chapter 4 16.college.cengage.com/mathematics/larson/precalculus_limits/1e2/... · vocabulary check...
TRANSCRIPT
Precalculus with Limits, Answers to Section 4.1 1
Chapter 4Section 4.1 (page 290)
Vocabulary Check (page 290)1. Trignometry 2. angle 3. coterminal4. radian 5. acute; obtuse6. complementary; supplementary 7. degree8. linear 9. angular 10.
1. 2 radians 2. 5.5 radians 3. radians
4. radians 5. 1 radian 6. 6.5 radians
7. (a) Quadrant I (b) Quadrant III
8. (a) Quadrant III (b) Quadrant III
9. (a) Quadrant IV (b) Quadrant III
10. (a) Quadrant IV (b) Quadrant II
11. (a) Quadrant III (b) Quadrant II
12. (a) Quadrant IV (b) Quadrant II
13. (a) (b)
14. (a) (b)
15. (a) (b)
16. (a) (b)
17. (a) (b)
18. (a) (b)
19. (a) (b)
20. (a) (b)
21. (a) Complement: Supplement:
(b) Complement: none; Supplement:
22. (a) Complement: Supplement:
(b) Complement: none; Supplement:
23. (a) Complement:
Supplement:
(b) Complement: none; Supplement:
24. (a) Complement: none; Supplement:
(b) Complement:
Supplement:
25. 26. 27. 28.
29. 30.
31. (a) Quadrant II (b) Quadrant IV32. (a) Quadrant I (b) Quadrant III33. (a) Quadrant III (b) Quadrant I34. (a) Quadrant II (b) Quadrant IV35. (a) (b)
150°
x
y
x30°
y
10�165�
�330��60�120�210�
� � 1.5 � 1.64
�
2� 1.5 � 0.07;
� � 3 � 0.14
� � 2 � 1.14
� � 1 � 2.14
�
2� 1 � 0.57;
�
12
11�
125�
12;
�
4
2�
3
�
6;
28�
15, �
32�
15
7�
4, �
�
4
25�
12, �
23�
128�
3, �
4�
3
�
6, �
23�
6
19�
6, �
5�
6
17�
6, �
7�
6
13�
6, �
11�
6
7π
x
y
4
x
y
−3
x
y
116π
x
y
x
y
52π
− 74π
y
x
− 23π
x
y
54π
x
y
�4
�3
A �12r2�
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(Continued)
36. (a) (b)
37. (a) (b)
38. (a) (b)
39. (a) (b)40. (a) (b)41. (a) (b)42. (a) (b)43. (a) Complement: Supplement:
(b) Complement: none; Supplement:44. (a) Complement: Supplement:
(b) Complement: Supplement:45. (a) Complement: ; Supplement:
(b) Complement: none; Supplement:46. (a) Complement: none; Supplement:
(b) Complement: none; Supplement:
47. (a) (b) 48. (a) (b)
49. (a) (b) 50. (a) (b)
51. (a) (b) 52. (a) (b)53. (a) (b) 54. (a) (b)55. 2.007 56. 1.525 57. 58.
59. 9.285 60. 6.021 61. 62. 0.00963. 64. 65.66. 67. 68.69. 70.71. (a) (b)72. (a) (b)73. (a) (b)74. (a) (b)75. (a) (b)76. (a) (b)77. (a) (b)78. (a) (b)
79. radians 80. radians 81. radians
82. radian 83. radian 84. radian
85. radians 86. 2 radians
87. 88.89. 3 meters 90.
91.
92.93. 12.27 square feet 94. 5.64 square miles95. 591.3 miles 96. 686.2 miles97. 98.99. 100.
101. (a) 728.3 revolutions per minute(b) 4576 radians per minute
102. (a) radians per minute; radians per minute(b) 850 revolutions per minute
103. (a)
(b) feet per minute feet per minute104. (a)
(b)105. (a) radians per minute
(b) centimeters per minute106.
107.
108. (a) feet per second; miles per hour
(b) (c)
(d) The functions are both linear.109. False. A measurement of radians corresponds to two
complete revolutions from the initial to the terminal sideof an angle.
4�
d �7�
7920td �
7�
7920n
�1014�
3
A � 476.39� square meters � 1496.62 square meters
140°
35
A � 175� square inches � 549.8 square inches�2400�, 6000���400�, 1000��200� feet per minute � 628.3 feet per minute 8� radians per minute � 25.13 radians per minute
� 9869.849425��3 � 32,672.56 radians per minute
10,400� radians per minute
1700�3400�
275�5
12 radian0.063 radian � 3.59�0.071 radian � 4.04�
18� square millimeters � 56.55 square millimeters
8�
3 square inches � 8.38 square inches
5� centimeters � 15.71 centimeters3� feet � 9.42 feet15� inches � 47.12 inches
5029
47
29�
45
327
2910
65
0� 47� 11.4��0� 21� 18�
�3� 34� 48�2� 30�
0� 27��345� 7� 12�
�145�48�240� 36�
�408.272��135.01�
330.007�85.308�
2.2�245.167�
�128.5�54.75�
�32.659��114.592�
864.000��756.000�1170.000�
337.500�81.818�25.714�
�0.014
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20��105�210�270�
4�
5�
3�
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�
9
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6�
6
10�
50�
30�
101�11�
116�26�;177�87�;
65�
162�72�;590�, �130�300�, �60�
180�, �540�600�, �120�
300�, �60�480�, �240�
324�, �396�405�, �315�
−600°
x
y
−750°x
y
480°
x
y
405°
x
y
−120°
x
y
−270°
x
y
Precalculus with Limits, Answers to Section 4.1 2
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Precalculus with Limits, Answers to Section 4.1 3
(Continued)
110. True. Let and represent coterminal angles, and let represent an integer.
111. False. The terminal side of the angle lies on the -axis.112. (a) The vertex is at the origin and the initial side is on the
positive axis.(b) Clockwise rotation of the terminal side(c) Two angles in standard position where the terminal
sides coincide(d) The magnitude of the angle is between and
113. Increases. The linear velocity is proportional to the radius.114. Radian. 115. The arc length is increasing. If is constant, the length of
the arc is proportional to the radius
116. Answers will vary. 117. 118.
119. 120.
121. 122.
123. 124.
3
2
1
−1
−2
−3
21−3−4−5x
y
y = x5
y = − (x + 3)5
321−2−3
6
5
4
3
1
−1
−2
−3
x
y
y = x5
y = 2 − x5
4
2
−2
−6
32−2−3 1
y = x5
y = x5 − 4
x
y
432−2
3
2
1
−1
−2
−3
x
yy = x5
y = (x − 2)5
4�132�10
5�24
�22
�s � r� �.�
1 radian � 57.3�
180�.90�
x-
x � � n�360�� � � n�360��
n
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Section 4.2 (page 299)
Vocabulary Check (page 299)1. unit circle 2. periodic3. period 4. odd; even
1.
2.
3.
4.
5. 6. 7.
8. 9. 10.
11. 12.
13. 14.
15. 16.
17. 18.
19. 20.
21.
is undefined.
23.
24.
25.
is undefined.
is undefined.
26.
is undefined.
is undefined.
27.
28.
29. 30.
31.
32.
33.
34. sin 19�
6� sin
7�
6� �
12
cos�15�
2 � cos �
2� 0
sin 9�
4� sin
�
4�
�22
cos 8�
3� cos
2�
3� �
12
cos 5� � cos � � �1sin 5� � sin � � 0
cot 7�
4� �1tan
7�
4� �1
sec 7�
4� �2cos
7�
4�
�22
csc 7�
4� ��2sin
7�
4� �
�22
cot4�
3 ��33
tan4�
3 � �3
sec4�
3 � �2cos4�
3 � �12
csc4�
3 � �2�3
3sin4�
3 � ��32
cot 3�
2� 0tan
3�
2
sec 3�
2cos
3�
2� 0
csc 3�
2� �1sin
3�
2� �1
cot��
2 � 0tan� �
2sec�
�
2cos��
2 � 0
csc��
2 � �1sin��
2 � �1
cot 5�
6� ��3tan
5�
6� �
�33
sec 5�
6� �
2�33
cos 5�
6� �
�32
csc 5�
6� 2sin
5�
6�
12
cot 3�
4� �1tan
3�
4� �1
sec 3�
4� ��2cos
3�
4� �
�22
csc 3�
4� �2sin
3�
4�
�22
tan�3�
2 cos�
3�
2 � 0
sin�3�
2 � 1
tan 5�
3� ��3tan
11�
6� �
�33
cos 5�
3�
12
cos 11�
6�
�32
sin 5�
3� �
�32
sin 11�
6� �
12
tan�4�
3 � ��3tan�7�
4 � 1
cos�4�
3 � �12
cos�7�
4 ��22
sin�4�
3 ��32
sin�7�
4 ��22
tan��
4 � �1tan��
6 � ��33
cos��
4 ��22
cos��
6 ��32
sin��
4 � ��22
sin��
6 � �12
tan �
3� �3tan
�
4� 1
cos �
3�
1
2cos
�
4�
�22
sin �
3�
�3
2sin
�
4�
�22
��1, 0��0, �1�
12
, ��32 �
12
, ��32 �
�22
, ��22
��32
, �121
2, �32 �2
2, �22
cot � �43tan � �
34
sec � � �54cos � � �
45
csc � � �53sin � � �
35
cot � � �125tan � � �
512
sec � �1312cos � �
1213
csc � � �135sin � � �
513
cot � �125tan � �
512
sec � �1312cos � �
1213
csc � �135sin � �
513
cot � � �815tan � � �
158
sec � � �178cos � � �
817
csc � �1715sin � �
1517
Precalculus with Limits, Answers to Section 4.2 4
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tan��2�� � 0cos��2�� � 1sin��2�� � 0
333202CB04_AN.qxd 1/1/70 09:39 AM Page 4
Precalculus with Limits, Answers to Section 4.2 5
(Continued)
35.
36.
37. (a) (b) 38. (a) (b)
39. (a) (b) 40. (a) (b)
41. (a) (b) 42. (a) (b)43. 0.7071 44. 1.7321 45. 1.0378 46. 0.642147. 48. 49. 1.394050. 51. 52.53. (a) (b) 54. (a) 0.7 (b)55. (a) 0.25 or 2.89 (b) 1.82 or 4.4656. (a) 4.0 or 5.4 (b) 0.72 or 5.5657. (a)
(b) (c) The displacement decreases.58. (a) 0.25 foot (b) 0.02 foot (c) foot59. False. means that the function is odd, not
that the sine of a negative angle is a negative number.60. True. The tangent function has a period of 61. (a) -axis symmetry (b)
(c)
62. Answers will vary. 63.
64. 65.
66.
67. 68.
69. 70.y
x−4 46 82−2−6
−2
y
x−1−5−6 1 2
−1
−2
−3
−4
1
2
3
4
−2
y
x−4 4 6 8
−2
−4
−6
−8
2
4
6
8
2−2
y
x−2−4−6 2 4 6 8 10
−2
−4
−6
−8
4
6
8
2
f �1�x� �2�2x � 1�
x � 1
f �1�x� � �x2 � 4, x ≥ 0f �1�x� � 3�4�x � 1�f �1�x� �
23 �x � 1�
cos�� � t1� � �cos t1
sin t1 � sin�� � t1�y�.
sin��t� � �sin t�0.25
t � 5.5
�0.8�0.4�1�0.7833�1.4486�4.4014
�0.8011�0.1288
�45�
45�
45
45
�43�
34�5�
15
�83�
38�3�
13
cos�8�
3 � cos 4�
3� �
12
sin�9�
4 � sin 7�
4� �
�22
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34
12
14
333202CB04_AN.qxd 1/1/70 09:39 AM Page 5
Section 4.3 (page 308)
Vocabulary Check (page 308)1. (a) v (b) iv (c) vi (d) iii (e) i (f) ii2. opposite; adjacent; hypotenuse3. elevation; depression
1. sin csc
cos sec
tan cot
2. sin csc
cos sec
tan cot
3. sin csc
cos sec
tan cot
4. sin csc
cos sec
tan cot
5. sin csc
cos sec
tan cot
The triangles are similar, and corresponding sides areproportional.
6. sin csc
cos sec
tan cot
The triangles are similar, and corresponding sides areproportional.
7. sin csc
cos sec
tan cot
The triangles are similar, and corresponding sides areproportional.
8.
The triangles are similar, and corresponding sides areproportional.
9. cos
tan cot
csc
10.
11. sin csc
cos cot
tan
12. sin csc
cos sec
tan
13. sin sec
cos cot
csc
14.
tan � � �35
cot � ��3535
cos � �16
csc � �6�35
35sin � �
�356
6
1
θ
35
� ��10
3
� �1
3� �
�10
10
� � �10� �3�10
10
3
1
10
θ
� �1
5
� ��26
5� �
5�26
26
� � �26� ��26
2626
1
5
θ
� � �3
� ��3
3� �
1
2
� �2�3
3� �
�3
2
1
2 3
θ
csc � �7�612
cot � �5�612
tan � �2�6
5
sec � �75
sin � �2�6
7
5
7 2 6
θ
� �4
3
� ��7
3� �
3�7
7
sec � �4�7
7� �
�7
4
θ
7
4 3
cot � � 2tan � �1
2
sec � ��5
2cos � �
2�55
csc � � �5sin � ��5
5
� �43� �
34
� �54� �
45
� �53� �
35
� �158� �
815
� �1715� �
1517
� �178� �
817
� � 2�2� ��2
4
� �3�2
4� �
2�2
3
� � 3� �1
3
� � 1� � 1
� � �2� ��2
2
� � �2� ��2
2
� �409� �
940
� �4140� �
4041
� �419� �
941
� �125� �
512
� �1312� �
1213
� �135� �
513
� �43� �
34
� �54� �
45
� �53� �
35
Precalculus with Limits, Answers to Section 4.3 6
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Precalculus with Limits, Answers to Section 4.3 7
(Continued)
15. sin csc
cos sec
tan
16.
17. 18. 19. 20.
21. 22. 23. 24.
25. 26.
27. (a) (b) (c) (d)
28. (a) 2 (b) (c) (d)
29. (a) (b) (c) (d)
30. (a) (b) (c) (d)
31. (a) 3 (b) (c) (d)
32. (a) (b) (c) (d)
33–42. Answers will vary.
43. (a) 0.1736 (b) 0.1736
44. (a) 0.4348 (b) 0.4348
45. (a) 0.2815 (b) 3.5523
46. (a) 0.9598 (b) 0.9609
47. (a) 1.3499 (b) 1.3432
48. (a) 0.9964 (b) 1.0036
49. (a) 5.0273 (b) 0.1989
50. (a) 1.7946 (b) 0.5572
51. (a) 1.8527 (b) 0.9817
52. (a) 2.6695 (b) 0.0699
53. (a) (b)
54. (a) (b)
55. (a) (b)
56. (a) (b)
57. (a) (b)
58. (a) (b)
59. 60. 61.
62. 63. 443.2 meters; 323.3 meters
64. (a) (b)
(c) 270 feet
65. 66. 137.6 feet
67. (a) 371.1 feet (b) 341.6 feet
(c) Moving down line at 61.8 feet per second
Dropping vertically at 24.2 feet per second
68. 1.3 miles 69.
70. 6.57 centimeters
71. (a)
(b) sin (c) 19.9 meters
(d) The side of the triangle labeled will become shorter.(e)
h
85� �h
20
h20
85°
�x2, y2� � �28, 28�3��x1, y1� � �28�3, 28�
30� ��
6
tan � �63
�h
135
h
3
6132
Not drawn to scale
20�2
32�3
39�330�3
45� ��
460� �
�
3
45� ��
460� �
�
3
60� ��
360� �
�
3
45� ��
460� �
�
3
45� ��
445� �
�
4
30� ��
630� �
�
6
�26
5
1
5
�26
26
1
5
1
3
�2
42�2
3
2�6
52�6
�6
12
1
5
�132
23
3�13
132�13
13
�3�3
2
�3
3
�3
3
�3
2
1
2�3
30�; �
645�;
�
4
45�; �22
30�; �32
45�; �
460�;
�
3
45�; �260�; �3�
4; �22
�
6;
12
tan � �4�273
273
cot � ��273
4cos � �
�27317
sec � �17�273
273sin � �
417
417
273
θ
� �2
3
� ��13
3� �
3�13
13
� ��13
2� �
2�13
13
2
3
13
θ
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Angle,
Height 19.7 18.8 17.3 15.3
50�60�70�80��
Angle,
Height 12.9 10.0 6.8 3.5
10�20�30�40��
333202CB04_AN.qxd 1/1/70 09:39 AM Page 7
(Continued)
(f) As
72.
73. True, csc 74. True, sec
75. False,
76. True, for all 77. False,78. False, tan 79. Corresponding sides of similar triangles are proportional.80. Yes, is equal to You can find the value of the
hypotenuse by the Pythagorean Theorem, then you can findwhich is equal to
81. (a)
(b) (c) As approaches 0, sin approaches 0.
82. (a)
(b) Increasing function (c) Decreasing function(d) As the angle increases, the length of the side opposite
the angle increases relative to the length of thehypotenuse and the length of the side adjacent to the angle decreases relative to the length of thehypotenuse. Thus, the sine increases and the cosinedecreases.
83. 84.
85. 86.14
, x � 0, 122�x2 � 5x � 10��x � 2��x � 2�2
2t � 34 � t
, t � ±32
, �4x
x � 2, x � ±6
���
hyp�adj.sec �,
opp�adj.tan �
25� � �tan 5���tan 5��.1.7321 � 0.0349.
�.cot2 � � csc2 � � �1
�2
2�
�2
2� 1.
x � csc�90� � x�.x �1
sin x.
cot 20� � 2.75sec 20� � 1.06csc 20� � 2.92tan 20� � 0.36cos 20� � 0.94sin 20� � 0.34
20 h
θ
� → 0�, h → 0.
Precalculus with Limits, Answers to Section 4.3 8
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0.1 0.2 0.3 0.4 0.5
sin 0.0998 0.1987 0.2955 0.3894 0.4794�
�
sin 0 0.3090 0.5878 0.8090 0.9511 1
cos 1 0.9511 0.8090 0.5878 0.3090 0�
�
90�72�54�36�18�0��
333202CB04_AN.qxd 1/1/70 09:39 AM Page 8
Precalculus with Limits, Answers to Section 4.4 9
Section 4.4 (page 318)
Vocabulary Check (page 318)
1. 2. 3. 4. 5.
6. 7. reference
1. (a) sin (b) sin
cos cos
tan tan
csc csc
sec sec
cot cot
2. (a) sin
cos
tan
csc
sec
cot
3. (a) sin (b) sin
cos cos
tan tan
csc csc
sec sec
cot
4. (a) sin (b) sin
cos cos
tan tan
csc
sec sec
cot
5. sin csc
cos sec
tan cot
6. sin csc
cos sec
tan cot
7. sin csc
cos sec
tan cot
8. sin csc
cos sec
tan cot
9.
10.
11. Quadrant III 12. Quadrant I
13. Quadrant II 14. Quadrant IV
15. sin csc
cos sec
tan cot
16. sin csc
cos sec
tan cot
17. sin csc
cos sec
tan cot
18. sin csc
cos sec
tan cot
19. sin csc
cos sec
tan cot � � �3� � �1
3
� ��10
3� �
3�10
10
� � ��10� � ��10
10
� � �8
15� � �158
� �178� �
817
� � �1715� � �
1517
� � �8
15� � �158
� �178� �
817
� � �1715� � �
1517
� �43� �
34
� � �54� � �
45
� � �53� � �
35
� � �43� � �
34
� � �54� � �
45
� �53� �
35
cot � � �1431
tan � � �3114
sec � ��1157
14cos � �
14�11571157
csc � � ��1157
31sin � � �
31�11571157
cot � � �3568
tan � � �6835
sec � � ��5849
35cos � � �
35�58495849
csc � ��5849
68sin � �
68�58495849
� �5
2� �
2
5
� � ��29
5� � �
5�29
29
� � ��29
2� � �
2�29
29
� � �2
5� � �
5
2
� � ��29
2� � �
2�29
29
� ��29
5� �
5�29
29
� �815� �
158
� �178� �
817
� �1715� �
1517
� �724� �
247
� �257� �
725
� �2524� �
2425
cot � � �1� � 3
� � �2� ��10
3
csc � � ��2� � �10
� � �1� �1
3
� ��2
2� �
3�10
10
� � ��2
2� �
�10
10
cot � � �4� � �3
� � ��17
4� � �
2�3
3
� � �17� � �2
� � �1
4� �
�3
3
� � �4�17
17� � �
�3
2
� ��17
17� � �
1
2
� �12
5
� � �13
12
� � �13
5
� �5
12
� � �12
13
� � �5
13
� � �8
15� �43
� �178� �
54
� � �1715� �
53
� � �158� �
34
� �817� �
45
� � �1517� �
35
cot �
cos �rx
yx
csc �yr
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(b) sin
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tan csc sec cot � � �1
� � ��2� � �2� � �1
� � ��2
2
� ��2
2
333202CB04_AN.qxd 1/1/70 09:39 AM Page 9
(Continued)
20. sin csc
cos sec
tan cot
21. sin csc
cos sec
tan cot
22. is undefined.
is undefined.
23. is undefined.
is undefined.
24. sin csc cos sec is undefined.tan cot
25. sin csc
cos sec
tan cot
26. sin csc
cos sec
tan cot
27. sin csc
cos sec
tan cot
28. sin csc
cos sec
tan cot
29. 0 30. 31. Undefined 32. 33. 1
34. Undefined 35. Undefined 36. 0
37. 38.
39. 40.
41. 42.
43. 44.
45. 46. sin
cos cos
tan tan
47. sin 48.
cos cos��405�� ��2
2750� �
�3
2
sin��405�� � ��2
2750� �
1
2
300� � ��3225� � 1
300� �1
2225� � �
�2
2
300� � ��3
2sin 225� � �
�2
2
′θ
x
y
113π
′θ
3.5
x
y
�� ��
3�� � 3.5 � �
′θ
74π
x
y
′θ
23π
x
y
�� ��
4�� �
�
3
′θ−145°
x
y
′θ
−245°
x
y
�� � 35��� � 65�
′θ
309°
x
y
′θ
203°
x
y
�� � 51��� � 23�
�1�1
� � �34� � �
43
� �53� �
35
� � �54� � �
45
� �1
2� � 2
� � ��5� � ��5
5
� � ��5
2� � �
2�5
5
� � 3� �1
3
� � ��10
3� � �
3�10
10
� � ��10� � ��10
10
� � �1� � �1
� � ��2� � ��2
2
� � �2� ��2
2
� � 0� is undefined.�� � 0� � �1� � �1
cot �tan � � 0
sec � � �1cos � � �1
csc �sin � � 0
cot �tan � � 0
sec � � �1cos � � �1
csc �sin � � 0
� � ��3
3� � ��3
� � �2� � �1
2
� �2�3
3� �
�3
2
� � ��15� � ��15
15
� � �4�15
15� � �
�15
4
� � 4� �1
4
Precalculus with Limits, Answers to Section 4.4 10
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333202CB04_AN.qxd 1/1/70 09:39 AM Page 10
Precalculus with Limits, Answers to Section 4.4 11
(Continued)
tan
49. 50.
51. 52. sin
cos
tan
53. 54.
55. 56. sin
cos
tan
57. 58.
is undefined.
59. 60. 61. 62. 63.
64. 65. 0.1736 66. 67.
68. 2.0000 69. 70.71. 3.2361 72. 73. 4.6373 74. 0.224575. 0.3640 76. 77.78. 1.0436 79. 80. 4.4940
81. (a) (b)
82. (a) (b)
83. (a) (b)
84. (a) (b)
85. (a) (b)
86. (a) (b)
87. (a)
(b) February:March:May:June:August:September:November:
(c) Answers will vary.
88. (a) 26,134 units (b) 31,438 units (c) 21,452 units(d) 26,756 units
89. (a) 2 centimeters (b) 0.14 centimeter(c) centimeters
90. (a) 2 centimeters (b) 0.11 centimeter(c) centimeters
91. 0.79 ampere
92. (a) 12 miles (b) 6 miles (c) 6.9 miles
93. False. In each of the four quadrants, the signs of the secantfunction and cosine function will be the same, becausethese functions are reciprocals of each other.
94. False. For in Quadrant II, For inQuadrant III, For in Quadrant IV,
95. As increases from to , decreases from 12 cm to 0 cm and increases from 0 cm to 12 cm. Therefore,
increases from 0 to 1 and decreases from 1 to 0. Thus, and increaseswithout bound. When , the tangent is undefined.
96. Determine the trigonometric function of the referenceangle and prefix the appropriate sign.
97. -intercepts:
-intercept:Domain: all real numbers
98. Intercept:-intercept:
Domain: all real numbers x�5
2, 0�x�0, 0�
−1−2−3 1 2 3 4 5−1
−2
−3
−4
1
2
y
x(0, 0) , 05
2( (
x�0, �4�y
�1, 0�, ��4, 0�x
−2−6−8 2 4 6 8−2
−4
−8
2
4
6
8
y
x(1, 0)(−4, 0)
(0, −4)
� � 90�tan � � y�x
cos � � x�12sin � � y�12y
x90�0��
�� � 360� � �.��� � � � 180�.
��� � 180� � �.�
�1.2
�1.98
N � 46.8�, F � 6.5�
N � 68.6�, F � 41.7�
N � 75.5�, F � 55.6�
N � 72.5�, F � 59.5�
N � 63.4�, F � 48.6�
N � 41.6�, F � 13.9�
N � 34.6�, F � �1.4�
F � 36.641 sin�0.502t � 1.831� � 25.610N � 22.099 sin�0.522t � 2.219� � 55.008
240� �4�
3, 300� �
5�
360� �
�
3, 120� �
2�
3
150� �5�
6, 330� �
11�
645� �
�
4, 225� �
5�
4
120� �2�
3, 240� �
4�
360� �
�
3, 300� �
5�
3
135� �3�
4, 315� �
7�
460� �
�
3, 120� �
2�
3
135� �3�
4, 225� �
5�
445� �
�
4, 315� �
7�
4
210� �7�
6, 330� �
11�
630� �
�
6, 150� �
5�
6
�0.4142�0.6052�0.3640
�0.1405�28.6363�1.4826
�0.3420�1.4142�65
4
8
5��3�
�13
2
�10
10
4
5
tan�25�
4 � �1tan�3�
2 cos�
25�
4 ��22
cos�3�
2 � 0
sin�25�
4 � ��22
sin�3�
2 � 1
10�
3� �3tan
11�
4� �1
10�
3� �
1
2cos
11�
4� �
�22
10�
3� �
�3
2sin
11�
4�
�22
tan��
2 is undefined.tan��
6 � ��33
cos��
2 � 0cos��
6 ��32
sin��
2 � �1sin��
6 � �12
�
4� 1tan
4�
3� �3
�
4�
�2
2cos
4�
3� �
12
�
4�
�2
2sin
4�
3� �
�32
tan��840�� � �3tan��150�� ��33
cos��840�� � �12
cos��150�� � ��32
sin��840�� � ��32
sin��150�� � �12
tan��405�� � �1750� �
�3
3
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333202CB04_AN.qxd 1/1/70 09:39 AM Page 11
(Continued)
99. -intercept:-intercept:
Domain: all real numbers
100. -intercepts:
-intercept:Domain: all real numbers
101. -intercept:-intercept:
Vertical asymptote:
Horizontal asymptote:
Domain: all real numbersexcept
102. -intercepts:
-intercept:Vertical asymptote:
Slant asymptote:
Domain: all real numbersexcept
103. -intercept:Horizontal asymptote:
Domain: all real numbers
104. -intercept:Horizontal asymptote:
Domain: all real numbers
105. -intercepts:Vertical asymptote:Domain: all real numbers
except
106. -intercept:-intercept:
Vertical asymptote:
Domain: all real numbers such that x > �2
xx � �2
�0, 0.301�y��1, 0�x
3
2
−1
−2
−3
321−1−3
(−1, 0)(0, 0.301)
x
y
x � 0x
x � 0�±1, 0�x
−12 −9 −6 −3 3 6 9 12
6
9
12
(−1, 0) (1, 0)x
y
xy � 2
�0, 5�y
x321−1−2−3−4−5
7
6
5
3
1
2
(0, 5)
y
xy � 0
�0, 12�y
−2 −1 1 2 3 4−1
2
3
4
5
10, 12 ))
x
y
x � �5x
y � x � 5
x � �5
�0, �15�y
�1, 0�, ��1, 0�x
−8−12 4
−8
−16
−24
8
y
x(−1, 0)
(1, 0)
0, 15
−( (
x � �2x
y � 0
x � �2
�0, �74�y
�7, 0�x
−8 −2 2 4 6 8
2
4
y
x
74
0, ( (−
(7, 0)
x�0, �3�y
�1, 0�, ��1, 0�x
−2−3−4 2 3 4
−3
−4
1
2
3
4
y
x(−1, 0)
(0, −3)
(1, 0)
x�0, 8�y��2, 0�x
−6 −4−8 2 4 6 8
−4
10
12
y
x
(0, 8)
(−2, 0)
Precalculus with Limits, Answers to Section 4.4 12
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333202CB04_AN.qxd 1/1/70 09:39 AM Page 12
Precalculus with Limits, Answers to Section 4.5 13
Section 4.5 (page 328)
Vocabulary Check (page 328)
1. cycle 2. amplitude 3.
4. phase shift 5. vertical shift
1. Period: 2. Period: 3. Period:
Amplitude: 3 Amplitude: 2 Amplitude:4. Period: 5. Period: 6 6. Period: 4
Amplitude: 3 Amplitude: Amplitude:
7. Period: 8. Period: 9. Period:
Amplitude: 3 Amplitude: 1 Amplitude: 3
10. Period: 11. Period: 12. Period:
Amplitude: Amplitude: Amplitude:
13. Period: 1 14. Period: 20Amplitude: Amplitude:
15. is a shift of units to the right.16. is a shift of units to the left.17. is a reflection of in the axis.18. is a reflection of in the axis.19. The period of is twice the period of 20. The period of is one-third the period of 21. is a shift of three units upward.22. is a shift of two units downward.23. The graph of has twice the amplitude of the graph of 24. The period of is the period of 25. The graph of is a horizontal shift of the graph of
units to the right.26. is a shift of two units upward.
27. 28.
29. 30.
31. 32.
33. 34.
35. 36.
37. 38.
39. 40. y
x
−2
1
2
π4
y
xπ4π2π−2
−1
−2
2
y
xππ 2π−2 −π
−2
−4
4
y
xπ π2
π2
1
−1
23
43
1323
43
−
−
−
y
xππ 2π−2 −π
−1
−2
1
2
y
xπ π32 2
−− π2
π32
−4
1
2
3
4
−2
2
ππ 2
f
g
x
y
−3
3
f
g
xππ 2
y
−8
2
4f
g
1 x
y
−1
1
2
3
4
5
f
g
x−π π3
y
−2
2
π
f
g
x
y
−1
g
f
xππ 2
y
− 2
2
π6
fg
x
y
x
−π π32 2
5
43
−5
f
g
y
fg�
fgf.1
3gf.g
fgfg
f.gg.f
x-fgx-f g
f �gf �g
23
14
52
12
13
8�3��
4
�
53�2�
32
12
6�
52
4�2�
3�
2�
b
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333202CB04_AN.qxd 1/1/70 09:39 AM Page 13
(Continued)
41. 42.
43. 44.
45. 46.
47. 48.
49. 50.
51. 52.
53. 54.
55. 56.
57. 58.
59. 60.
61. 62.
63. 64.
65. 66.
67. 68.
69. 70. a � 2, b ��
2, c � �
�
2a � 2, b � 1, c � �
�
4
a � 2, b �12, c � 0a � �3, b � 2, c � 0
a � �1, d � �3a � �4, d � 4
a � 2, d � �1a � 2, d � 1
−0.03
−0 02
0.03
0.02
−20
−0.12
20
0.12
−6
−6
6
2
−3
−1
3
3
−12
−8
12
8
−6 6
−4
4
2
3
x
y
π
−4
−3
−2
−1
1
2
3
4
π π4x
y
−4
2
4
6
10
x
y
π3π2π−2 − ππ
−8
2
4
ππ 2x
y
−7
−6
−5
−4
1
−x
y
πππ 2
0.20.1−0.1 0
1.8
2.2
x
y
−12 4 12
−24−20−16−12−8
48
1216
t
y
–3 –2 –1 1 2 3−1
1
2
4
5
x
y
−6
−4
−2−
2
6
x
y
πππ 2
−6
−4
2
4
6
− ππx
y
2
−2
−1
x
y
−π π32 2
−3
−2
1
2
3
ππ−x
y
1284−4−12
12
8
4
−12
x
y
−1 2 3
−3
−2
2
3
x
y
2
1
−2
62−2−6x
y
1 2
−2
1
2
x
y
Precalculus with Limits, Answers to Section 4.5 14
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333202CB04_AN.qxd 1/1/70 09:39 AM Page 14
Precalculus with Limits, Answers to Section 4.5 15
(Continued)
71. 72.
73. (a) 6 seconds (b) 10 cycles per minute(c)
74. (a) 4 seconds (b) 15 cycles per minute(c)
75. (a)
(b) (c)
The model is a good fit. The model is a good fit.
(d) Tallahassee: Chicago:
The constant term gives the annual average tempera-ture.
(e) 12; yes; one full period is one year.
(f ) Chicago; amplitude; the greater the amplitude, thegreater the variability in temperature.
76. (a) seconds (b) 50 heartbeats per minute
77. (a) second (b) 440 cycles per second
78. (a)–(c)
(b)The model is a good fit.
(d) 29 days(e) 0.44
79. (a) 365; answers will vary.(b) 30.3 gallons; the constant term(c)
80. (a) 20 seconds; it takes 20 seconds to complete one revolution on the Ferris wheel.
(b) 50 feet; the diameter of the Ferris wheel is 100 feet.(c)
81. False. The graph of translates thegraph of exactly one period to the left so thatthe two graphs look identical.
82. False. The function has an amplitude that isone-half that of For the amplitudeis
83. True. Because is a
reflection in the -axis of
84. Answers will vary.85. 86.
Conjecture: Conjecture:
sin x � �cosx ��
2sin x � cosx ��
2
2
1
−2
f = g
x
y
π π32 2
π32
−
2
1
−2
f = g
xπ π32 2
− π32
y
y � sinx ��
2.x
y � �cos xcos x � sinx ��
2,
�a�.y � a cos bx,y � cos x.
y �12 cos 2x
f �x� � sin xf �x� � sin�x � 2��
00 20
110
124 < t < 252
00
365
60
y �12 �
12 sin�0.21x � 0.92�
y
x10 20 30 40
0.2
0.4
0.6
0.8
1.0
Perc
ent o
f m
oon’
sfa
ce il
lum
inat
ed
Day of the year
1440
65
56.55�77.90�;
00
12
100
00
12
100
C�t� � 56.55 � 26.95 cos�
6t � 3.67
1 3 5 7
−2
−3
1
2
3
t
v
t2 4 8 10
0.25
0.50
0.75
1.00
−0.25
−1.00
v
x � �, ��x � �
�
6, �
5�
6,
7�
6,
11�
6
−2
2
2
�−2�
−2
2
2
�−2�
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333202CB04_AN.qxd 1/1/70 09:39 AM Page 15
(Continued)
87. (a)
The graphs appear to coincide from to
(b)
The graphs appear to coincide from to
(c)
The interval of accuracy increased.88. (a) 0.4794, 0.4794 (b) 0.8417, 0.8415 (c) 0.5, 0.5
(d) 0.8776, 0.8776 (e) 0.5417, 0.5403(f) 0.7074, 0.7071The error increases as moves farther away from 0.
89. 90.
91. 92.
93. 94.
95. 96. 97. Answers will vary.ln�x2�2x�ln 3xy4
log2�x3y�log10 �xy
12 ln z �
12 ln�z2 � 1�3 ln t � ln�t � 1�
2 log2 x � log2�x � 3�12 log10�x � 2�
x
−2
2�−2�
2
−2
−2 2� �
2
�x7
7!, �
x 6
6!
�
2.�
�
2
−2
−2 2� �
2
�
2.�
�
2
−2
−2 2� �
2
Precalculus with Limits, Answers to Section 4.5 16
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333202CB04_AN.qxd 1/1/70 09:39 AM Page 16
Precalculus with Limits, Answers to Section 4.6 17
Section 4.6 (page 339)
Vocabulary Check (page 339)1. vertical 2. reciprocal 3. damping4. 5. 6.7.
1. e, 2. c, 3. a, 1 4. d,5. f, 4 6. b, 4
7. 8.
9. 10.
11. 12.
13. 14.
15. 16.
17. 18.
19. 20.
21. 22.
23. 24.
25. 26. y
x
−1
−2
π32
π2
2ππ
1
2
3
4
x
− ππ322
y
4
3
2
1
πx
y
−4 4
2
4
6
x
y
2
3
ππ−
y
x
3
xπ π
y
−
2
4
6
2−2x
y
1
2
3
x
y
π−2 2π
2
4
6
ππ 2x
y
2
4
6
xπ
y
2
4
6
x
y
π4
π4
π2
−
1−1−2−3 2 3−1
x
y
8
6
4
2
−2− π4
π4
x
y
x
4
3
2
1
−3
−4
21−1−2
y
π
3
2x
y
1
2
3
xπ
y
− π
2
−8
−4
x
y
x
4
3
2
1
− π3
π3
y
−3
1
2
3
ππ−x
y
1
2
3
xπ π
y
−
2�2��
2�
�� , �1� � �1, �x � n��
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333202CB04_AN.qxd 1/1/70 09:39 AM Page 17
(Continued)
27. 28.
29. 30.
31. 32.
33. 34.
35. 36.
37. 38.
39. 40.
41. 42.
43. 44.
45. 46.
47. 48.
49. Even 50. Odd
51. (a) (b)
(c) approaches 0 and approaches because thecosecant is the reciprocal of the sine.
52. (a)
(b) (c) The intervals are the same.
53.
The expressions are equivalent except that when is undefined.
54. 55.
The expressions are The expressions areequivalent. equivalent.
56.
The expressions are equivalent.
57. d, 58. a,
59. b, 60. c, g → 0 as x → 0.g → 0 as x → 0.
f → 0 as x → 0.f → 0 as x → 0.
−1
−
3
�2
3�2
3
−4
2�−2�
4
−4
−2 2
4
� �
y1
sin x � 0,
−2
3−3
2
��1, 13�;��1, 13�−3
g
f
3
1−1
� gf
�
6< x <
5�
6
1
−1
2
3
f
g
xπ ππ32 4
π4
y
�2�
3, �
�
3,
4�
3,
5�
3�
7�
4, �
5�
4,
�
4,
3�
4
�5�
3, �
�
3,
�
3,
5�
3�
4�
3, �
2�
3,
2�
3,
4�
3
�7�
4, �
3�
4,
�
4,
5�
4�
4�
3, �
�
3,
2�
3,
5�
3
�5�
3, �
2�
3,
�
3,
4�
3�
7�
4, �
3�
4,
�
4,
5�
4
−6
−2
6
2
−6
−0.6
6
0.6
−4
−� �
4
−3
− �2
�2
3
−3
�2
3�2
3−
3
−3
− �2
3�2
3
3
−3
−2
3
2
−4
�2
�2
−
4
−3
3
�4
3�4
3−
−5
5�−5�
5
−2
xπ32
− π32
y
xπ2
1
2
y
y
x1 2 3 4
1
2
3
y
x
−1
1
2
3
4
π 2π−π 3π
Precalculus with Limits, Answers to Section 4.6 18
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Precalculus with Limits, Answers to Section 4.6 19
(Continued)
61. 62.
The functions are equal. The functions are equal.
63. 64.
The functions are equal. The functions are equal.
65. 66.
As As
67. 68.
As As
69. 70.
As As
71. 72.
As As
73. 74.
As oscillates As oscillates.between 1 and
75. 76.
77. (a)
(b) As the predator population increases, the number ofprey decreases. When the number of prey is small, thenumber of predators decreases.
(c) 24 months; 24 months
78.
79. (a) 12 months; 12 months(b) Summer; winter (c) 1 month
80. (a)
(b) approaches 0 as increases.
81. True. For a given value of the coordinate of is thereciprocal of the coordinate of sin x.y-
csc xy-x,
ty
0 4�
−0.6
0.6
L:H:
Month (1 ↔ January)
Law
n m
ower
sal
es(i
n th
ousa
nds
of u
nits
) 150135120105907560453015
2 4 6 8 10 12t
S
R :C :
00 100
50,000
R
C
x
20
40
60
80
Angle of camera
Dis
tanc
e
d
0 π2
π4
π4
π2
− −
Gro
und
dist
ance
x
14
10
6
2
−2
−6
−10
−14Angle of elevation
d
π π π32 4
π4
d � 27 sec xd � 7 cot x
�1.x → 0, h�x�x → 0, f �x�
−1
−� �
2
−2
−� �
2
x → 0, y → .x → 0, g�x� → 1.
−1
−6� 6�
1
−1
6�−6�
2
x → 0, y → .x → 0, y → .
0
−2
6�
6
0
−2
8�
6
x → , h�x� → 0.x → , f �x� → 0.
−8
−1
8
1
−9 9
−6
6
x → , f �x� → 0.x → , g�x� → 0.
−3 6
−3
3
−1
8−8
1
−1
−3 3 6−6
2
3
x
y
–1
2
3
y
xππ−
−4
2
4
xππ−
y
−3 −2 −1 1 2 3
−3
−2
−1
1
2
3
x
y
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333202CB04_AN.qxd 1/1/70 09:39 AM Page 19
(Continued)
82. True. is equal to and if the recip-rocal of is translated units to the left, then
83. As approaches from the left, approaches As approaches from the right, approaches
84. As approaches from the left, approaches As approaches from the right, approaches
85. (a)
0.7391(b) 1, 0.5403, 0.8576, 0.6543, 0.7935, 0.7014, 0.7640,
0.7221, 0.7504, 0.7314, . . . ; 0.7391
86. 87.
The graphs appear to The graphs appear tocoincide on the interval coincide on the interval
88. (a)
(b)
(c)
89. 90.
91.
92.
93.
94.
95. 96.97. 2 98. �65 � 8.062
e10 � 4 � 22,022.466±�e3.2 � 1 � ±4.851
14 � e68
2� �1.702 � 1029
2 � e73
3� 1.684 � 1031
1365
log10 5log10 1.00041096 � 10.732
�ln 2 � �0.693
13
log10 98log10 8
� 0.735ln 54
2� 1.994
�17
sin�7�x� �19
sin�9�x�� y4 �
4
� sin��x� �1
3 sin�3�x� �
1
5 sin�5�x�
−3 3
−2
2
�17
sin�7�x�� y3 �
4
� sin��x� �1
3 sin�3�x� �
1
5 sin�5�x�
−3 3
−2
2
y2
−3 3
−2
2
y1
�1.1 ≤ x ≤ 1.1.�1.1 ≤ x ≤ 1.1.
−6
−
6
�2
3�2
3
−6
− �2
3�2
3
6
3
−2
−3
2
� .f�x .f�x
� .f��2x .f��2x
1
sinx ��
2�
1cos x
� sec x.
��2y � sin xy � 1�cos x,y � sec x
Precalculus with Limits, Answers to Section 4.6 20
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333202CB04_AN.qxd 1/1/70 09:39 AM Page 20
Precalculus with Limits, Answers to Section 4.7 21
Section 4.7 (page 349)
Vocabulary Check (page 349)1.2.
3.
1. 2. 0 3. 4. 5. 6.
7. 8. 9. 10. 11.
12. 13. 14. 15. 0 16. 0
17. 18.
19. 1.29 20. 0.47 21. 22. 2.3523. 24. 1.50 25. 0.32 26. 1.3127. 1.99 28. 29. 0.74 30. 1.2331. 0.85 32. 1.91 33. 1.29 34.
35. 36.
37. 38.
39. 40.
41. 42.
43. 0.3 44. 25 45. 46. 47. 0
48. 49. 50. 51. 52.
53. 54. 55. 56.
57. 58. 59. 60.
61. 62. 63.
64. 65. 66.
67. 68.
69. 70.
Asymptotes: Asymptote:
71.
72. 73. 74.
75. 76.
The graph of is a The graph of is ahorizontal shift one unit horizontal stretch of theto the right of graph of
77. 78.
79. 80.
81. 82.
83. 84.
−0.5 0.5
2�
−2�0
2�
−1
−4 −2 2 4x
π
y
−2 1 2
π
y
v
−4 −2 2 4x
π
y
−4 −2 2 4
π
π
y
x
−
−4 −3 −2 −1t
π
y
−2 −1 1 2
π
π2
x
y
f.f.
gg
1 2−2x
π
π
y
−
−1 1 2 3
π
π
y
x
−
�4x � x2
x � 2�x � 1�
�x2 � 2x � 10
x6
�9
�x2 � 81, x < 0
9
�x2 � 81, x > 0;
x � 0y � ±1
−3
−2
3
2
−3 3
−2
2
�r2 � �x � h�2
r
�x 2 � 2
x
x�9 � x 2
x
1
�2x � x2
�1 � x 2�9x2 � 1�1 � 4x 2
x
�x2 � 1
1
x
8
5
�5
3
�3�7
7
�34
5�
13
5
12
13
2�5
5
�5
553
3
5�
2
�0.2�0.1
� � arctan 1
x � 1, x � 1� � arccos
x � 32x
� � arctan x � 1
10� � arcsin
x � 2
5
� � arccos 4
x� � arctan
x
4
�, 2�
3, �3
2�
�
3, �
�3
3, 1
�1.50�0.13
�1.25�0.85
−2
g
f
2
�2
�2
−
−1
1.5−1.5
fg
1
��
6
�
3
�
4
2�
3
�
3�
�
3�
�
4
5�
6
��
4
�
6
�
2
�
3
�
6
y � tan�1 x; � < x < ; ��
2< y <
�
2
y � arccos x; 0 ≤ y ≤ �
y � sin�1 x; �1 ≤ x ≤ 1
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333202CB04_AN.qxd 1/1/70 09:39 AM Page 21
(Continued)
85. 86.
87. 88.
89.
The graph implies that the identity is true.
90.
The graph implies that the identity is true.
91. (a) (b) 0.13, 0.25
92. (a) (b)
93. (a)
(b) 2 feet (c) As increases, approaches 0.94. (a) (b) 12.94 feet95. (a) (b) 24.4 feet
96. (a) (b)
97. (a) (b)
98. False. is not in the range of the arcsine.
99. False. is not in the range of the arctangent.
100. False. The graphs are not the same.
101. Domain:Range:
102. Domain:Range:
103. Domain:Range:
104. (a) (b) 0 (c) (d)
105. (a) (b) (c) 1.25 (d) 2.03
106.
As increases to infinity, approaches , but has nomaximum.a � 87.54
f3�gx
00
6
g
f
12
�
2
�
4
�
6
5�
6
�
4
x−2 −1 21
−
y
π2
π2
����2, 0� � �0, ��2��� , �1� � �1, �
x−2 −1 1 2
y
π
π2
�0, ��2� � ��� 2, ���� , �1� � �1, �
x−1−2 21
π2
y
π
�0, ���� , �
5�
4
5�
6
14.0�, 31.0�� � arctan x
20
40.6�, 80.5�� � arctan 6x
� � 26.0�
� � 32.9�
x � 0;
0
−0.5
6
1.5
21.8�, 58.0�� � arctan s
750
� � arcsin 5
s
−6
6−6
6
5 sin�� t � arctan 43�
−6
−2
6
� 2�
3�2 sin2t ��
4
5
−2
−4
4
5
−2
−4
4
−4 4
−2�
�2
−2 4
�−
�
Precalculus with Limits, Answers to Section 4.7 22
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333202CB04_AN.qxd 1/1/70 09:39 AM Page 22
Precalculus with Limits, Answers to Section 4.7 23
(Continued)
107. (a)
(b) The domains and ranges of the functions are restricted.The graphs of and differ because of thedomains and ranges of and
108. (a)–(e) Answers will vary. 109. 1279.284110. 0.051 111. 117.391 112.
113.
114.
115.
116.
117. Eight people 118. 3 miles per hour
119. (a) $21,253.63 (b) $21,275.17
(c) $21,285.66 (d) $21,286.01
120. 2008: $458,504
2�2
tan � �2�2
1
cot � ��24
cos � �13
csc � �3�2
4sin � �
2�23
3
θ
�11
csc � �6�11
11
cot � �5�11
11tan � �
�115
sec � �65
sin � ��11
66
5
θ
�5
csc � ��52
cot � �12
cos � ��55
sec � � �5sin � �2�5
5
2
θ
�7
csc � �43
cot � ��73
tan � �3�7
7
sec � �4�7
7cos � �
�74
4 3
θ
2.718 � 10�8
f �1.ff �1
� ff � f �1
−2
−� �
2
−2
−� �
2
f �1� ff � f �1
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333202CB04_AN.qxd 1/1/70 09:39 AM Page 23
Section 4.8 (page 359)
Vocabulary Check (page 359)1. elevation; depression 2. bearing3. harmonic motion
1. 2. 3.
4. 5. 6.
7. 8. 9.
10. 11. 2.56 inches 12. 1.62 meters
13. 19.99 inches 14. 2.80 feet 15. 107.2 feet16. 1648.5 feet 17. 19.7 feet 18. 81.2 feet19. (a)
(b) (c) 19.9 feet20. 123.5 feet 21. 2236.8 feet 22.
23. (a) (b) (c)
24. 25.26. (a) 5099 feet (b) 117.7 seconds 27. 0.73 mile28. 29. 554 miles north; 709 miles east30. (a) 429.26 miles north; 2434.44 miles east
(b)31. (a) 58.18 nautical miles west; 104.95 nautical miles south
(b) S W; 32. (a) 21.4 hours
(b) 5.86 nautical miles east; 239.93 nautical miles south(c)
33. (a) (b) 68.82 meters34. 35.36. 37. 1933.3 feet 38. 11.8 kilometers39.
40.41. 42. 43. 44.45. 29.4 inches 46. 25 inches 47.48. 9.06 centimeters 49.50. feet, feet, feet
51. 52.
53. 54.
55. (a) 4 (b) 4 (c) 4 (d)
56. (a) (b) 10 (c) (d)
57. (a) (b) 60 (c) 0 (d)
58. (a) (b) 396 (c) 0 (d)
59. 60.
61. (a) (b) (c)
62. (a)
(b)
7.0 (minimum length)
�
32
�
8
t
1
−1
y
π π38 8
π4
π2
d �7
4 cos
� t
5� � 528�
1792
164
1120
116
140
12
12
116
d � 2 cos� t
5 d � 3 cos4� t
3 d � 3 sin� t
3 d � 4 sin�� t�
c � 13b � 7.2a � 21.6a � 12.2, b � 7
y � �3r54.7�35.3�52.1�78.7�
� 1.025 miles or � 5412 feet
� 3.23 miles or � 17,054 feet208�
N 56.31� Wd � 5.46 kilometersN 58� E178.6�
distance � 130.9 nautical miles36.7�
280�
6.8�; 2516.3 feet
2.06�71.34�
35.8�tan � �121
2
1713
θ
17 ft
12 ft
1
1
3
2
56.3�
h � 50�tan 47�40� � tan 35��
50 ft
47° 40′
35°
h
x
y
A � 24�48�
c � 33.85b � 30.73
B � 77�45�B � 8.03�B � 17.92�
b � 420.70A � 81.97�A � 72.08�
a � 91.34a � 9.36a � 49.48B � 44.42�B � 59.04�B � 81.6�
A � 45.58�A � 30.96�c � 277.24b � 24.49c � 11.66b � 274.27A � 19�A � 36�B � 70�
c � 25.38b � 12.14c � 10.64a � 8.26a � 8.82a � 3.64
Precalculus with Limits, Answers to Section 4.8 24
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0.1 23.0
0.2 13.1
0.3 9.9
0.4 8.43
cos 0.42
sin 0.4
3cos 0.3
2sin 0.3
3cos 0.2
2sin 0.2
3cos 0.1
2sin 0.1
L1 � L2L2L1�
0.5 7.6
0.6 7.2
0.7 7.0
0.8 7.13
cos 0.82
sin 0.8
3cos 0.7
2sin 0.7
3cos 0.6
2sin 0.6
3cos 0.5
2sin 0.5
L1 � L2L2L1�
333202CB04_AN.qxd 1/1/70 09:39 AM Page 24
Precalculus with Limits, Answers to Section 4.8 25
(Continued)
(c)
(d) 7.0 (minimum length)
63. (a)
(b)
83.14 square feet(c)(d)
square feet when The answers are the same.
64. (a)
(b) or
The model is a good fit.(c) 12. Yes, sales of outerwear are seasonal.(d) Maximum displacement from average sales of $8
million65. False. The tower is leaning, so it is not perfectly vertical
and does not form a right angle with the ground.66. False. One period is the time for one complete cycle of the
motion.67. No. means 24 degrees east of north.68. Air navigation is always measured clockwise from the
north. Nautical navigation measures the acute angle a pathmakes with a fixed north-south line.
69. 70.
71. 72.y
x−1−2−3 2 3
−1
−2
−3
1
2
3
y
x−1−2 1 2 3 4 5
−1
1
2
3
4
6
7
y � �43x �
13y � �
45x �
225
y
x−1−2−3 2 3
−1
−2
−3
1
2
3
y
x−1−2−3−4 1 2 3 4
−1
1
2
3
5
6
7
y � �12x �
16y � 4x � 6
N 24� E
S � 8 � 6.3 sin�
6t �
�
2S � 8 � 6.3 cos�
6t
t2 4 121086
3
6
9
12
15
Ave
rage
sal
es(i
n m
illio
ns o
f do
llars
)
Month (1 ↔ January)
S
� � 60�� 83.1
0900
100
A � 64�1 � cos ���sin ��
−12
−2� 2�
12
L � L1 � L2 �2
sin ��
3
cos �
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Base 1 Base 2 Altitude Area
8 59.7
8 72.7
8 80.5
8 83.1
8 80.7
8 74.0
8 64.08 sin 90�8 � 16 cos 90�
8 sin 80�8 � 16 cos 80�
8 sin 70�8 � 16 cos 70�
8 sin 60�8 � 16 cos 60�
8 sin 50�8 � 16 cos 50�
8 sin 40�8 � 16 cos 40�
8 sin 30�8 � 16 cos 30�
Base 1 Base 2 Altitude Area
8 82.73
8 83.04
8 83.11
8 83.14
8 83.11
8 83.048 sin 62�8 � 16 cos 62�
8 sin 61�8 � 16 cos 61�
8 sin 60�8 � 16 cos 60�
8 sin 59�8 � 16 cos 59�
8 sin 58�8 � 16 cos 58�
8 sin 56�8 � 16 cos 56�
333202CB04_AN.qxd 1/1/70 09:39 AM Page 25
Review Exercises (page 365)
1. 0.5 radian 2. 4.5 radians3. (a) 4. (a)
(b) Quadrant II (b) Quadrant I
(c) (c)
5. (a) 6. (a)
(b) Quadrant II (b) Quadrant I
(c) (c)
7. (a) 8. (a)
(b) Quadrant I (b) Quadrant IV(c) (c)
9. (a) 10. (a)
(b) Quadrant III (b) Quadrant IV(c) (c)
11. 8.378 12. 13. 14. 3.44315. 16. 17.18. 19. 48.17 inches 20. 11.52 meters21. (a) radians per minute
(b) inches per minute22. 212.1 inches per second23.
24. 25.
26. 27. 28.
29.
30.
31.
32. is undefined.
is undefined.
33.
34. 35.
36. 37.
38. 39. 3.2361 40.
41.
cot � �54
sec � ��41
5
csc � ��41
4
tan � �45
cos � �5�41
41
sin � �4�41
41
�0.3420�1.1368
�75.3130cos�13�
3 � cos 5�
3�
12
sin�17�
6 � sin 7�
6� �
12
cos 4� � cos 0 � 1
sin 11�
4� sin
3�
4�
�22
cot 2�tan 2� � 0
sec 2� � 1cos 2� � 1
csc 2�sin 2� � 0
cot�2�
3 ��33
tan�2�
3 � �3
sec�2�
3 � �2cos�2�
3 � �12
csc�2�
3 � �2�3
3sin�
2�
3 � ��32
cot �
4� 1tan
�
4� 1
sec �
4� �2cos
�
4�
�22
csc �
4� �2sin
�
4�
�22
cot 7�
6� �3tan
7�
6�
�33
sec 7�
6� �
2�33
cos 7�
6� �
�32
csc 7�
6� �2sin
7�
6� �
12
�12
, �32 �
�32
, 12�
�22
, �22
�12
, �32 Area � 55.31 square millimeters
Area � 339.29 square inches
400�662
3�
326.586�
�200.535��330.000�128.571�
�0.589�2.225
315�, �45�250�, �470�
x
−405°
y
−110°
x
y
640�, �80�430�, �290�
x
280°
y
70°
y
x
�
3, �
17�
32�
3, �
10�
3
x
−
y
233π
− 43π
x
y
20�
9, �
16�
93�
4, �
5�
4
x
y
29π
114π
x
y
Precalculus with Limits, Answers to Review Exercises 26
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42.
cot � � 1
sec � � �2
csc � � �2
tan � � 1
cos � ��22
sin � ��22
333202CB04_AN.qxd 1/1/70 09:39 AM Page 26
Precalculus with Limits, Answers to Review Exercises 27
(Continued)
43. 44.
45. (a) 3 (b) (c) (d)
46. (a) (b) (c) (d)
47. (a) (b) (c) (d)
48. (a) (b) (c) (d) 5
49. 0.6494 50. 5.2408 51. 0.5621 52. 5.3860
53. 3.6722 54. 0.2045 55. 71.3 meters 56. 19.5 feet
57.
58.
59.
60.
61.
62.
63.
64.
65. 66.
67. 68.
69. 70.
cot � � ��3cot � � �2�21
21
sec � �2�3
3sec � � �
5
2
csc � � �2csc � �5�21
21
tan � � ��33
tan � � ��21
2
cos � ��32
sin � ��21
5
cot � �45
cot � � ��55
3
sec � � ��41
4sec � � �
8�55
55
csc � � ��41
5csc � �
8
3
cos � � �4�41
41tan � � �
3�55
55
sin � � �5�41
41cos � � �
�55
8
cot � � ��52
cot � � �5�11
11
sec � � �3�5
5csc � � �
6�11
11
tan � � �2�5
5tan � � �
�11
5
cos � � ��53
cos � �5
6
sin � �23
sin � � ��11
6
cot � �23
tan � �32
sec � � ��13
2cos � � �
2�1313
csc � � ��13
3sin � � �
3�1313
cot � �1
4tan � � 4
sec � � �17cos � ��17
17
csc � ��17
4sin � �
4�17
17
cot � �34tan � �
43
sec � �53cos � �
35
csc � �54sin � �
45
cot � � �19
tan � � �9
sec � � ��82cos � ���82
82
csc � ��82
9sin � �
9�8282
cot � � 5tan � �15
sec � � ��26
5cos � � �
5�2626
csc � � ��26sin � � ��2626
cot � �4
15tan � �
15
4
sec � ��241
4cos � �
4�241
241
csc � ��241
15sin � �
15�241
241
cot � � �34tan � � �
43
sec � �53cos � �
35
csc � � �54sin � � �
45
cot � �34tan � �
43
sec � �53cos � �
35
csc � �54sin � �
45
�612
2�615
�1515
4�1515
�154
14
�174
�1717
�1714
�24
3�24
2�23
cot � �2�14
5cot � �
�33
sec � �9�14
28sec � � 2
csc � �95
csc � �2�3
3
tan � �5�14
28tan � � �3
cos � �2�14
9cos � �
12
sin � �59
sin � ��32
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333202CB04_AN.qxd 1/1/70 09:39 AM Page 27
(Continued)
71. 72.
73. 74.
75.
76.
77.
78.
79.
80.
81.
82.
83. 84. 85. 0.0584
86. 0.0878 87. 3.2361 88. 4.3813
89. 90.
91. 92.
93. 94.
95. 96.
97. (a) (b) 264 cycles per second
98. (a)
(b) 12. Yes. One period is one year.(c) 1.41. 1.41 represents the maximum change in time
from the average time of sunset.�d � 18.09�
014
12
22
y � 2 sin 528�x
−4
−3
1
2
3
4
tπ
y
−4
−3
−2
−1
1
3
4
tπ
y
321−2−3x
−1
−2
−3
−5
−6
−1
y
4
3
2
−1
−2
xπππ 2
y
−
−8
−6
−4
8
x
y
π8π4
−6
−2
2
4
6
xπ6
y
x
2
−1
−2
2πππ−
y
2
1
−2
x
− ππ322
y
�0.1425�0.7568
tan 315� � �1sin 315� � ��22
; cos 315� ��22
;
tan��240�� � ��3
sin��240�� ��32
; cos��240�� � �12
;
tan��150�� ��33
cos��150�� � ��32
;sin��150�� � �12
;
tan 495� � �1cos 495� � ��22
;sin 495� ��22
;
tan�5�
4 � �1
cos�5�
4 � ��22
;sin�5�
4 ��22
;
tan�7�
3 � ��3
cos�7�
3 �12
;sin�7�
3 � ��32
;
tan �
4� 1cos
�
4�
�22
;sin �
4�
�22
;
tan �
3� �3cos
�
3�
12
;sin �
3�
�32
;
′θ
x
y
173π
′θx
y
65π−
�� ��
3�� �
�
5
′θ
635°
x
y
′θ
264°
x
y
�� � 85��� � 84�
Precalculus with Limits, Answers to Review Exercises 28
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Precalculus with Limits, Answers to Review Exercises 29
(Continued)
99. 100.
101. 102.
103. 104.
105. 106.
107. 108.
As As
109. 110. 111. 0.41 112. 0.21
113. 114. 1.10 115. 116.
117. 118. 119. 1.24 120. 2.66
121. 122. 1.45123. 124.
125. 126.
127. 128. 129. 130.
131. 132. 133.
134. 9.6 feet 135. 1221 miles,
136.
137. False. The sine or cosine function is often useful for modeling simple harmonic motion.
138. True. The inverse sine, is defined byif and only if where
and
139. False. For each there corresponds exactly one value of
140. False. is not in the range of the arctangent function.
141. d; The period is and the amplitude is 3.
142. a; The period is and, because the graph isreflected in the -axis.
143. b; The period is 2 and the amplitude is 2.
144. c; The period is and the amplitude is 2.
145. The function is undefined because 146. (a)
(b) tan� ��
2 � �cot �
sec � � 1�cos �.
4�
xa < 0,2�
2�
3��4
y.�
���2 ≤ y ≤ ��2.�1 ≤ x ≤ 1sin y � xy � arcsin x
y � arcsin x,
d � 0.75 cos2�t3
85.6�
66.8�1
�1 � �x � 1�2
�4 � x2
x
�512
135
43
45
−1.5 1.5
�2
−�2
−4 4
�2
−�2
−1.5 1.5
�3
0
−1.5 1.5
�
−�
�0.98
�
6�
�
4�
6�0.46
��
2�
�
6
x → � , f �x� → � x → � , f �x� → �
−2
−300
300
� 2�−9
−6
9
6
tπ
2
y
4
3
2
1
−3
−4
x
− ππ322
y
tπ
1
y
ππ−−1
−2
−3
−4
x
y
t
3
2
1
y
ππ−x
4
3
2
1
−3
−4
ππ−
y
t
1
2
3
y
π2
π2
−
4
3
2
1
πx
y
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0.1 0.4 0.7
tan 9.9666 2.3652 1.1872
9.9666 2.3652 1.1872����cot �
���� ��
2�
1.0 1.3
tan 0.6421 0.2776
0.6421 0.2776���cot �
��� ��
2�
333202CB04_AN.qxd 1/1/70 09:39 AM Page 29
(Continued)
147. The ranges of the other four trigonometric functions areor
148. (a) The displacement is increased.(b) The friction damps the oscillations more quickly.(c) The frequency of the oscillations increases.
149. (a) (b)The area function increases more rapidly.
150. Answers will vary.
03
30
A s
00
0 6
4
A s
s � 10�, � > 0s � 0.8r, r > 0A � 50�, � > 0;A � 0.4r2, r > 0;
�� , �1� � �1, �.�� , �
Precalculus with Limits, Answers to Review Exercises 30
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333202CB04_AN.qxd 1/1/70 09:39 AM Page 30
Precalculus with Limits, Answers to Chapter Test 31
Chapter Test (page 369)
1. (a) (b)
(c)
2. 3000 radians per minute 3. square feet
4. sin csc
cos sec
tan cot
5. For For
sin
cos
csc
sec
cot
6.
7. Quadrant III 8. 9. 1.33, 1.81
10. 11.
12. 13.
14. 15.
Period: 2 Not periodic
16. 17.
18. 19.
20. d � �6 cos �t
310.1�
1−2 2
−
x
π
π
y
�5
2a � �2, b �
12
, c � ��
4
0
−2
32
6
−6
−4
6
4
α1
2
3
4
− π π2
π−π2
y
−4
−3
−2
−1
1
3
4
π2x
y
cot � � �8
15cot � � �34
csc � �1715sec � �
53
tan � � �158csc � � �
54
cos � � �817tan � � �
43
sin � �1517sin � � �
45
150�, 210�
′θ
290°
x
y
�� � 70�
cot � �23
� �2
3
sec � � ��13
2� �
�13
2
csc � � ��13
3� �
�13
3
cos � � �2�13
13� �
2�13
13
sin � � �3�13
13� �
3�13
13
� ≤ � <3�
2:0 ≤ � <
�
2:
� � �1
3� � �3
� � ��10� � ��10
10
� ��10
3� �
3�10
10
� 709.04
225�
13�
4, �
3�
4
54π
x
y
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Problem Solving (page 371)
1. (a) radians or (b) feet
2. Gear 1:
Gear 2:
Gear 3:
Gear 4:
Gear 5:
3. (a) 4767 feet (b) 3705 feet(c) feet,
4. (a) Answers will vary. (b) The ratios are equal.(c) No; No(d) Yes, because all six trigonometric functions are ratios
of right triangles.5. (a) (b)
Even Even
6. (a) is even. (b) is even.
7.
8. (a)
(b) answers will vary.(c) 20 millimeters; answers will vary.
(d) 80 beats per minute (e)
9. (a)
(b) (c)
10. (a) (b) Period of
Period of
(c) Yes, because the sine and cosine functions are periodic.
11. (a) 3.35, 7.35 (b)(c) Yes. There is a difference of nine periods between the
values.
12. (a) Equal; two-period shift(b) Not equal; is a horizontal translation and
is a period change.(c) Not equal; For example,
13. (a) (b)(c)(d) As you move closer to the rock, must get smaller and
smaller. The angles and will decrease along withthe distance so will decrease.
14. (a) The approximation isaccurate over the interval
(b) The accuracy improved.
−2
2
�2
−�2
x9
9
�1 ≤ x ≤ 1.
−2
2
�2
−�2
dy,�2�1
d� 1.75 feet
x � 1.71 feet; y � 3.46 feet40.5�
sin�12�� � 2��� � sin�1
2��.f �1
2t�f �t �
12c�
�0.65
g : �
f : 2� ;
−6
−
6
� �
g
f
I�7369� � 0.945E�7369� � 0.901P�7369� � 0.631
−2
7348 7377
2
P
I
E
−2
7300 7380
2
P IE
Period �1516
seconds; 32�
15
Period �34 seconds;
070
5
130
h � 51 � 50 sin8�t ��
2h�x�h�x�
−1
−2
3
2� �
−1
−2
3
2� �
tan 63� �w � 3705
3000
w � 2183
48�
19 radians� 454.7�,
5�
2 radians450�,
24�
11 radians� 392.7�,
24�
13 radians� 332.3�,
3�
2 radians270�,
� 816.42990�11�
2
Precalculus with Limits, Answers to Problem Solving 32
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