. 39. 40.
41. 42. 43.
44. 45. 46.
47. 48. 1 49.
50. 51. 52. sin4 xsec4 xcos x � 2
sec x � 1sin2 x tan2 x
cos2 xsin2 xcsc � sec �
cos u � sin u2 sec �sec �
sec t1 � sin ysin x
tan xsin2 �
cos 2 xsin2 xcot �
cos xcos2 �
sin �csc �
� � 0sec �
�csc � � �1
� � 1cot � � 0
tan �tan �
� � 0cos � � 0
� � 1sin � � �1
cot � � 2�6� �1
2
csc � � �5� � ��5
sec � � �5�612
� � ��5
2
tan � ��612
tan � � 2
cos � � �2�6
5� � �
�5
5
sin � � �15
� � �2�5
5
cot x ��1515
cot x � �2�2
csc x �4�15
15sec x � �
3�24
sec x � 4csc x � 3
tan x � �15tan x � ��24
cos x �14
cos x � �2�2
3
sin x ��15
4sin x �
13
x �4
3cot � � �
2�55
x �5
4csc � � �
3�55
x �5
3sec � �
32
x �3
4tan � � �
�52
cos x �45
cos � �23
x �3
5sin � � �
�53
� � ��10
3cot x �
12
5
� � �10csc x � �13
5
cot � � �3sec x � �1312
� � �1
3tan x �
512
� � �3�10
10cos x � �
12
13
sin � ��1010
sin x � �5
13
� �4
3cot � � �1
csc � �53
csc � � ��2
� �5
4sec � � �2
tan � �34
tan � � �1
� �4
5cos � �
�22
� �3
5sin � � �
�22
x � �3cot x � ��33
sec x � �2�3
3sec x � �2
x � �2csc x �2�3
3
tan x ��33
tan x � ��3
cos x � ��32
cos x � �12
x � �1
2sin x �
�32
�tan ucos ucsc ucos usec2 ucot2 ucsc ucot ucos utan u
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d.
Precalculus with Limits, Answers to Section 5.1 2
(Continued)
53. 54.
55. 56.
57. 58.
59. 60. 61.
62. 63. 64.
65. 66.
67. 68.
69.
70.
71.
72.
73. 74. 75.
76. 77. 78.
79. 80. 81.
82. 83.
84.
85.
86.
87. 88.
89. 90.
91. 92.
93. 94. 0
95. (a)
(b) csc22�
7� cot2
2�
7� 1.6360 � 0.6360 � 1
csc2 132 � cot 2 132 � 1.8107 � 0.8107 � 1
ln�csc t sec t�ln�tan x�ln�cot x�0 < � < �0 ≤ � <
�
2,
3�
2< � < 2�
�
2≤ � ≤
3�
20 ≤ � ≤ �
10 sin � � �5�3; sin � � ��32
; cos � �12
4 sin � � 2�2; sin � ��22
; cos � ��22
6 cos � � 3; sin � � ±�32
; cos � �12
3 cos � � 3; sin � � 0; cos � � 110 sec �
5 sec �2 tan �3 tan �
8 sin �3 sin �sec �
tan xcot xcsc x
y1� y2
00 �
2
1200
y1 � y2
10 �
2
12
y1 � y2
00 �
2
6
y1 � y2
00 �
2
1
tan4 x �csc x � 1�3�sec x � tan x�
5�sec x � tan x�1 � cos y
�cot x2 sec x�2 cot 2 x
2 csc2 x9 cos2 x4 cot2 x
�11 � 2 sin x cos x
tan2 x�sec x � 1�cot2 x�csc x � 1�sec2 x � tan2 xsin2 x � cos2 x
0.2 0.4 0.6 0.8 1.0
0.0428 0.2107 0.6871 2.1841 8.3087
0.0428 0.2107 0.6871 2.1841 8.3087y2
y1
x
1.2 1.4
50.3869 1163.6143
50.3869 1163.6143y2
y1
x
0.2 0.4 0.6 0.8 1.0
1.2230 1.5085 1.8958 2.4650 3.4082
1.2230 1.5085 1.8958 2.4650 3.4082y2
y1
x
1.2 1.4
5.3319 11.6814
5.3319 11.6814y2
y1
x
0.2 0.4 0.6 0.8 1.0
0.0403 0.1646 0.3863 0.7386 1.3105
0.0403 0.1646 0.3863 0.7386 1.3105y2
y1
x
1.2 1.4
2.3973 5.7135
2.3973 5.7135y2
y1
x
0.2 0.4 0.6 0.8 1.0
0.1987 0.3894 0.5646 0.7174 0.8415
0.1987 0.3894 0.5646 0.7174 0.8415y2
y1
x
1.2 1.4
0.9320 0.9854
0.9320 0.9854y2
y1
x
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d.Precalculus with Limits, Answers to Section 5.1 3
(Continued)
96. (a)
(b)
97. (a)
(b)
98. (a)
(b)
99.
100. Answers will vary.
101. True. For example,
102. False. A cofunction identity can be used to transform atangent function so that it can be represented by a cotan-gent function.
103. 1, 1 104. 1, 1 105. 106.
107. Not an identity because
108. Not an identity because
109. Not an identity because
110. Not an identity because
111. An identity because sin
112. Not an identity because
113–114. Answers will vary.
115. 116.
117. 118.
119. 120.
121. 122.
123. 124.y
x
−3
−2
−1
3
4
1
5
πππ 2−
y
x
−2
−4
−3
2
3
1
4
π3 π22
π2
y
x
−1
−2
−3
3
1 3−1−3
y
x1 3
−1
−2
1
2
x�x2 � 5x � 1�x2 � 25
�5x2 � 8x � 28�x2 � 4��x � 4�
3�2x � 1�x � 4
x2 � 6x � 8�x � 5��x � 8�
4z � 12�z � 9x � 25
1sin �
1
� �1
sin �� 1
5
cos �
1
5 cos �
sin k�
cos k�� tan k�
cot � � ±�csc2 � � 1
cos � � ±�1 � sin2 �
0, ���, 0
sin��x� � �sin x.
� tan �
sin��12� � �sin�1
2� � �0.4794
sin��250� � �sin 250 � 0.9397
cos��
2� 0.8� � sin 0.8 � 0.7174
cos�90 � 80� � sin 80 � 0.9848
�tan 3.1�2 � 1 � �sec 3.1�2 � 1.0017
�tan 346�2 � 1 � �sec 346�2 � 1.0622
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d.
Precalculus with Limits, Answers to Section 5.2 4
Section 5.2 (page 387)
Vocabulary Check (page 387)1. identity 2. conditional equation 3.4. 5. 6.7. 8.
1–38. Answers will vary.39. (a) (b)
Identity
(c) Answers will vary.
40. (a) (b)
Identity
(c) Answers will vary.
41. (a) (b)
Not an identity
(c) Answers will vary.
42. (a) (b)
Not an identity
(c) Answers will vary.
43. (a) (b)
Identity
(c) Answers will vary.
44. (a) (b)
Identity
(c) Answers will vary.45. (a) (b)
Not an identity
(c) Answers will vary.46. (a) (b)
Not an identity
(c) Answers will vary.47–50. Answers will vary. 51. 152. 1 53. 2 54. 2 55. Answers will vary.56. (a) Answers will vary.
(b)
(c) Greatest: Least: (d) Noon57. 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
Possible answer:
60. The equation is not an identity because
Possible answer:
61. 62. 63.
64. 65. 66.
67. 68. 14�1 ± �7�1 ± �5
�5 ± �532
�3 ± �21�9 � 46i
�8 � 4i�21 � 20i2 � �3 � �26 �i
3�
4
±�sec2 � � 1.tan � �
7�
4
±�1 � cos2 � .sin � �
�.
9010;
−5
3
−2� 2�
−3
−2� 2�
3
y2 y1
−1
−2� 2�
1
−1
−2� 2�
5
−5
−� �
5
−1
−2� 2�
5
y2
y1
−1
−2� 2�
3
−5
−5
5
5
sec u�csc usin ucos2 ucot u
tan u
28.36 13.74 8.66 5.96 4.20s
5040302010�
2.89 1.82 0.88 0s
90807060�
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d.Precalculus with Limits, Answers to Section 5.3 5
Section 5.3 (page 396)
Vocabulary Check (page 396)1. general 2. quadratic 3. extraneous
1–6. Answers will vary. 7.
8. 9.
10. 11.
12. 13.
14.
15. 16.
17. 18.
19.
20.
21. 22.
23. 24.
25. 26.
27. No solution 28. 0
29. 30.
31. 32.
33. 34.
35. 36.
37. 38.
39. 40.
41. 42.
43. 44.
45. 2.678, 5.820
46. 0.785, 2.356, 3.665, 3.927, 5.498, 5.760
47. 1.047, 5.236 48. 0.524, 2.618
49. 0.860, 3.426 50. 4.917
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.
60.
61. 62.
63. (a) (b)
Maximum:Minimum:
64. (a)
Maximum:
Maximum:
Minimum:
Minimum:65. 1 66. 0.739
67. (a) All real numbers except
(b) -axis symmetry; Horizontal asymptote:
(c) Oscillates (d) Infinitely many solutions
(e) Yes, 0.6366
68. (a) All real numbers except
(b) -axis symmetry (c) approaches 1.
(d) Four solutions: ±�, ±2�
yy
x � 0x
y � 1y
x � 0x
�4.7124, �3��1.5708, 1��2.6180, 1.5��0.5236, 1.5�
0
−3
2�
3
�3.9270, �1.4142��0.7854, 1.4142�
5�
4� 3.9270
�
4� 0.7854
−3
0 2�
3
�
6,
5�
6�
3,
5�
3
arctan��2� � �, arctan��2� � 2�, �
4,
5�
4
�
4,
5�
4, arctan 5, arctan 5 � �
�2 � 8n, 2 � 8n�2 � 6n, 2 � 6n
34 � n�1 � 4n
8�
3� 4n�,
10�
3� 4n�
�
2� 4n�,
7�
2� 4n�
�
12�
n�
2,
5�
12�
n�
2�
12�
n�
3
2�
3� n�,
5�
6� n�
�
6� n�,
5�
6� n�
�
4,
5�
4�
2
�
3,
5�
3
�
6,
5�
6,
7�
6,
11�
6
7�
6,
3�
2,
11�
6�,
�
3,
5�
3
�
3,
5�
3
�
3,
5�
3, �
�
2,
3�
2,
2�
3,
4�
30, �,
�
6,
5�
6,
7�
6,
11�
6
0, �0, �
2, �,
3�
2
�
4�
n�
2,
2�
3� 2n�,
4�
3� 2n�
n�
3,
�
4� n�
�
9�
n�
3,
2�
9�
n�
3�
8�
n�
2,
3�
8�
n�
2
�
3� n�,
2�
3� n�
�
3� n�,
2�
3� n�
�
6� n�,
5�
6� n�,
�
3� n�,
2�
3� n�
n�, 3�
2� 2n�
�
3� n�,
2�
3� n�
�
6� n�,
5�
6� n�
2�
3� n�
�
3� 2n� ,
2�
3� 2n�
7�
6� 2n�,
11�
6� 2n�
2�
3� 2n�,
4�
3� 2n�
(b)
3�
2� 4.7124
5�
6� 2.618
�
2� 1.5708
�
6� 0.5236
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d.
Precalculus with Limits, Answers to Section 5.3 6
(Continued)
69. 0.04 second, 0.43 second, 0.83 second
70. 1.96 seconds
71. February, March, and April
72. January, October, November, December
73. 74.
75. (a) Between seconds and seconds
(b) 5 times: seconds
76. (a)
(b) 1
(c) The constant term, 5.45
(d) years
(e) 2010
77. (a) (b)
78. (a)
(b)
For the approximation appears to be good.
(c) 3.46, 8.81
3.46 is close to the zero of in the interval
79. True. The first equation has a smaller period than the sec-ond equation, so it will have more solutions in the interval
80. False. There is no value of for which
81. 1 82. 3
83. 84.
85. 86.
87. 88.
89. 90. 30 feet
91. Answers will vary.
1.36
tan��1410� ��33
tan��1845� � �1
cos��1410� ��32
cos��1845� ��22
sin��1410� �12
sin��1845� � ��22
tan�600� � �3tan 390 ��33
cos�600� � �12
cos 390 ��32
sin�600� � ��32
sin 390 �12
c � 5.0b � 50.1
a � 15.4a � 54.8
C � 19C � 24
sin x � 3.4.x
0, 2��.
0, 6.f
3.5 ≤ x ≤ 6,
−4
0 6
fg
4
103
A � 1.12
0.6 < x < 1.1
−2
0 �2
2
�13.37
y
t
8
6
4
2
2 4 6 8 10 12 14
Une
mpl
oym
ent r
ate
Year (0 ↔ 1990)
t � 16, 48, 80, 112, 144
t � 24t � 8
1.936.9, 53.1
−4
0 10
4
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d.Precalculus with Limits, Answers to Section 5.4 7
Section 5.4 (page 404)
Vocabulary Check (page 404)1.
2. 3.
4.
5. 6.
1. (a) (b)
2. (a) (b)
3. (a) (b)
4. (a) (b)
5. (a) (b)
6. (a) (b)
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
tan��165� � 2 � �3
cos��165� � ��24
�1 � �3 �
sin��165� � ��24
��3 � 1�
tan��105� � 2 ��3
cos��105� ��2
4�1 � �3 �
sin��105� � ��2
4��3 � 1�
tan 285 � ��2 � �3 �
cos 285 ��2
4��3 � 1�
sin 285 � ��2
4��3 � 1�
tan�� �
12� � �2 � �3
cos�� �
12� ��2
4��3 � 1�
sin�� �
12� ��2
4�1 � �3 �
tan 17�
12� 2 � �3
cos 17�
12�
�2
4�1 � �3 �
sin 17�
12� �
�2
4��3 � 1�
tan 7�
12� �2 ��3
cos 7�
12�
�2
4�1 � �3 �
sin 7�
12�
�2
4��3 � 1�
tan 11�
12� �2 � �3
cos 11�
12� �
�2
4��3 � 1�
sin 11�
12�
�2
4��3 � 1�
tan 255 � 2 ��3
cos 255 ��2
4�1 � �3 �
sin 255 � ��2
4��3 � 1�
tan 195 � 2 � �3
cos 195 � ��2
4��3 � 1�
sin 195 ��2
4�1 � �3 �
tan 165 � �2 ��3
cos 165 � ��2
4��3 � 1�
sin 165 ��2
4��3 � 1�
tan 105 � �2 � �3
cos 105 ��2
4�1 � �3 �
sin 105 ��2
4��3 � 1�
��2 � �32
�2 � �64
��3 � 1
2
1
2
�2 � 1
2��6 � �2
4
�2 � 1
2
�2 � �6
4
�2 � �32
�6 � �24
�1 � �22
��2 � �64
tan u � tan v1 � tan u tan v
cos u cos v � sin u sin v
sin u cos v � cos u sin v
tan u � tan v1 � tan u tan v
cos u cos v � sin u sin v
sin u cos v � cos u sin v
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d.
Precalculus with Limits, Answers to Section 5.4 8
(Continued)
18.
19.
20.
21.
22.
23. 24. 25.
26. 27. 28.
29. 30. 31.
32. 33. 34.
35. 36. 37.
38. 39. 40.
41. 42. 43.
44. 45. 46.
47. 48. 49. 50.
51. 1 52. 53. 0
54. 55–64. Answers will vary.
65. 66. 67.
68. 69. 70. 71.
72. 73. 74.
75. (a)
(b) feet (c) cycle per second
76. Answers will vary.
77. False.
78. False.
79. False.
80. True.
81– 84. Answers will vary.
85. (a) (b)
86. (a) (b)
87. (a) (b)
88. (a) (b)
89. 90. 91. Proof
92. (a) All real numbers except
(b)
x � 0x
�5�2
2 sin � �
5�2
2 cos �2 cos �
�2 cos�2� ��
4��2 sin�2� ��
4�13 cos�3� � 1.1760�13 sin�3� � 0.3948�
5 cos�2� � 0.6435�5 sin�2� � 0.9273�
�2 cos�� ��
4��2 sin�� ��
4�
sin�x ��
2� � sin x cos �
2� cos x sin
�
2� �cos x
cos�x ��
2� � cos x cos �
2� sin x sin
�
2� sin x
cos�u ± v� � cos u cos v � sin u sin v
sin�u ± v� � sin u cos v ± cos u sin v
1
�
5
12
y �5
12 sin�2t � 0.6435�
0, ��
4,
7�
40,
�
3, �,
5�
3
5�
4,
7�
4�
3,
5�
3�
2tan �
�cos ��cos x�sin x
x � x�1 � x2
�x2 � 1
2x2 � �1 � x2
�4x2 � 1
117125
53
11744�
44117
45
35�
1663
6556
6533�
6316
�3365
1665
5665
�6365
�33
�1
�22
�32
�22
��32
cos�3x � 2y�tan 3x
cos 12�
35sin 1.8tan 80
tan 239sin 190cos 40
tan 5�
12� �3 � 2
cos 5�
12�
�2
4��3 � 1�
sin 5�
12�
�2
4�1 � �3 �
tan��13�
12 � � �2 � �3
cos��13�
12 � � ��2
4��3 � 1�
sin��13�
12 � ��2
4��3 � 1�
tan��7�
12� � 2 � �3
cos��7�
12� ��24
�1 � �3 �
sin��7�
12� � ��24
��3 � 1 �
tan 13�
12� 2 � �3
cos 13�
12� �
�24
�1 � �3 �
sin 13�
12�
�24
�1 � �3 �
tan 15 � 2 � �3
cos 15 ��24
�1 � �3 �
sin 15 ��24
��3 � 1�
0.01 0.02 0.05
�0.521�0.509�0.504g�h�
�0.521�0.509�0.504f �h�
h
0.1 0.2 0.5
�0.691�0.583�0.542g�h�
�0.691�0.583�0.542f �h�
h
333202CB05_AN.qxd 1/1/70 09:40 AM Page 8
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d.Precalculus with Limits, Answers to Section 5.4 9
(Continued)
(c)
(d) As the function approaches and thefunction approaches
93. 94.95.
96. (a) and (b) Answers will vary.
97.
98.
99. Because is not one-to-one, does not exist.
100. 101.
102. 103.
104. x2 � 10x
6x � 33x2
4x � 3f �1�x� � x2 � 16
f �1f
f �1�x� � �8x � 7
f �1�x� �x � 15
5
sin2�� ��
4� � sin2�� ��
4� � 1
−3
−2� 2�
3
1515
�0.5.g�h��0.5f �h�h → 0,
−2
−3 3
2
333202CB05_AN.qxd 1/1/70 09:40 AM Page 9
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d.
Precalculus with Limits, Answers to Section 5.5 10
Section 5.5 (page 415)
Vocabulary Check (page 415)1. 2.3.
4. 5.
6.
7.
8.
9.
10.
1. 2. 3. 4. 5.
6. 7. 8. 9.
10. 11.
12. 13.
14. 15.
16. 17.
18. 19.
20. 21. 22.
23. sin 24. sin
cos cos
tan tan
25. sin 26. sin
cos cos
tan tan
27. sin 28. sin
cos cos
tan tan
29.
30.
31.
32.
33.
34.
35. 36.
37. 38.
39. 40. 4
41. 42. sin
cos
tan
43. sin
cos
tan
44. sin 45. sin
cos cos
tan tan
46. sin 47.
cos
tan
48. 49.
50. 51. sin
cos
tan u
2�
8 ��89
5tan
u2
�12
u
2� ��89 � 8�89
178cos
u2
�2�5
5
u
2��89 � 8�89
178sin
u2
��55
tan u2
� 5tan 7�
12� �2 � �3
cos u2
��2626
cos 7�
12� �
12�2 � �3
sin u2
�5�26
26sin
7�
12�
12�2 � �3
tan 3�
8� �2 � 1
�
12� 2 � �3
cos 3�
8�
12�2 � �2
�
12�
1
2�2 � �3
sin 3�
8�
12�2 � �2
�
12�
1
2�2 � �3
�
8� �2 � 167 30� � 1 ��2
�
8�
1
2�2 � �267 30� �
1
2�2 � �2
�
8�
1
2�2 � �267 30� �
1
2�2 � �2
112 30� � �1 � �2
112 30� � �12�2 � �2
112 30� �12�2 � �2
165 � �3 � 2 tan 75 � 2 ��3
165 � �12�2 ��3 cos 75 �
12�2 � �3
165 �12�2 � �3 sin 75 �
12�2 � �3
�17
�174
1
4
�1717
4�1717
116�1 � cos 2x � cos 4x � cos 2x cos 4x�
116�1 � cos 2x � cos 4x � cos 2x cos 4x�
1128�3 � 4 cos 4x � cos 8x�
18�1 � cos 4x�
1128�35 � 48 cos 2x � 28 cos 4x �16 cos 2x cos 4x � cos 8x�
18�3 � 4 cos 2x � cos 4x�
2u � �4�2
7 2u �
4�21
17
2u �7
9 2u � �
17
25
2u � �4�2
3 2u � �
4�21
25
2u � �8152u �
247
2u �15172u �
725
2u � �8172u �
2425
2u � 4�5 2u � �24
7
2u � �1
9 2u � �
7
25
2u � �4�5
9 2u �
24
25
cos 2x4 cos 2x3 cos 2x
3 sin 2x0, �
4,
�
2,
3�
4, �,
5�
4,
3�
2,
7�
4
0, �
2, �,
3�
2
�
6,
�
2,
5�
6,
3�
2
�
2,
�
6,
5�
6,
7�
6,
3�
2,
11�
6
�
2,
7�
6,
11�
6
0, 2�
3,
4�
3
�
2,
3�
2,
�
4,
3�
4,
5�
4,
7�
4
�
12,
5�
12,
13�
12,
17�
12
�
2,
7�
6,
3�
2,
11�
6
0, �
3, �,
5�
3158
178
1715
815
817
1517
14
�1717
�2 sin�u � v2 � sin�u � v
2 �2 sin�u � v
2 � cos�u � v2 �
12sin�u � v� � sin� u � v�
12cos�u � v� � cos�u � v�
1 � cos usin u
�sin u
1 � cos u
±�1 � cos u2
tan2 u
cos2 u � sin2 u � 2 cos2 u � 1 � 1 � 2 sin2 ucos2 u2 sin u cos u
333202CB05_AN.qxd 1/1/70 09:40 AM Page 10
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d.Precalculus with Limits, Answers to Section 5.5 11
(Continued)
52. sin 53. sin
cos cos
tan tan
54. sin 55.
cos
tan
56. 57. 58.
59. 60.
61. 62.
63. 64.
65. 66.
67. 68.
69. 70.
71. 72.
73.
74. 75.
76. 77.
78. 79.
80.
81. 82.
83. 84.
85. 86.
87.
88.
89. 90.
91. 92. 93. 94.95–110. Answers will vary.
111. 112.
113. 114.
115. 116.
x
2
1
−1
−2
ππ 2
y
x
2
1
−1
−2
ππ 2
y
−3
−� �
3
−3
−2� 2�
3
−3
−2� 2�
3
−3
−2� 2�
3
3665
413
144169
25169
0
−1
2�
1
0
−2
2�
2
0, �
2, �,
3�
2,
�
4,
3�
4,
5�
4,
7�
4
�
6,
5�
6
0
−2
2�
2
0, �
4,
�
2,
3�
4, �,
5�
4,
3�
2,
7�
4
0
−2
2�
2
0, �
4,
�
2,
3�
4, �,
5�
4,
3�
2,
7�
4
��2��2
�3 � 12
�3 � 12
2 sin x cos �
2� 0�2 sin � sin
�
2
2 cos�� � �� cos � � �2 cos�� � ��
2 cos � sin �2 sin 3x cos 2x
2 cos 4x cos 2x2 sin 2� cos �
2 cos 4� sin �12�cos 2� � cos 2�� �
12�1 � cos 2��
12 �sin 2� � sin 2��
12�sin 2x � sin 2y�1
2�cos 2y � cos 2x�
12 �cos 2� � cos 6��5
2�cos 8� � cos 2��
32�cos � � cos 5��1
2 �sin 10� � sin 2��
3�sin 60 � sin 30�5�cos 60 � cos 90�
2�sin 7�
6� sin
�
2�3�sin �
2� sin 0�
0
−3
2�
3
0
−2
2�
2
0, �
2,
3�
2�
3, �,
5�
3
−2
0 2�
1
0
−2
2�
2
0, �
3,
5�
3�
��sin�x � 1
2 ����tan 4x��cos 2x�
u
2�
3�5
5
u
2�
�70
14
�sin 3x�u
2�
3�14
14
u
2� �3
u
2� �3 � �10
u
2� �
�10
10
u
2� �
1
2�10 � 3�10
5
u
2�
3�10
10
u
2�
1
2 �10 � 3�10
5
333202CB05_AN.qxd 1/1/70 09:40 AM Page 11
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d.
Precalculus with Limits, Answers to Section 5.5 12
(Continued)
117. 118. 119.
120. (a)
(b)
The area is maximum when
121. (a) (b) 0.4482
(c) 760 miles per hour; 3420 miles per hour
(d)
122.
123. False. For
124. False. when is in the second
quadrant.
125. (a) (b)
Maximum:
126. (a) (b)
Minima:
Maxima:
127. (a) (b)
(c) (d)
(e) No. There is often more than one way to rewrite atrigonometric expression.
128. (a)
(b) (c) Answers will vary.
129. (a)
(b) (c) Midpoint:
130. (a)
(b) (c) Midpoint:
131. (a)
(b) (c) Midpoint:
132. (a)
(b) (c) Midpoint:
133. (a) Complement: supplement:
(b) No complement; supplement: 18
12535;
��13, � 5
12�Distance �16�233
y
x
( )13
, 23
( )32
−1−2 1
−1
−2
1
−1, −
�23, 32�Distance �
23�13
y
x−1 1 2
1
2
3
( )43
, 52
( )0, 12
�1, 72�Distance � �269
y
x−4−6 2 4 6 8 10
−2
2
4
6
8
10
12
(−4, −3)
(6, 10)
�2, 3�Distance � 2�10
y
x−1−2−3 1 2 3 4 5
−1
−2
1
2
3
5
6
(−1, 4)
(5, 2)
g�x� � sin 2x
−2
−2� 2�
2
1 �12 sin2 2x1 � 2 sin2 x cos2 x
2 cos4 x � 2 cos2 x � 114�3 � cos 4x�
� 7�
6,
3
2� , � 11�
6,
3
2�
��
2, �3� , � 3�
2, 1�
�
2,
7�
6,
3�
2,
11�
6
−3
0 2�
2
��, 3�
�
020 �
4
u2
sin u2
��1 � cos u2
� 2 sin u cos u.
� �2��sin u� cos u
� �2 sin��u� cos��u�
sin 2u � �sin��2u�
u < 0,
x � 2r�1 � cos ��
� � 2 sin�1� 1M�
�
� � ��2.
A � 50 sin �
A � 100 sin �
2 cos
�
2
23.852x2 � 12x�1 � x2
333202CB05_AN.qxd 1/1/70 09:41 AM Page 12
Precalculus with Limits, Answers to Section 5.5 13
(Continued)
134. (a) No complement; supplement:
(b) Complement: supplement:
135. (a) Complement: supplement:
(b) Complement: supplement:
136. (a) Complement: supplement: 2.19
(b) No complement; supplement: 0.38
137. September: $235,000; October: $272,600
138. gallons
139. �127 feet
�15.7
0.62;
11�
20�
20;
17�
184�
9;
10212;
71
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d.
333202CB05_AN.qxd 1/1/70 09:41 AM Page 13
Review Exercises (page 420)1. 2. 3.4. 5. 6.
7. 8.
10.
11. 12. 13. 1 14.
15. 16. 17. 18. 1
19. 20.
21. 22.
23–32. Answers will vary.
33. 34.
35. 36.
37. 38.
39. 40.
41. 42. 0
43. 44.
45.
46. No solution 47. 48.
49.
50.
51.
52.
53.
54.
55. 56. 57.
58. 59.
60. 61.
62. 63.
64. 65–70. Answers will vary.
71. 72.
73. 74.
75. 76.
77. 78.
79. 80.
81. 82.1 � cos 2x
23 � 4 cos 2x � cos 4x
4�1 � cos 2x�
1 � cos 6x2
1 � cos 4x1 � cos 4x
−1
−2� 2�
4
−2
−2� 2�
2
tan 2u � �43tan 2u � �
247
cos 2u �35cos 2u � �
725
sin 2u � �45sin 2u �
2425
0, ��
6,
11�
6
3�
2�
4,
7�
4
�960 � 507�71121
152
�5�7 � 36�12�7 � 1552
152
�5�7 � 36�960 � 507�71121
�352�5 � 4�7 �tan��47�
tan 35cos 165sin 15
tan 19�
12� �2 � �3
cos 19�
12�
�24
��3 � 1�
sin 19�
12� �
�24
��3 � 1�
tan 25�
12� 2 � �3
cos 25�
12�
�24
��3 � 1�
sin 25�
12�
�24
��3 � 1�
tan 345 � �2 � �3
cos 345 ��24
�1 � �3 �
sin 345 ��24
�1 � �3 �
tan 285 � �2 � �3
cos 285 ��24
��3 � 1�
sin 285 � ��24
��3 � 1�
3�
4,
7�
4, arctan��5� � �, arctan��5� � 2�
� � arctan 3
arctan��4� � �, arctan��4� � 2�, arctan 3,
�
2,
3�
20, �
0, �
8,
3�
8,
5�
8,
7�
8,
9�
8,
11�
8,
13�
8,
15�
8
0, �
3,
2�
3, �,
4�
3,
5�
3�
8,
3�
8,
9�
8,
11�
8
0, �
2, �
�
6,
�
2,
5�
60,
2�
3,
4�
3
�
6� n�,
5�
6� n�
�
3� n�,
2�
3� n�
�
3� 2n�,
5�
3� 2n�
�
6� n�
�
3� 2n�,
5�
3� 2n�
�
3� 2n�,
2�
3� 2n�
1 � sin x�2 tan2 �
2 tan2 x � 2 sec x tan x � 1sec x � 2 sin x
cot2 xtan u sec ucot �
cos2 xsec � csc �sin2 x
cot � ��520
sec � � 9
csc � �9�520
tan � � 4�5
cos � �19
cot � �32
cot x �43
csc � ��13
2sec x �
54
cos � �3�13
13csc x �
53
sin � �2�13
13tan x �
34
�sec x�cot xcot xcos xcsc xsec x
Precalculus with Limits, Answers to Review Exercises 14
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d.
9.
cot x � �1sec x � �2csc x � ��2tan x � �1
cos x ��22
333202CB05_AN.qxd 1/1/70 09:41 AM Page 14
Precalculus with Limits, Answers to Review Exercises 15
(Continued)
83.
84.
85.
86. 87.
88.
89. 90.
91. 92. 93.
94. 95.
96. 97.
98. 99.
100. 101.
102. (a)
(b)
Volume is maximum when
103.
104.
105. feet 106. cycles per second
107. False. If then The sign ofdepends on the quadrant in which lies.
108. False. Using the sum and difference formula,
109. True.
110. True by the product-to-sum formula
111. Reciprocal identities:
Quotient identities:
Pythagorean identities:
112. No. For an equation to be an identity, the equation must be true for all real numbers has an infinitenumber of solutions but is not an identity.
113. for all
114. (a) (b)
115.
116.
117.
118. �3.1395, �2.0000, �0.4378, 2.0000
�1.8431, 2.1758, 3.9903, 8.8935, 9.8820
y1 � 1 � y2
y1 � y2 � 1
54.73561649.91479, 59.86118
x�1 ≤ sin x ≤ 1
sin � �12x.
1 � cot2 � � csc2 �1 � tan2 � � sec2 �,
sin2 � � cos2 � � 1,
tan � �sin �
cos �, cot � �
cos �
sin �
cot � �1
tan �sec � �
1cos �
,csc � �1
sin �,
tan � �1
cot �,cos � �
1sec �
,sin � �1
csc �,
� �2 sin 2x
� �2�2 sin x cos x�
� �4 sin x cos x
4 sin��x� cos��x� � 4��sin x� cos x
sin x cos y � cos x sin y.sin�x � y� �
��2cos���2�cos���2� > 0.���2� < � < �,
4��12�10
y �12�10 sin�8t � arctan 13�
−2
0 2�
2
� � ��2.
V �12 sin � cubic meters
V � sin �
2 cos
�
2 cubic meters
� � 15 or �
122 cos x sin
�
4
�2 sin x sin �
62 cos
5�
2 cos
�
2
2 cos 3� sin �2�sin 5� � sin ��
12�cos 2� � cos 8��3cos��30� � cos 60
12
sin �
3tan 3x��cos 5x�
tan u2
�7
�35�
�355
tan u2
�3�5
5
cos u2
�� 512
��15
6cos
u2
��7014
sin u2
�� 712
��21
6sin
u2
�3�14
14
tan u2
��89 � 8
5
cos u2
� ���89 � 8
2�89���178�8�89 � 89�
178
sin u2
���89 � 8
2�89��178�8�89 � 89�
178
tan u2
�13
tan��17�
12 � � �2 � �3
cos u
2�
3�1010
cos��17�
12 � � �12�2 � �3
sin u2
��1010
sin��17�
12 � �12�2 � �3
tan 19�
12� �2 � �3
cos 19�
12�
12�2 � �3
sin 19�
12� �
12�2 � �3
tan 15 � 2 � �3
cos 15 �12�2 � �3
sin 15 �12�2 � �3
tan��75� � �2 � �3
cos��75� �12�2 � �3
sin��75� � �12�2 � �3
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333202CB05_AN.qxd 1/1/70 09:41 AM Page 15
Chapter Test (page 423)
1. 2. 1 3. 1 4.
5.
6. 7–12. Answers will vary.
13.
14.
15.
16.
17.
18.
19.
20.
21. 22.
23.
24. Day 123 to day 223
25.
1.83 minutes
1.52 minutes
1.20 minutes
0.89 minute
0.58 minute
t � 0.26 minute
sin 2u �45 , tan 2u � �
43, cos 2u � �
35
�2 � �6
4�2.938, �2.663, 1.170
�
6,
5�
6,
3�
2
�
6,
5�
6,
7�
6,
11�
6
�
6,
�
2,
5�
6,
3�
2
0, 3�
4, �,
7�
4
�2 cos 7�
2 sin
�
2
2�sin 6� � sin 2��
tan 2�
116�
10 � 15 cos 2x � 6 cos 4x � cos 6x1 � cos 2x �
y1 � y2
−3
−2� 2�
3
3�
2< � < 2�� � 0,
�
2< � ≤ �,
cot � �2
3
sec � � ��13
2
csc � � ��13
3
cos � � �2�13
13
csc � sec �sin � � �3�13
13
Precalculus with Limits, Answers to Chapter Test 16
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Precalculus with Limits, Answers to Problem Solving 17
Problem Solving (page 427)1. (a) (b)
2–3. Answers will vary.
4. (a)
(b)
is periodic with period
(c)
(d) Maximum:
Minimum:
5. 6.
7.
8. (a)
(b) (c) Maximum:
Minimum:
9. (a) (b) SpringEquinox and Fall Equinox
(c) Seward; The amplitudes:6.4 and 1.9
(d) 365.2 days
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)
11. (a) (b)
(c)
(d)
12. (a) (b)
13. (a)
(b)
14. (a)
(b)
15. (a) (b) 233.3 times per second
00 1
15
cos 4� � cos4 � � 6 sin2 � cos2 � � sin4 �
cos 3� � cos � � 4 sin2 � cos �
�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
tan�u � v � w�
� cos u cos v sin w� cos u sin v cos w
� sin u sin v sin w� sin u cos v cos w
sin�u � v � w�
� � 76.5n �12�cot
�
2� �3�
5�
4≤ x ≤ 2�0 ≤ x ≤
�
4,
3�
2< x < 2�
�
2< x < �,
2�
3≤ x ≤
4�
3�
6≤ x ≤
5�
6
00 24
70
t � 91, t � 274;
00 365
20
� � 90
� � 0
00 90
550
F �0.6 W cos �
sin 12
tan �
2�
sin �1 � cos �
cos �
2��1 � cos �
2
sin �
2��1 � cos �
2
y �164 v0
2 sin2 �u � v � w
�0.00357, �1.19525��0.00024, 1.19524�
�0.00382, 0��0.00285, 0�,�0, 0�, �0.00096, 0�, �0.00191, 0�
1262.p
p5: 1
1310p2: 1
524
p6: 1
1572p3: 1
786p1: 1
262
p6(t)
−1.4
−0.006 0.006
1.4
p5(t)
−1.4
−0.006 0.006
1.4
p3(t)
−1.4
−0.006 0.006
1.4
p2(t)
−1.4
−0.006 0.006
1.4
−1.4
−0.006 0.006
1.4
p1(t)
p6�t� �16 sin�3144 �t�
p5�t� �15 sin�2620 �t�
p3�t� �13 sin�1572 �t�
p2�t� �12 sin�1048 �t�
p1�t� � sin�524 �t�
cot � � ±cos �
�1 � cos2 �csc � �
1sin �
sec � �1
cos �sec � � ±
1�1 � sin2 �
csc � � ±1
�1 � cos2 �cot � � ±
�1 � sin2 �sin �
tan � � ±�1 � cos2 �
cos �tan � � ±
sin ��1 � sin2 �
sin � � ±�1 � cos2 �cos � � ±�1 � sin2 �
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