CHAPTER 4 Trigonometric Functions · 274 Chapter 4 Trigonometric Functions 12. (a) Coterminal...
Transcript of CHAPTER 4 Trigonometric Functions · 274 Chapter 4 Trigonometric Functions 12. (a) Coterminal...
C H A P T E R 4Trigonometric Functions
Section 4.1 Radian and Degree Measure . . . . . . . . . . . . . . . . 272
Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . 281
Section 4.3 Right Triangle Trigonometry . . . . . . . . . . . . . . . . 289
Section 4.4 Trigonometric Functions of Any Angle . . . . . . . . . . 300
Section 4.5 Graphs of Sine and Cosine Functions . . . . . . . . . . . 317
Section 4.6 Graphs of Other Trigonometric Functions . . . . . . . . . 329
Section 4.7 Inverse Trigonometric Functions . . . . . . . . . . . . . . 339
Section 4.8 Applications and Models . . . . . . . . . . . . . . . . . . 350
Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
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C H A P T E R 4Trigonometric Functions
Section 4.1 Radian and Degree Measure
272
You should know the following basic facts about angles, their measurement, and their applications.
■ Types of Angles:
(a) Acute: Measure between and
(b) Right: Measure
(c) Obtuse: Measure between and
(d) Straight: Measure
■ and are complementary if They are supplementary if
■ Two angles in standard position that have the same terminal side are called coterminal angles.
■ To convert degrees to radians, use radians.
■ To convert radians to degrees, use 1 radian
■ one minute of
■ one second of
■ The length of a circular arc is where is measured in radians.
■ Speed
■ Angular speed � ��t � s�rt
� distance�time
�s � r�
1�� 1�60 of 1� � 1�36001� �
1�� 1�601� �
� �180����.1� � ��180
� � � 180�.� � � 90�.�
180�.
180�.90�
90�.
90�.0�
1. The angle shown is approximately 2 radians.
Vocabulary Check
1. Trigonometry 2. angle 3. standard position 4. coterminal
5. radian 6. complementary 7. supplementary 8. degree
9. linear 10. angular
2. The angle shown isapproximately radians.4
3. (a) Since lies in Quadrant IV.
(b) Since lies in
Quadrant II.
5�
2<
11�
4< 3�,
11�
4
3�
2<
7�
4< 2�,
7�
44. (a) Since lies in
Quadrant IV.
(b) Since lies in
Quadrant II.
3�
2<
13�
9< �,
13�
9
�
2<
5�
12< 0,
5�
2
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Section 4.1 Radian and Degree Measure 273
9. (a)
(b)
x
y
π3
2
2�
3
x
116π
y11�
6
7. (a)
(b)
x
y
π3
4
4�
3
43π
x
y13�
4
5. (a) Since lies in Quadrant IV.
(b) Since lies in
Quadrant III.
� < 2 < �
2; 2
�
2< 1 < 0; 1 6. (a) Since 3.5 lies in Quadrant III.
(b) Since 2.25 lies in Quadrant II.�
2< 2.25 < �,
� < 3.5 <3�
2,
8. (a)
(b)
x
y
52π−
5�
2
x
74
− π
y
7�
4
10. (a) 4
(b)
x
y
−3
3
x
4
y
11. (a) Coterminal angles for
�
6 2� �
11�
6
�
6� 2� �
13�
6
�
6: (b) Coterminal angles for
2�
3 2� �
4�
3
2�
3� 2� �
8�
3
2�
3:
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274 Chapter 4 Trigonometric Functions
12. (a) Coterminal angles for
(b) Coterminal angles for
5�
4 2� �
3�
4
5�
4� 2� �
13�
4
5�
4:
7�
6 2� �
5�
6
7�
6� 2� �
19�
6
7�
6: 13. (a) Coterminal angles for
(b) Coterminal angles for
2�
15 2� �
32�
15
2�
15� 2� �
28�
15
2�
15:
9�
4� 4� �
7�
4
9�
4� 2� �
�
4
9�
4: 14. (a) Coterminal angles for
(b) Coterminal angles for
8�
45 2� �
82�
45
8�
45� 2� �
98�
45
8�
45:
7�
8 2� �
9�
8
7�
8� 2� �
23�
8
7�
8:
15. Complement:
Supplement: � �
3�
2�
3
�
2
�
3�
�
6
17. Complement:
Supplement: � �
6�
5�
6
�
2
�
6�
�
3
16. Complement: Not possible; is greater than
(a)Supplement: � 3�
4�
�
4
�
2.
3�
4
18. Complement: Not possible; is greater than
Supplement: � 2�
3�
�
3
�
2.
2�
3
19. Complement:
Supplement: � 1 � 2.14
�
2 1 � 0.57 20. Complement: None
Supplement: � 2 � 1.14
�2 >�
2�
21.
The angle shown is approximately 210�.
22.
The angle shown is approximately 45�.
23. (a) Since lies in Quadrant II.
(b) Since lies inQuadrant IV.
282�270� < 282� < 360�,
150�90� < 150� < 180�,
25. (a) Since lies in Quadrant III.
(b) Since lies in Quadrant I.336� 30�
360� < 336� 30� < 270�,
132� 50�180� < 132� 50� < 90�,
24. (a) Since lies in Quadrant I.
(b) Since lies in Quadrant I.8.5�0� < 8.5� < 90�,
87.9�0� < 87.9� < 90�,
26. (a) Since lies in Quadrant II.
(b) Since liesin Quadrant IV.
12.35�90� < 12.35� < 0�,
245.25�270� < 245.25� < 180�,
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Section 4.1 Radian and Degree Measure 275
27. (a)
30�x
y30� (b)
150°
x
y150�
28. (a)
(b)
x
−120°
y
120�
x
−270°
y
270� 29. (a)
(b)
780°
x
y
780�
x
405°
y
405� 30. (a)
(b)
x
y
−600�
600�
− °450
x
y
450�
31. (a) Coterminal angles for
(b) Coterminal angles for
36� 360� � 396�
36� � 360� � 324�
36�:
52� 360� � 308�
52� � 360� � 412�
52�: 33. (a) Coterminal angles for
(b) Coterminal angles for
230� 360� � 130�
230� � 360� � 590�
230�:
300� 360� � 60�
300� � 360� � 660�300�:32. (a) Coterminal angles for
(b) Coterminal angles for
390� � 360� � 30�
390� � 720� � 330�
390�:
114� 360� � 246�
114� � 360� � 474�
114�:
34. (a) Coterminal angles for
(b) Coterminal angles for
740� � 720� � 20�
740� � 1080� � 340�
740�:
445� � 360� � 85�
445� � 720� � 275�
445�: 35. Complement:
Supplement: 180� 24� � 156�
90� 24� � 66� 36. Complement: Not possible
Supplement: 180� 129� � 51�
37. Complement:
Supplement: 180� 87� � 93�
90� 87� � 3� 38. Complement: Not possible
Supplement: 180� 167� � 13�
39. (a)
(b) 150� � 150�� �
180�� �5�
6
30� � 30�� �
180�� ��
6
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276 Chapter 4 Trigonometric Functions
44. (a)
(b) 3� � 3��180�
� � � 540�
4� � 4��180�
� � � 720�
46. (a)
(b)28�
15�
28�
15 �180�
� � � 336�
15�
6�
15�
6 �180�
� � � 450�45. (a)
(b) 13�
60�
13�
60 �180�
� � � 39�
7�
3�
7�
3 �180�
� � � 420�
47. 115� � 115� �
180�� � 2.007 radians 48. radians83.7� � 83.7�� �
180�� � 1.461
50. radians46.52� � 46.52�� �
180�� � 0.81249. 216.35� � 216.35� �
180�� � 3.776 radians
51. 0.78� � 0.78� �
180�� � 0.014 radians
53.�
7�
�
7�180�
� � � 25.714� 55. 6.5� � 6.5��180�
� � � 1170�
61. 85� 18� 30� � 85� � �1860�� � � 30
3600�� � 85.308�
63. 125� 36� � 125� � 363600�� � 125.01�
65. 280.6� � 280� � 0.6�60�� � 280� 36�
67. 345.12� � 345� 7� 12�
57. 2 � 2�180�
� � � 114.592�
59. 64� 45� � 64� � �4560�
�� 64.75�
52. radians395� � 395�� �
180�� � 6.894
54.8�
13�
8�
13�180�
� � � 110.769�
56. 4.2� � 4.2��180�
� � � 756�
58. 0.48 � 0.48�180�
� � � 27.502�
60. 124� 30� � 124.5�
62. 408� 16� 25� � 408.274�
64. 330� 25� � 330.007�
66. 115.8� � 115� 48� 68. 310.75� � 310� 45�
43. (a)
(b) 7�
6�
7�
6 �180�
� � � 210�
3�
2�
3�
2 �180�
� � � 270�
40. (a)
(b) 120� � 120�� �
180�� �2�
3
315� � 315�� �
180�� �7�
4
42. (a) (b) 144� � 144�� �
180�� �4�
5270� � 270�� �
180�� � 3�
2
41. (a)
(b) 240� � 240�� �
180�� � 4�
3
20� � 20�� �
180�� � �
9
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Section 4.1 Radian and Degree Measure 277
69.
� 20.34� � 20� 20� 24�
0.355 � 0.355�180�
� � 70.
� 45� 3� 47�
� 45� � 3� � 0.786�60� �
� 45� � �0.0631��60��
� 45.0631�
0.7865 � 0.7865�180�
� �
72.
radians � �3112 � 2 7
12
31 � 12�
s � r�71.
radians� �65
6 � 5�
s � r� 73.
radians3� �327 � 44
7
32 � 7�
3s � r�
74.
Because the angle represented is clockwise, this angle is radians.45
� �6075 �
45 radians
60 � 75�
s � r�
78. radian� �sr
�1022
�511
75. The angles in radians are:
90� ��
2
60� ��
3
45� ��
4
30� ��
6
0� � 0
330� �11�
6
270� �3�
2
210� �7�
6
180� � �
135� �3�
4
76. The angles in degrees are:
5�
6� 150�
2�
3� 120�
�
3� 60�
�
4� 45�
�
6� 30�
� � 180�
7�
4� 315�
5�
3� 300�
4�
3� 240�
5�
4� 225�
77.
radians � �815
8 � 15�
s � r� 79.
radians � �7029
� 2.414
35 � 14.5�
s � r�
80. kilometers, kilometers
� �s
r�
160
80� 2 radians
s � 160r � 80
82. feet,
s � r� � 9��
3� � 3� feet
� � 60� ��
3r � 9
81. in radians
inches s � 14�180�� �
180� � 14� � 43.982
s � r�, �
83. in radians
meters meters� 56.55 s � 27�2�
3 � � 18�
s � r�, �
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278 Chapter 4 Trigonometric Functions
90.
s � r� � 4000�1.0105� � 4042.0 miles
� � 31� 46� � 26� 8� � 57� 54� � 1.0105 rad
r � 4000 miles
89.
s � r� � 4000�0.28537� � 1141.48 miles
� 16� 21� 1� � 0.28537 radian
� � 42� 7� 33� 25� 46� 32�88. r �s�
�8
330����180�� �48
11� inches � 1.39 inches
91.
� 4� 2� 33.02�
� �sr
�4506378
� 0.07056 radian � 4.04�
93. � 23.87�� �s
r�
2.5
6�
25
60�
5
12 radian
95. (a) single axel:
(b) double axel:
(c) triple axel: � 7� radians 312 revolutions � 1260�
� 4� � � � 5� radians
212 revolutions � 720� � 180� � 900�
� 2� � � � 3� radians
112 revolutions � 360� � 180� � 540�
96. Linear speed �st
�r�
t�
�6400 � 1250�2�
110� 436.967 km�min
92.
� �sr
�4006378
� 0.0627 rad � 3.59�
r � 6378, s � 400
94. � �sr
�245
� 4.8 rad � 275.02�
97. (a)
Angular speed
(b) Radius of saw blade
Radius in feet
Speed
� 0.3125�80�� � 78.54 ft�sec
�s
t�
r�
t� r
�
t� r�angular speed�
�3.75
12� 0.3125 ft
�7.5
2� 3.75 in.
� �2���40� � 80� rad�sec
Revolutions
Second�
2400
60� 40 rev�sec 98. (a)
Angular speed
(b) Radius of saw blade
Radius in feet
Speed
� 0.3021�160�� � 151.84 ft�sec
�s
t�
r�
t� r
�
t� r�angular speed�
�3.625
12 � 0.3021 ft
�7.25
2� 3.625 in.
� �2���80� � 160� rad�sec
Revolutions
Second�
4800
60� 80 rev�sec
86. r �s�
�3
4��3�
94�
meters � 0.72 meter 87. r �s�
�82
135����180�� �3283�
miles � 34.80 miles
84. centimeters,
centimeters cm� 28.27s � r� � 12�3�
4 � � 9�
� �3�
4r � 12 85. r �
s�
�36
��2�
72�
feet � 22.92 feet
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Section 4.1 Radian and Degree Measure 279
99. (a)
Angular speed
Radius of wheel
Speed
(b) Let spin balance machine rate.
70 � r�angular speed� � r�2� �x
�1�60�� �5
25,344120�x ⇒ x � 941.18 rev�min
x �
�5
25,344� 57,600� �
125�
11� 35.70 miles�hr
�s
t�
r�
t� r
�
t� r�angular speed�
�25�2
�12 in.�ft��5280 ft�mi��
5
25,344 miles
� 2� �28,800� � 57,600� rad�hr
Revolutions
Hour�
480
�1�60�� 28,800 rev�hr
100. (a)
(b) Speed
For the outermost track, 6000� cm�min
6�400�� ≤ linear speed ≤ 6�1000��
�s
t�
r�
t� r
�
t� r�angular speed� � 6�angular speed�
400� rad�min ≤ angular speed ≤ 1000� rad�min
2��200� ≤ angular speed ≤ 2��500�
200 ≤revolutions
minute≤ 500
101. False, 1 radian so one radian
is much larger than one degree.
� �180
� �� � 57.3�,
103. True:2�
3�
�
4�
�
12�
8� � 3� � �
12� � � 180�
102. No, is coterminal with 180�.1260�
104. (a) An angle is in standard position when the origin is the vertex and the initial sidecoincides with the positive -axis.
(b) A negative angle is generated by a clockwise rotation.
(c) Angles that have the same initial and terminal sides are coterminal angles.
(d) An obtuse angle is between and 180�.90�
x
105. If is constant, the length of the arc isproportional to the radius and hence increasing.
�s � r��,� 106. Let A be the area of a circular sector of radius
and central angle Then
A
�r2�
�
2� ⇒ � �
1
2 r2�.
�.r
107. square metersA �1
2r2� �
1
2�10�2 �
�
3�
50
3� 108. Because
Hence, A �1
2r2� �
1
2152�12
15� � 90 ft2.
s � r�, � �12
15.
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280 Chapter 4 Trigonometric Functions
109.
(a) Domain:
Domain:
The area function changes more rapidly for because it is quadratic and the arc length function is linear.
(b) Domain:
Domain: 0 < � < 2� s � r� � 10�
00
2
320
A
s
�
0 < � < 2� r � 10 ⇒ A �12�102�� � 50�
r > 1
r > 0 s � r� � r�0.8�
r > 0 � � 0.8 ⇒ A �12r2�0.8� � 0.4r2
00
12
8
As
A �12r2�, s � r�
113.
−2 2 3 4 5 6
−2
−3
−4
1
2
3
4
x
y
115.
x
y
−2−3−4 1 2 3 4
−3
1
3
4
5
117.
x
y
−2−3−4 1 2 3 4
−2
−4
−5
1
2
3
110. If a fan of greater diameter is installed, the angular speed does not change.
111. Answers will vary. 112. Answers will vary.
114.
x
y
−2−3−4 1 2 3 4
−2
−3
−6
1
2
116.
x
y
−3−4−5 1 2 3
−2
−3
−4
1
2
3
4
118.
x
y
−2 1 2 3 5 6
−2
−3
−4
1
2
3
4
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Section 4.2 Trigonometric Functions: The Unit Circle
Section 4.2 Trigonometric Functions: The Unit Circle 281
2.
cot � �x
y�
12�13
5�13�
12
5
sec � �1
x�
1
12�13�
13
12
csc � �1
y�
1
5�13�
13
5
tan � �y
x�
5�13
12�13�
5
12
cos � � x �12
13
sin � � y �5
13
�x, y� � �12
13,
5
13�
4.
cot � �x
y�
�4�5
�3�5�
4
3
sec � �1
x�
1
�4�5� �
5
4
csc � �1
y�
1
�3�5� �
5
3
tan � �y
x�
�3�5
�4�5�
3
4
cos � � x � �4
5
sin � � y � �3
5
�x, y� � ��4
5, �
3
5�
6. t ��
3 ⇒ �1
2, �3
2 �
1.
csc � �1y
�1715
sec � �1x
� �178
cot � �xy
� �815
tan � �yx
� �158
cos � � x � �817
sin � � y �1517
3.
csc � �1y
� �135
sec � �1x
�1312
cot � �xy
� �125
tan � �yx
� �512
cos � � x �1213
sin � � y � �513
Vocabulary Check
1. unit circle 2. periodic 3. odd, even
5. corresponds to ��2
2, �2
2 �.t ��
4
■ You should know how to evaluate trigonometric functions using the unit circle.
■ You should know the definition of a periodic function.
■ You should be able to recognize even and odd trigonometric functions.
■ You should be able to evaluate trigonometric functions with a calculator in both radian and degree mode.
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282 Chapter 4 Trigonometric Functions
10. corresponds to �12
, ��32 �.t �
5�
3
12. t � � ⇒ ��1, 0�
18. corresponds to
tan �
3�
y
x�
�3�2
1�2� �3
cos �
3� x �
1
2
sin �
3� y �
�3
2
�1
2, �3
2 �.t ��
3
9. corresponds to ��12
, �32 �.t �
2�
3
11. corresponds to �0, �1�.t �3�
2
13. corresponds to ��22
, �22 �.t � �
7�
414. corresponds to ��
12
, �32 �.t � �
4�
3
15. corresponds to �0, 1�.t � �3�
216. corresponds to �1, 0�.t � �2�
17. corresponds to
tan t �y
x� 1
cos t � x ��2
2
sin t � y ��2
2
��2
2, �2
2 �.t ��
4
19. corresponds to
tan t �y
x�
1�3
��3
3
cos t � x � ��3
2
sin t � y � �1
2
���3
2, �
1
2�.t �7�
620. corresponds to
tan t �y
x� �1
cos t � x � ��2
2
sin t � y ��2
2
���2
2, �2
2 �.t � �5�
4
21. corresponds to
tan t �y
x� ��3
cos t � x � �1
2
sin t � y ��3
2
��1
2, �3
2 �.t �2�
322. corresponds to
tan t �y
x� ��3
cos t � x �1
2
sin t � y � ��3
2
�1
2, �
�3
2 �.t �5�
3
8. t �5�
4 ⇒ ��
�2
2, �
�2
2 �7. corresponds to ���32
, �12�.t �
7�
6
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Section 4.2 Trigonometric Functions: The Unit Circle 283
24. corresponds to
tan t �y
x� �
�3
3
cos t � x ��3
2
sin t � y � �1
2
��3
2, �
1
2�.t �11�
623. corresponds to
tan t �yx
� �3
cos t � x �12
sin t � y ��32
�12
, �32 �.t � �
5�
3
25. corresponds to
tan t �y
x� �
�3
3
cos t � x ��3
2
sin t � y � �1
2
��3
2, �
1
2�.t � ��
626. corresponds to
tan��3�
4 � �yx
� 1
cos��3�
4 � � x � ��22
sin��3�
4 � � y � ��22
���2
2, �
�2
2 �.t � �3�
4
27. corresponds to
tan t �yx
� 1
cos t � x ��22
sin t � y ��22
��22
, �22 �.t � �
7�
428. corresponds to
tan t �yx
� ��3
cos t � x � �12
sin t � y ��32
��12
, �32 �.t � �
4�
3
29. corresponds to
is undefined.tan t �y
x
cos t � x � 0
sin t � y � 1
�0, 1�.t � �3�
2
31. corresponds to
cot t �xy � �1tan t �
y
x � �1
sec t �1
x� ��2cos t � x � �
�2
2
csc t �1
y� �2sin t � y �
�2
2
���2
2, �2
2 � .t �3�
4
30. corresponds to
tan��2�� �y
x�
0
1� 0
cos��2�� � x � 1
sin��2�� � y � 0
�1, 0�.t � �2�
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284 Chapter 4 Trigonometric Functions
32. corresponds to
cot t �xy
� ��3
sec t �1x
� �2�3
� �2�3
3
csc t �1y
� 2
tan t �yx
� �1�3
� ��33
cos t � x � ��32
sin t � y �12
���32
, 12�.t �
5�
633. corresponds to
is undefined.
is undefined. cot t �xy � 0tan t �
y
x
sec t �1
xcos t � x � 0
csc t �1
y� 1sin t � y � 1
�0, 1�.t ��
2
35. corresponds to
cot � ��33
tan t �y
x � �3
sec � � �2cos t � x � �1
2
csc t � �2�3
3sin t � y � �
�3
2
��12
, ��32 �.t � �
2�
334. corresponds to
undefined
undefined
cot 3�
2�
x
y�
0
�1� 0
sec 3�
2�
1
x�
1
0 ⇒
csc 3�
2�
1
y�
1
�1� �1
tan 3�
2�
y
x�
�1
0 ⇒
cos 3�
2� x � 0
sin 3�
2� y � �1
�0, �1�.t �3�
2
36. corresponds to
tan��7�
4 � �yx
� 1
cos��7�
4 � � x ��22
sin��7�
4 � � y ��22
��2
2, �2
2 �.t � �7�
4
cot��7�
4 � �xy
� 1
sec��7�
4 � �1x
� �2
csc��7�
4 � �1y
� �2
37. sin 5� � sin � � 0
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Section 4.2 Trigonometric Functions: The Unit Circle 285
39. cos 8�
3� cos
2�
3� �
1
238. Because
cos 7� � cos�6� � �� � cos � � �1
7� � 6� � � :
40. Because
sin 9�
4� sin�2� �
�
4� � sin �
4�
�2
2
9�
4� 2� �
�
4:
42. Because
� sin�5�
6 � �12
sin��19�
6 � � sin��4� �5�
6 �
�19�
6� �4� �
5�
6:
44. Because :
� cos 4�
3� �
1
2
cos��8�
3 � � cos��4� �4�
3 �
�8�
3� �4� �
4�
3
41. cos��13�
6 � � cos���
6� � cos�11�
6 � ��32
43. sin��9�
4 � � sin���
4� � ��2
2
45.
(a)
(b) csc��t� � �csc t � �3
sin��t� � �sin t � �1
3
sin t �1
3
47.
(a)
(b) sec��t� �1
cos��t�� �5
cos t � cos��t� � �1
5
cos��t� � �1
546. cos
(a)
(b) sec��t� �1
cos��t��
1
cos t� �
4
3
cos��t� � cos t � �3
4
t � �3
4
48.
(a)
(b) csc t �1
sin�t��
1
�sin��t�� �
8
3
sin t � �sin��t� � �3
8
sin��t� �3
849.
(a)
(b) sin�t � �� � �sin t � �4
5
sin�� � t� � sin t �4
5
sin t �4
5
50.
(a)
(b) cos�t � �� � �cos t � �4
5
cos�� � t� � �cos t � �4
5
cos t �4
551. sin
7�
9� 0.6428
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286 Chapter 4 Trigonometric Functions
56. cot 3.7 �1
tan 3.7� 1.6007
52. tan 2�
5� 3.0777 53. cos
11�
5� 0.8090 54. sin
11�
9� �0.6428
55. csc 1.3 � 1.0378 57. cos��1.7� � �0.1288
59. csc 0.8 �1
sin 0.8� 1.3940
76.
(a)
(b)
y�0� �1
4e�0 cos�0� �
1
4 foot
y�t� �1
4e�t cos 6t
70. (a)
(b) cos 2.5 � x � �0.8
sin 0.75 � y � 0.7
72. (a)
(b)
t � 0.72 or t � 5.56
cos t � 0.75
t � 4.0 or t � 5.4
sin t � �0.75
58. cos��2.5� � �0.8011 60. sec 1.8 �1
cos 1.8� �4.4014
62. sin��13.4� � �0.740461. sec 22.8 �1
cos 22.8� �1.4486
69. (a)
(b) cos 2 � �0.4
sin 5 � �1
63. cot 2.5 �1
tan 2.5� �1.3386
64. tan 1.75 � �5.5204 65. csc��1.5� �1
sin��1.5� � �1.0025
66. tan��2.25� � 1.2386 67. sec��4.5� �1
cos��4.5� � �4.7439
68. csc��5.2� �1
sin��5.2� � 1.1319
71. (a)
(b)
t � 1.82 or 4.46
cos t � �0.25
t � 0.25 or 2.89
sin t � 0.25 73.
amperes � 0.79
I�0.7� � 5e�1.4 sin 0.7
I � 5e�2t sin t
74. At
I � 5e�2�1.4� sin 1.4 � 0.30 amperes.
t � 1.4, 75.
(a)
(b)
(c) y�12� �
14 cos 3 � �0.2475 ft
y�14� �
14 cos 32 � 0.0177 ft
y�0� �14 cos 0 � 0.2500 ft
y�t� �14 cos 6t
(c) The maximum displacements are decreasingbecause of friction, which is modeled by the
term.
(d)
t ��
12,
�
4
6t ��
2,
3�
2
cos 6t � 0
14
e�t cos 6t � 0
e�t
0.50 1.02 1.54 2.07 2.59
0.09 0.03 �0.02�0.05�0.15y
t
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Section 4.2 Trigonometric Functions: The Unit Circle 287
78. True
80. True
82. (a) The points and are symmetric about the origin.
(b) Because of the symmetry of the points, you canmake the conjecture that
(c) Because of the symmetry of the points, you canmake the conjecture that cos�t1 � �� � �cos t1.
sin�t1 � �� � �sin t1.
�x2, y2��x1, y1�
84.
Therefore, sin t1 � sin t2 � sin�t1 � t2�.
sin 1 � 0.8415
sin�0.25� � sin�0.75� � 0.2474 � 0.6816 � 0.9290
77. False. sin��4�
3 � ��32
> 0
79. False. 0 corresponds to �1, 0�.
81. (a) The points have -axis symmetry.
(b) since they have the same-value.
(c) since the -values haveopposite signs.
x�cos t1 � cos �� � t1�
ysin t1 � sin �� � t1�
y
83.
Thus, cos 2t � 2 cos t.
2 cos 0.75 � 1.4634cos 1.5 � 0.0707,
85.
θ
θ−
(x, y)
(x, −y)
x
y
sec � �1
cos ��
1cos���� � sec����
cos � � x � cos����
87. is odd.
h��t� � f ��t�g��t� � �f �t�g�t� � �h�t�
h�t� � f �t�g�t� 88. and
Both and are odd functions.
The function is even.h�t� � f�t�g�t�
� sin t tan t � h�t�
� ��sin t���tan t�
h��t� � sin��t� tan��t�
h�t� � f�t�g�t� � sin t tan t
gf
g�t� � tan tf�t� � sin t
86.
cot � �xy
� �cot����
tan � �yx
� �tan����
csc � �1y
� �csc����
sin � � y � �sin����
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288 Chapter 4 Trigonometric Functions
90.
f�1�x� � 3�4�x � 1�
y � 3�4�x � 1�
4�x � 1� � y3
x � 1 �14 y3
x �14 y3 � 1
y �14x3 � 1
−9 9
−6
6
f f−1
f�x� �14 x3 � 189.
f �1�x� �23�x � 1�
23�x � 1� � y
2x � 2 � 3y
2x � 3y � 2
x �12�3y � 2�
y �12�3x � 2�
−9 9
−6
6
ff−1
f �x� �12�3x � 2�
92.
f�1�x� �x
2 � x, x < 2
x
2 � x� y, x < 2
x � y�2 � x�
x � 2y � xy
xy � x � 2y
x �2y
y � 1
y �2x
x � 1, x > �1
−3 9
−4
4
f�x� �2x
x � 1, x > �191.
f�1�x� � �x2 � 4, x ≥ 0
�x2 � 4 � y, x ≥ 0
x2 � 4 � y2
x2 � y2 � 4
x � �y2 � 4
−2 10
−1
7
ff−1
y � �x2 � 4
f �x� � �x2 � 4, x ≥ 2, y ≥ 0
94.
Asymptotes:
10864
4
6
8
10
−6
−8
x
y
x � �3, x � 2, y � 0
f �x� �5x
x2 � x � 6�
5x�x � 3��x � 2�93.
Asymptotes:
2 4 5 6 7 8−2 −1
1
345678
−2−3−4
−3−4x
y
x � 3, y � 2
f �x� �2x
x � 3
95.
Asymptotes: x � �2, y �12
x � 2 �x � 5
2�x � 2�,
f �x� �x2 � 3x � 10
2x2 � 8�
�x � 2��x � 5�2�x � 2��x � 2�
x
y
−1−5−6−7 1−1
−2
−3
−4
1
2
3
4
−2
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Section 4.3 Right Triangle Trigonometry 289
96.
Slant asymptote:
Vertical asymptotes: x � 3.608, x � �1.108
y �x2
�74 4
6
8
10
−8
−10
−2−4−6−8 6 8 10
2
y
x
f �x� �x3 � 6x2 � x � 1
2x2 � 5x � 8�
x2
�74
�15�x � 4�
4�2x2 � 5x � 8�
97.
Domain: all real numbers
Intercepts:
No asymptotes
�0, �4�, �1, 0�, ��4, 0�
0 � x2 � 3x � 4 � �x � 4��x � 1� ⇒ x � 1, �4
y � x2 � 3x � 4 98.
Domain: all
Intercepts:
Asymptote: x � 0
�1, 0�, ��1, 0�
ln x4 � 0 ⇒ x4 � 1 ⇒ x � ±1
x � 0
y � ln x4
99.
Domain: all real numbers
Intercept:
Asymptote: y � 2
�0, 5�
f�x� � 3x�1 � 2 100.
Domain: all real numbers
Intercepts:
Asymptotes: x � �2, y � 0
�7, 0�, �0, �74�
x � �2
f�x� �x � 7
�x2 � 4x � 4� �x � 7
�x � 2�2
Section 4.3 Right Triangle Trigonometry
■ You should know the right triangle definition of trigonometric functions.
(a) (b) (c)
(d) (e) (f)
■ You should know the following identities.
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
( j) (k)
■ You should know that two acute angles are complementary if and cofunctions of complementary angles are equal.
■ You should know the trigonometric function values of and or be able to construct triangles fromwhich you can determine them.
60�,30�, 45�,
� � � � 90�,� and �
1 � cot2 � � csc2 �1 � tan2 � � sec2 �
sin2 � � cos2 � � 1cot � �cos �
sin �tan � �
sin �
cos �
cot � �1
tan �tan � �
1
cot �sec � �
1
cos �
cos � �1
sec �csc � �
1
sin �sin � �
1
csc �
cot � �adj
oppsec � �
hyp
adjcsc � �
hyp
opp
Adjacent side
Opp
osite
sid
eHyp
otenu
se
θ
tan � �opp
adjcos � �
adj
hypsin � �
opp
hyp
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290 Chapter 4 Trigonometric Functions
Vocabulary Check
1. (a) iii (b) vi (c) ii (d) v (e) i (f) iv
2. hypotenuse, opposite, adjacent 3. elevation, depression
1.
cot � �adjopp
�43
sec � �hypadj
�54
csc � �hypopp
�53
tan � �oppadj
�34
cos � �adjhyp
�45
sin � �opphyp
�35
adj � �52 � 32 � �16 � 4
35
θ
3.
cot � �adj
opp�
15
8
sec � �hyp
adj�
17
15
csc � �hyp
opp�
17
8
tan � �opp
adj�
8
15
cos � �adj
hyp�
15
17
sin � �opp
hyp�
8
17
hyp � �82 � 152 � 17
θ15
8
2.
sin
cos
tan
csc
sec
cot � �adj
opp�
12
5
� �hyp
adj�
13
12
� �hyp
opp�
13
5
� �opp
adj�
5
12
� �adj
hyp�
12
13
� �opp
hyp�
5
13
b � �132 � 52 � �169 � 25 � 12
513
bθ
4.
sin
cos
tan
csc
sec
cot � �2
3
� ��13
2
� ��13
3
� �18
12�
3
2
� �12
6�13�
2�13
13
� �18
6�13�
3�13
13
C � �182 � 122 � �468 � 6�13
12
18 c
θ
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Section 4.3 Right Triangle Trigonometry 291
6.
sin
cos
tan
csc
sec
cot � �adj
opp�
�161
8
� �hyp
adj�
15�161
�15�161
161
� �hyp
opp�
15
8
� �opp
adj�
8�161
�8�161
161
� �adj
hyp�
�161
15
� �opp
hyp�
8
15
158
θ
adj � �152 � 82 � �161
sin
cos
tan
csc
sec
cot � �adj
opp�
�161
2 4�
�161
8
� �hyp
adj�
7.5
��161�2� �15
�161�
15�161
161
� �hyp
opp�
7.5
4�
15
8
� �opp
adj�
4
��161�2� �8
�161�
8�161
161
� �adj
hyp�
�161
2 7.5�
�161
15
� �opp
hyp�
4
7.5�
8
15
7.5 4θ
adj � �7.52 � 42 ��161
2
The function values are the same because the triangles are similar, and corresponding sides are proportional.
5.
cot � �adj
opp�
8
6�
4
3
sec � �hyp
adj�
10
8�
5
4
csc � �hyp
opp�
10
6�
5
3
tan � �opp
adj�
6
8�
3
4
cos � �adj
hyp�
8
10�
4
5
sin � �opp
hyp�
6
10�
3
5
θ8
10
opp � �102 � 82 � 6
cot � �adj
opp�
2
1.5�
4
3
sec � �hyp
adj�
2.5
2�
5
4
csc � �hyp
opp�
2.5
1.5�
5
3
tan � �opp
adj�
1.5
2�
3
4
cos � �adj
hyp�
2
2.5�
4
5
sin � �opp
hyp�
1.5
2.5�
3
5
θ2
2.5opp � �2.52 � 22 � 1.5
The function values are the same since the triangles are similar and the corresponding sides are proportional.
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292 Chapter 4 Trigonometric Functions
7.
cot � �adj
opp� 2�2
sec � �hyp
adj�
3
2�2�
3�2
4
csc � �hyp
opp� 3
tan � �opp
adj�
1
2�2�
�2
4
cos � �adj
hyp�
2�2
3
θ 31
sin � �opp
hyp�
1
3
adj � �32 � 12 � �8 � 2�2
8.
sin
cos
tan
csc
sec
cot � �adj
opp�
2
1� 2
� �hyp
adj�
�5
2
� �hyp
opp�
�5
1� �5
� �opp
adj�
1
2
� �adj
hyp�
2�5
�2�5
5
� �opp
hyp�
1�5
��5
5
1
2
θhyp � �12 � 22 � �5
sin
cos
tan
csc
sec
cot � �6
3� 2
� �3�5
6�
�5
2
� �3�5
3� �5
� �3
6�
1
2
� �6
3�5�
2�5
�2�5
5
� �3
3�5�
1�5
��5
5
3
6
θ
hyp � �32 � 62 � 3�5
The function values are the same because the triangles are similar, and corresponding sides are proportional.
cot � �adj
opp�
4�2
2� 2�2
sec � �hyp
adj�
6
4�2�
3
2�2�
3�2
4
csc � �hyp
opp�
6
2� 3
tan � �opp
adj�
2
4�2�
1
2�2�
�2
4
cos � �adj
hyp�
4�2
6�
2�2
3
θ
62
sin � �opp
hyp�
2
6�
1
3
adj � �62 � 22 � �32 � 4�2
The function values are the same since the triangles are similar and the corresponding sides are proportional.
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Section 4.3 Right Triangle Trigonometry 293
10.
sin
cos
tan
csc
sec � �hyp
adj�
�26
5
� �hyp
opp�
�26
1� �26
� �opp
adj�
1
5
� �adj
hyp�
5�26
�5�26
26
� �opp
hyp�
1�26
��26
26
1
5
26θ
hyp � �52 � 12 � �26
12.
sin
tan
csc
sec
cot � �3
2�10�
3�10
20
� �7
3
� �7
2�10�
7�10
20
� �2�10
3
� �2�10
7
θ
102
3
7
opp � �72 � 32 � �40 � 2�10
9. Given:
csc � �hypopp
�65
sec � �hypadj
�6
�11�
6�1111
cot � �adjopp
��11
5
tan � �oppadj
�5
�11�
5�1111
cos � �adjhyp
��11
6
adj � �11
52 � �adj�2 � 62
56
11
θ
sin � �56
�opphyp
11. Given:
csc � �4
�15�
4�1515
cot � �1
�15�
�1515
tan � � �15
cos � �14
sin � ��15
4
opp � �15
�opp�2 � 12 � 42
15
θ
4
1
sec � � 4 �41
�hypadj
14.
sin
cos
tan
sec
cot � �1
tan ��
�273
4
� �1
cos ��
17�273
�17�273
273
� �opp
adj�
4�273
�4�273
273
� �adj
hyp�
�273
17
� �opp
hyp�
4
17
417
273
θ
adj � �172 � 42 � �27313. Given:
csc � ��10
3
cot � �1
3
sec � � �10
cos � ��10
10
sin � �3�10
10
hyp � �10
32 � 12 � �hyp�2
3
1
10
θ
tan � � 3 �3
1�
opp
adj
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.
294 Chapter 4 Trigonometric Functions294 Chapter 4 Trigonometric Functions
16.
cos
tan
csc
sec
cot � �1
tan ��
�55
3
� �1
cos ��
8�55
�8�55
55
� �1
sin ��
8
3
� �opp
adj�
3�55
�3�55
55
� �adj
hyp�
�55
8
adj � �82 � 32 � �553
8
55
θ
sin � �38
15. Given:
csc � ��97
4
sec � ��97
9
tan � �49
cos � �9
�97�
9�9797
sin � �4
�97�
4�9797
hyp � �97
42 � 92 � �hyp�2
97
θ
9
4
cot � �94
�adjopp
Function (deg) (rad) Function Value��
26. tan1�3
630�
24. sin�22
445�
22. csc �2
445�
20. sec �2
445�
18. cos�22
445�
Function (deg) (rad) Function Value
17. sin
19. tan
21. cot
23. cos
25. cot 1
445�
�32
630�
�33
360�
�3
360�
12
630�
��
27. sin � �1
csc �28. cos � �
1sec �
29. tan � �1
cot �30. csc � �
1sin �
31. sec � �1
cos �32. cot � �
1tan � 33. tan � �
sin �cos �
34. cot � �cos �sin �
35. sin2 � � cos2 � � 1 36. 1 � tan2 � � sec2 � 37. sin�90� � �� � cos � 38. cos�90� � �� � sin �
39. tan�90� � �� � cot � 40. cot�90� � �� � tan � 41. sec�90� � �� � csc � 42. csc�90� � �� � sec �
43.
(a)
(c) cos 30� � sin 60� ��3
2
tan 60� �sin 60�
cos 60�� �3
sin 60� ��3
2, cos 60� �
1
2
(b)
(d) cot 60� �cos 60�
sin 60��
1
�3�
�3
3
sin 30� � cos 60� �1
2
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Section 4.3 Right Triangle Trigonometry 295
44.
(a)
(b)
(c)
(d) cot 30� �1
tan 30��
3�3
�3�3
3� �3
cos 30� �sin 30�
tan 30��
�1�2���3�3� �
3
2�3�
�3
2
cot 60� � tan�90� � 60�� � tan 30� ��3
3
csc 30� �1
sin 30�� 2
sin 30� �1
2, tan 30� �
�3
3
46.
(a)
(b) cot � �1
tan ��
1
2�6�
�6
12
cos � �1
sec ��
1
5
sec � � 5, tan � � 2�6
45.
(a)
(b)
(c)
(d) sec�90º � �� � csc � � 3
tan � �sin �
cos ��
1�3
�2�2 ��3�
�2
4
cos � �1
sec ��
2�2
3
sin � �1
csc ��
1
3
csc � � 3, sec � �3�2
4
47.
(a)
(b)
sin � ��15
4
sin2 � �15
16
sin2 � � �1
4�2
� 1
sin2 � � cos2 � � 1
sec � �1
cos �� 4
cos � �1
4
(c)
(d) sin�90� � �� � cos � �1
4
��15
15 �
1
�15
�1�4
�15�4
cot � �cos �
sin �
(c)
(d) sin � � tan � cos � � �2�6 ��1
5� �2�6
5
cot�90� � �� � tan � � 2�6
48.
(a)
(b)
(c)
(d) csc � � �1 � cot2 � ��1 �125
��26
5
tan�90� � �� � cot � �15
sec2 � � 1 � tan2 � ⇒ cos � � 1
�1 � tan2 ��
1�1 � 25
� 1
�26�
�2626
cot � �1
tan ��
15
�� lies in Quadrant I or III.�tan � � 5
50. csc � tan � �1
sin �
sin �cos �
�1
cos �� sec �49. tan � cot � � tan �� 1
tan �� � 1
51. tan � cos � � � sin �
cos �� cos � � sin � 52. cot � sin � �cos �
sin � sin � � cos �
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296 Chapter 4 Trigonometric Functions296 Chapter 4 Trigonometric Functions
53.
� sin2 �
� �sin2 � � cos2 �� � cos2 �
�1 � cos ���1 � cos �� � 1 � cos2 �
55.
� csc � sec � �1
sin �
1
cos �
�1
sin � cos �
sin �
cos ��
cos �
sin ��
sin2 � � cos2 �
sin � cos �
54.
� 1
�csc � � cot ���csc � � cot �� � csc2 � � cot2 �
56.
� 1 � cot2 � � csc2 �
� 1 �cot �
�1�cot ��
tan � � cot �tan �
�tan �tan �
�cot �tan �
57. (a)
(b) cos 87� � 0.0523
sin 41� � 0.6561
59. (a)
(b) csc 48º 7� �1
sin�48 �760�º � 1.3432
sec 42� 12� � sec 42.2� �1
cos 42.2�� 1.3499
58. (a)
(b) cot 71.5� �1
tan 71.5�� 0.3346
tan 18.5� � 0.3346
60. (a)
(b) sec�8� 50� 25� � �1
cos�8� 50� 25� � � 1.0120
� cos�8.840278� � 0.9881
cos�8� 50� 25� � � cos�8 �5060
�25
3600�
61. Make sure that your calculator is in radian mode.
(a)
(b) tan
8� 0.4142
cot
16�
1
tan��16�� 5.0273
63. (a)
(b) csc � � 2 ⇒ � � 30� �
6
sin � �1
2 ⇒ � � 30� �
6
62. (a)
(b) cos�1.25� � 0.3153
sec�1.54� �1
cos�1.54� � 32.4765
64. (a)
(b) tan � � 1 ⇒ � � 45� �
4
cos � ��2
2 ⇒ � � 45� �
4
65. (a)
(b) cot � � 1 ⇒ � � 45� �
4
sec � � 2 ⇒ � � 60� �
3
67. (a)
(b) sin � ��2
2 ⇒ � � 45� �
4
csc � �2�3
3 ⇒ � � 60� �
3
66. (a)
(b) cos � �1
2 ⇒ � � 60� �
3
tan � � �3 ⇒ � � 60� �
3
68. (a)
(b)
cos � �1�2
��2
2 ⇒ � � 45� �
4
sec � � �2
tan � �3�3
� �3 ⇒ � � 60� �
3
cot � ��3
3
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Section 4.3 Right Triangle Trigonometry 297
69.
cos 30� �105
r ⇒ r �
105cos 30�
�105�3�2
�210�3
� 70�3
tan 30� �y
105 ⇒ y � 105 tan 30� � 105
�33
� 35�3
70.
sin 30� �y
15 ⇒ y � 15 sin 30� � 15�1
2� �152
cos 30� �x
15 ⇒ x � 15 cos 30� � 15
�32
�15�3
2
71.
sin 60� �y
16 ⇒ y � 16 sin 60� � 16��3
2 � � 8�3
cos 60� �x
16 ⇒ x � 16 cos 60� � 16�1
2� � 8
72.
sin 60� �38r
⇒ r �38
sin 60��
38�3�2
�76�3
3
cot 60� �x
38 ⇒ x � 38 cot 60� � 38
1�3
�38�3
3
73.
r2 � 202 � 202 ⇒ r � 20�2
tan 45� �20x
⇒ 1 �20x
⇒ x � 20 74.
r2 � 102 � 102 ⇒ r � 10�2
tan 45� �y
10 ⇒ 1 �
y10
⇒ y � 10
75.
r2 � �2�5�2 � �2�5�2 � 20 � 20 � 40 ⇒ r � 2�10
tan 45� �2�5
x ⇒ 1 �
2�5x
⇒ x � 2�5
76.
r2 � �4�6�2 � �4�6�2 � 96 � 96 � 192 ⇒ r � 8�3
tan 45� �y
4�6 ⇒ 1 �
y
4�6 ⇒ y � 4�6
77. (a)
(b) and Thus,
(c) feeth �6�21�
5� 25.2
65
�h21
.tan � �h21
tan � �65
6
h
516
θ
78. (a)
(b)
(c) x � 30 sin 75� � 28.98 meters
sin 75� �x
30
x
75°
30
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298 Chapter 4 Trigonometric Functions298 Chapter 4 Trigonometric Functions
80.
Subtracting,
� 1.295 � 1.3 miles. �13
16.3499 � 6.3138
h �13
cot 3.5� � cot 9�
13h
� cot 3.5� � cot 9�
cot 3.5� �13 � c
h
not drawn to scale
9°3.5°13 c
h
cot 9� �ch
79.
feetw � 100 tan 58� � 160
tan 58� �w
100
tan � �oppadj
81. (a)
(b)
(c) rate down the zip line
vertical rate506
�253
ft�sec
50�26
�25�2
3 ft�sec
L2 � 502 � 502 � 2 502 ⇒ L � 50�2 feet
tan � �5050
� 1 ⇒ � � 45� 82. (a)
(b) feet above sea level
(c) minutes to reach top
Vertical rate
feet per minute � 173.8
�519.3
�896.5�300�
896.5300
1693.5 � 519.3 � 1174.2
x � 896.5 sin 35.4� � 519.3 feet
sin 35.4� �x
896.5
83.
�x1, y1� � �28�3, 28�
x1 � cos 30��56� ��3
2�56� � 28�3
cos 30� �x1
56
y1 � �sin 30���56� � �1
2��56� � 28
sin 30� �y1
56
30°
56
( , )x y1 1
�x2, y2� � �28, 28�3 �
x2 � �cos 60���56� � �1
2��56� � 28
cos 60� �x2
56
y2 � sin 60��56� � ��3
2 ��56� � 28�3
sin 60º �y2
56
60°
56
( , )x y2 2
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Section 4.3 Right Triangle Trigonometry 299
84.
csc 20� �10
y� 2.92sec 20� �
10
x� 1.06
cot 20� �x
y� 2.75tan 20� �
y
x� 0.36
cos 20� �x
10� 0.94sin 20� �
y
10� 0.34
x � 9.397, y � 3.420
86. False
� �2 � 1 sin 45� � cos 45� ��22
��22
87. True
for all �1 � cot2 � � csc2 �
85. True
sin 60� csc 60� � sin 60� 1
sin 60�� 1
88. No. so you can find ± sec �.tan2 � � 1 � sec2 �,
(b) Sine and tangent are increasing, cosine isdecreasing.
(c) In each case, tan � �sin �
cos �.
89. (a)
0 0.3420 0.6428 0.8660 0.9848
1 0.9397 0.7660 0.5000 0.1736
0 0.3640 0.8391 1.7321 5.6713tan �
cos �
sin �
80�60�40�20�0��
90.
1 0.9397 0.7660 0.5000 0.1736
1 0.9397 0.7660 0.5000 0.1736sin�90� � ��
cos �
80�60�40�20�0��are complementary angles.� and 90� � �
cos � � sin�90� � ��
91.
−6 6
−1
7 93.
−6 6
−1
7
95.
−2 10
−4
4 97.
−2 10
−4
4
92.−3 3
−3
1 94.−6 6
−6
2
96.
−2 10
−4
4 98.
−2 10
−4
4
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Section 4.4 Trigonometric Functions of Any Angle
300 Chapter 4 Trigonometric Functions
■ Know the Definitions of Trigonometric Functions of Any Angle.
If is in standard position, a point on the terminal side and then:
■ You should know the signs of the trigonometric functions in each quadrant.
■ You should know the trigonometric function values of the quadrant angles
■ You should be able to find reference angles.
■ You should be able to evaluate trigonometric functions of any angle. (Use reference angles.)
■ You should know that the period of sine and cosine is
■ You should know which trigonometric functions are odd and even.
Even: cos x and sec x
Odd: sin x, tan x, cot x, csc x
2�.
0, �
2, �, and
3�
2.
cot � �x
y, y � 0tan � �
y
x, x � 0
sec � �r
x, x � 0cos � �
x
r
csc � �r
y, y � 0sin � �
y
r
r � �x2 � y2 � 0,�x, y��
Vocabulary Check
1. 2. 3. 4.
5. 6. 7. referencecot �cos �
rx
yx
csc �yr
1. (a)
cot � �x
y�
4
3tan � �
y
x�
3
4
sec � �r
x�
5
4cos � �
x
r�
4
5
csc � �r
y�
5
3sin � �
y
r�
3
5
r � �16 � 9 � 5
�x, y� � �4, 3� (b)
cot � �x
y�
8
15tan � �
y
x�
15
8
sec � �r
x� �
17
8cos � �
x
r� �
8
17
csc � �r
y� �
17
15sin � �
y
r� �
15
17
r � �64 � 225 � 17
�x, y� � ��8, �15�
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Section 4.4 Trigonometric Functions of Any Angle 301
2. (a)
cot � �x
y�
12
�5� �
12
5
sec � �r
x�
13
12
csc � �r
y�
13
�5� �
13
5
tan � �y
x�
�5
12� �
5
12
cos � �x
r�
12
13
sin � �y
r�
�5
13� �
5
13
r � �122 � ��5�2 � 13
x � 12, y � �5 (b)
cot � �x
y�
�1
1� �1
sec � �r
x�
�2
�1� ��2
csc � �r
y�
�2
1� �2
tan � �y
x�
1
�1� �1
cos � �x
r�
�1�2
� ��2
2
sin � �y
r�
1�2
��2
2
r � ���1�2 � 12 � �2
x � �1, y � 1
4. (a)
cot � �x
y�
3
1� 3
sec � �r
x�
�10
3
csc � �r
y�
�10
1� �10
tan � �y
x�
1
3
cos � �x
r�
3�10
�3�10
10
sin � �y
r�
1�10
��10
10
r � �32 � 12 � �10
x � 3, y � 1 (b)
cot � �x
y�
2
�4� �
1
2
sec � �r
x�
2�5
2� �5
csc � �r
y�
2�5
�4� �
�5
2
tan � �y
x�
�4
2� �2
cos � �x
r�
2
2�5�
�5
5
sin � �y
r�
�4
2�5� �
2�5
5
r � �22 � ��4�2 � 2�5
x � 2, y � �4
3. (a)
cot � �x
y� �3tan � �
y
x�
�3
3
sec � �r
x�
�2�3
3cos � �
x
r�
��3
2
csc � �r
y� �2sin � �
y
r� �
1
2
r � �3 � 1 � 2
�x, y� � ���3, �1� (b)
cot � �x
y� �1tan � �
y
x� �1
sec � �r
x� ��2cos � �
x
r� �
�2
2
csc � �r
y� �2sin � �
y
r�
�2
2
r � �4 � 4 � 2�2
�x, y� � ��2, 2�
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.
302 Chapter 4 Trigonometric Functions
5.
cot � �x
y�
7
24tan � �
y
x�
24
7
sec � �r
x�
25
7cos � �
x
r�
7
25
csc � �r
y�
25
24sin � �
y
r�
24
25
r � �49 � 576 � 25
�x, y� � �7, 24�
7.
cot � �xy
� �512
sec � �rx
�135
csc � �ry
� �1312
tan � �yx
� �125
cos � �xr
�513
sin � �yr
� �1213
r � �52 � ��12�2 � �25 � 144 � �169 � 13
�x, y� � �5, �12�
6.
sin cos
tan csc
sec cot � �x
y�
8
15� �
r
x�
17
8
� �r
y�
17
15� �
y
x�
15
8
� �x
r�
8
17� �
y
r�
15
17
r � �82 � 152 � 17
x � 8, y � 15,
8.
sin
cos
tan
csc
sec
cot � �x
y� �
12
5
� �r
x� �
13
12
� �r
y�
13
5
� �y
x�
10
�24� �
5
12
� �x
r�
�24
26�
�12
13
� �y
r�
10
26�
5
13
r � ���24�2 � �10�2 � 26
x � �24, y � 10
9.
cot � �x
y� �
2
5
sec � �r
x� �
�29
2
csc � �r
y�
�29
5
tan � �y
x� �
5
2
cos � �x
r� �
2�29
29
sin � �y
r�
5�29
29
r � �16 � 100 � 2�29
�x, y� � ��4, 10� 10.
sin
cos
tan
csc
sec
cot � �x
y�
5
6
� �r
x� �
�61
5
� �r
y� �
�61
6
� �y
x�
�6
�5�
6
5
� �x
r�
�5�61
��5�61
61
� �y
r�
�6�61
��6�61
61
r � ���5�2� � ��6�2 � �61
x � �5, y � �6
©H
ough
ton
Miff
lin C
ompa
ny. A
ll rig
hts
rese
rved
.
Section 4.4 Trigonometric Functions of Any Angle 303
11.
cot � �xy
� �54
sec � �rx
� ���41
5
csc � �ry
��41
4
tan � �yx
�8
�10� �
45
cos � �xr
��10
2�41� �
5�4141
sin � �y
r�
8
2�41�
4�41
41
r � ���10�2 � 82 � �164 � 2�41
�x, y� � ��10, 8� 12.
sin
cos
tan
csc
sec
cot � �x
y� �
1
3
� �r
x� �10
� �r
y� �
�10
3
� �y
x�
�9
3� �3
� �x
r�
3
3�10�
�10
10
� �y
r�
�9
3�10�
�3�10
10
r � �32 � ��9�2 � �90 � 3�10
x � 3, y � �9
13. lies in Quadrant III or Quadrant IV.
lies in Quadrant II or Quadrant III.
and lies in Quadrant III.cos � < 0 ⇒ �sin � < 0
cos � < 0 ⇒ �
sin � < 0 ⇒ �
15. lies in Quadrant I or Quadrant III.
lies in Quadrant I or Quadrant IV.
and lies in Quadrant I.cos � > 0 ⇒ �cot � > 0
cos � > 0 ⇒ �
cot � > 0 ⇒ �
17.
in Quadrant II
cot � �x
y� �
4
3tan � �
y
x� �
3
4
sec � �r
x� �
5
4cos � �
x
r� �
4
5
csc � �r
y�
5
3sin � �
y
r�
3
5
⇒ x � �4�
sin � �y
r�
3
5 ⇒ x2 � 25 � 9 � 16
14. and
and
Quadrant IV
x
y< 0
r
x> 0
cot � < 0sec � > 0
16. and
and
Quadrant III
ry
< 0yx
> 0
csc � < 0tan � > 0
18.
in Quadrant III
sin csc
cos sec
tan cot � �4
3� �
y
x�
3
4
� � �5
4� �
x
r� �
4
5
� � �5
3� �
y
r� �
3
5
⇒ y � �3�
cos � �x
r�
�4
5 ⇒ �y� � 3
©H
ough
ton
Miff
lin C
ompa
ny. A
ll rig
hts
rese
rved
.
304 Chapter 4 Trigonometric Functions
19.
cot � �x
y� �
8
15
sec � �r
x�
17
8cos � �
x
r�
8
17
csc � �r
y� �
17
15sin � �
y
r� �
15
17
tan � �y
x�
�15
8 ⇒ r � 17
sin � < 0 ⇒ y < 0 20. csc
cot
sin csc
cos sec
tan cot � � ��15� �y
x� �
�15
15
� � �4�15
15� �
x
r� �
�15
4
� � 4� �y
r�
1
4
� < 0 ⇒ x � ��15
� �r
y�
4
1 ⇒ x � ±�15
21.
cot � �x
y� �
�3
3tan � �
y
x� ��3
sec � �r
x� �2cos � �
x
r� �
1
2
csc � �r
y�
2�3
3sin � �
y
r�
�3
2
sin � ≥ 0 ⇒ y � �3
sec � �r
x�
2
�1 ⇒ y2 � 4 � 1 � 3
23. is undefined
is undefined.
is undefined.cot �tan � �yx
�0x
� 0
sec � �rx
� �1cos � �xr
��rr
� �1
csc � �ry
sin � �yr
� 0
�
2≤ � ≤
3�
2⇒ � � �, y � 0, x � �r
⇒ � � n�.cot �
22. and
is undefined.
is undefined.cot �
sec � � �1
csc �
tan � �sin �cos �
� 0
cos � � cos � � �1
�
2≤ � ≤
3�
2 ⇒ � � �sin � � 0
24. is undefined and
is undefined.
is undefined.
cot � � 0
sec �
csc � � �1
tan �
cos � � 0
sin � � �1
� ≤ � ≤ 2� ⇒ � �3�
2.tan �
25. To find a point on the terminal side of use any point on the line that lies inQuadrant II. is one such point.
cot � � �1tan � � �1
sec � � ��2cos � � �1
�2� �
�2
2
csc � � �2sin � �1
�2�
�2
2
x � �1, y � 1, r � �2
��1, 1�y � �x�,
©H
ough
ton
Miff
lin C
ompa
ny. A
ll rig
hts
rese
rved
.
Section 4.4 Trigonometric Functions of Any Angle 305
27. To find a point on the terminal side of use anypoint on the line that lies in Quadrant III.
is one such point.
cot � ��1
�2�
1
2
sec � ��5
�1� ��5
csc � ��5
�2� �
�5
2
tan � ��2
�1� 2
cos � � �1
�5� �
�5
5
sin � � �2
�5� �
2�5
5
x � �1, y � �2, r � �5
��1, �2�y � 2x
�,26. Quadrant III,
sin
cos
tan
csc
sec
cot � �x
y�
�x
��1�3� x� 3
� �r
x�
��10x��3
�x� �
�10
3
� �r
y�
��10x��3
��1�3�x� ��10
� �y
x�
��1�3�x
�x�
1
3
� �x
r�
�x
��10x��3� �
3�10
10
� �y
r�
��x�3���10x��3
� ��10
10
r ��x2 �1
9x2 �
�10x
3
x > 0��x, �1
3x�
28.
Quadrant IV,
sin csc
cos sec
tan cot � � �3
4� �
y
x�
��4�3� x
x� �
4
3
� �5
3� �
x
r�
x
�5�3�x�
3
5
� � �5
4� �
y
r�
��4�3�x
�5�3� x� �
4
5
r ��x2 �16
9x2 �
5
3x
x > 0�x, �4
3x�
4x � 3y � 0 ⇒ y � �4
3x
29.
sec � �r
x�
1
�1� �1
�x, y� � ��1, 0� 31.
�0
�1� 0 cot�3�
2 � �xy
�x, y� � �0, �1�
33.
sec 0 �rx
�11
� 1
�x, y� � �1, 0�
35.
undefined cot � �xy
� �10
⇒
�x, y� � ��1, 0�
30. undefined
since corresponds to �0, 1�.�
2
tan �
2�
y
x�
1
0 ⇒
32. undefinedcsc 0 �r
y�
1
0 ⇒ 34. csc
3�
2�
1
sin 3�
2
�1
�1� �1
36. csc �
2�
1
sin �
2
�11
� 1
©H
ough
ton
Miff
lin C
ompa
ny. A
ll rig
hts
rese
rved
.
306 Chapter 4 Trigonometric Functions
37.
�� � 180� � 120� � 60�
x
y
120�=θ60�=θ ′
� � 120�
39. is coterminal with
x
y
−235�=θ
55�=θ ′
�� � 225� � 180� � 45�
225�.� � �135�
38.
x
y
225�=θ
45�=θ ′
�� � 225� � 180� � 45�
40.
x
y
−330�=θ 30�=θ ′
�� � �330� � 360� � 30�
41.
x
y
=
θ =
π3
π3
5
θ ′
� �5�
3, �� � 2� �
5�
3�
�
3
43. is coterminal with
x
y
=θ =
π6 π
65−
θ ′
�� �7�
6� � �
�
6
7�
6.� � �
5�
6
42.
x
y
= π4
θ = π4
3θ ′
�� � � �3�
4�
�
4
44.
x
y
= π3 θ = π
32−
θ ′
�� ��2�
3� � �
�
3
45.
�� � 208� � 180� � 28�= 208°
x
y
θ
θ
′ = 28°
� � 208� 46.
�� � 360� � 322� � 38�
x
y
= 322°
θ
θ
′ = 38°
� � 322�
©H
ough
ton
Miff
lin C
ompa
ny. A
ll rig
hts
rese
rved
.
Section 4.4 Trigonometric Functions of Any Angle 307
47.
�� � 360� � 292� � 68�
x
y
= −292° θθ ′ = 68°
� � �292� 48. lies inQuadrant III.
Reference angle:180� � 165� � 15�
x
y
−165�=
θ ′ 15�=
θ
� � �165�
49. is coterminal with
x
y
π5
11
π5
=
θ
θ
′ =
�� ��
5
�
5.� �
11�
5
51. lies in Quadrant III.
Reference angle:
x
y
−1.8=θ ′ 1.342=
θ
� � 1.8 � 1.342
� � �1.8
50.
x
y
π7
17
π7
3θ
θ
′ =
=
�� �17�
7�
14�
7�
3�
7
52. lies in Quadrant IV.
Reference angle:
x
y
θ ′= 1.358
θ = 4.5
4.5 � � � 1.358
�
53. Quadrant III
tan 225� � tan 45� � 1
cos 225� � �cos 45� � ��2
2
sin 225� � �sin 45� � ��2
2
�� � 45�,
55. is coterminal with Quadrant IV.
tan��750�� � �tan 30� � ��33
cos��750�� � cos 30� ��32
sin��750�� � �sin 30� � �12
�� � 360� � 330� � 30�
330�,� � �750�
54. Quadrant IV
sin
cos
tan 300� � �tan 60� � ��3
300� � cos 60� �1
2
300� � �sin 60� � ��3
2
� � 300�, �� � 360� � 300� � 60�,
56. Quadrant III
tan��495�� � tan 45� � 1
cos��495�� � �cos 45� � ��22
sin��495�� � �sin 45� � ��22
� � �495�, �� � 45�,
©H
ough
ton
Miff
lin C
ompa
ny. A
ll rig
hts
rese
rved
.
308 Chapter 4 Trigonometric Functions
57. Quadrant IV
tan�5�
3 � � �tan��
3� � ��3
cos�5�
3 � � cos��
3� �12
sin�5�
3 � � �sin��
3� � ��32
�� � 2� �5�
3�
�
3
� �5�
3, 58.
sin
cos
tan 3�
4� �1
3�
4� �
�2
2
3�
4�
�2
2
� �3�
4, �� � � �
3�
4�
�
4
59. Quadrant IV
tan���
6� � �tan �
6� �
�3
3
cos���
6� � cos �
6�
�3
2
sin���
6� � �sin �
6� �
1
2
�� ��
6,
61. Quadrant II
tan 11�
4� �tan
�
4� �1
cos 11�
4� �cos
�
4� �
�2
2
sin 11�
4� sin
�
4�
�2
2
�� ��
4,
60. ,
sin
cos
tan��4�
3 � � ��3
��4�
3 � ��1
2
��4�
3 � ��3
2
�� ��
3� �
�4�
3
62. is coterminal with
in Quadrant III.
sin
cos
tan 10�
3� tan
�
3� �3
10�
3� �cos
�
3� �
1
2
10�
3� �sin
�
3� �
�3
2
�� �4�
3� � �
�
3
4�
3.� �
10�
3
63. is coterminal with
Quadrant III
tan��17�
6 � � tan��
6� ��33
cos��17�
6 � � �cos��
6� � ��32
sin��17�
6 � � �sin��
6� � �12
�� �7�
6� � �
�
6,
7�
6.� � �
17�
664.
sin
cos
tan��20�
3 � � �3
��20�
3 � ��1
2
��20�
3 � ���3
2
� ��20�
3, �� �
�
3©
Hou
ghto
n M
ifflin
Com
pany
. All
right
s re
serv
ed.
Section 4.4 Trigonometric Functions of Any Angle 309
65.
in Quadrant IV.
cos � �4
5
cos � > 0
cos2 � �16
25
cos2 � � 1 �9
25
cos2 � � 1 � ��3
5�2
cos2 � � 1 � sin2 �
sin2 � � cos2 � � 1
sin � � �3
566.
sin � �1
csc ��
1�10
��10
10
csc � �1
sin �
�10 � csc �
csc � > 0 in Quadrant II.
10 � csc2 �
1 � ��3�2 � csc2 �
1 � cot2 � � csc2 �
cot � � �3
67.
cot � � ��3
cot � < 0 in Quadrant IV.
cot2 � � 3
cot2 � � ��2�2 � 1
cot2 � � csc2 � � 1
1 � cot2 � � csc2 �
csc � � �2
69.
tan � ��65
4
tan � > 0 in Quadrant III.
tan2 � �65
16
tan2 � � ��9
4�2
� 1
tan2 � � sec2 � � 1
1 � tan2 � � sec2 �
sec � � �9
4
68.
sec � �1
5�8�
8
5
cos � �1
sec � ⇒ sec � �
1
cos �
cos � �5
8
70.
Quadrant IV ⇒ csc � � ��41
5
csc � � ±�41
5
1 �1625
� csc2 �
1 � cot2 � � csc2 �
tan � � �54
⇒ cot � � �45
©H
ough
ton
Miff
lin C
ompa
ny. A
ll rig
hts
rese
rved
.
310 Chapter 4 Trigonometric Functions
71. is in Quadrant II.
cot � �1
tan ��
��212
sec � �1
cos ��
�5�21
��5�21
21
csc � �1
sin ��
52
tan � �sin �cs �
�2�5
��21�5�
�2�21
��2�21
21
cos � � ��1 � sin2 � � ��1 �425
� ��21
5
sin � �25
and cos � < 0 ⇒ �
72. is in Quadrant III.
sec � �1
cos �� �
73
csc � �1
sin ��
�7
2�10�
�7�1020
cot � �1
tan ��
3
2�10�
3�1020
tan � �sin �cs �
��2�10�7
�3�7�
2�103
sin � � ��1 � cos2 � � ��1 �949
���40
7�
�2�107
cos � � �37
and sin � < 0 ⇒ �
73. is in Quadrant II.
cot � �1
tan �� �
14
csc � �1
sin ��
174�17
��17
4
sin � � tan � cos � � ��4����1717 � �
4�1717
cos � �1
sec ��
�1�17
���17
17
sec � � ��1 � tan2 � � ��1 � 16 � ��17
tan � � �4 and cos � < 0 ⇒ � 74. is in Quadrant II.
csc � �1
sin ��
26�26
� �26
sin � � tan � cos � � ��15���5�26
26 � ��2626
cos � �1
sec ��
�5�26
��5�26
26
sec � � ��1 � tan2 � � ��1 �125
���26
5
tan � �1
cot ��
�15
cot � � �5 and sin � > 0 ⇒ �
©H
ough
ton
Miff
lin C
ompa
ny. A
ll rig
hts
rese
rved
.
Section 4.4 Trigonometric Functions of Any Angle 311
78. sec 235� �1
cos 235�� �1.7434
75. is in Quadrant IV.
cot � �1
tan ��
��52
tan � �sin �cos �
��2�3�5�3
��2�5
��2�5
5
sec � �1
cos ��
3�5
�3�5
5
cos � � �1 � sin2 � ��1 �49
��53
sin � �1
csc �� �
23
csc � � �32
and tan � < 0 ⇒ � 76. is in Quadrant III.
cot � �1
tan ��
3�7
�3�7
7
tan � �sin �cos �
���7�4�3�4
��73
csc � �1
sin �� �
4�7
��4�7
7
sin � � ��1 � cos2 � � ��1 �916
� ��74
cos � �1
sec �� �
34
sec � � �43
and cot � > 0 ⇒ �
77. sin 10� � 0.1736 79. tan 245� � 2.1445
81. cos��110�� � �0.342080. csc 320� �1
sin 320�� �1.5557 82.
� �1.1918
cot ��220�� �1
tan��220��
84. csc 0.33 �1
sin 0.33� 3.0860
86. tan 11�
9� 0.8391 88. cos��15�
14 � � �0.9749
83.
� 5.7588
sec��280�� �1
cos��280�� 85. tan�2�
9 � � 0.8391
87.
� �2.9238
csc��8�
9 � �1
sin��8�
9 �
89. (a) reference angle is or
and is in Quadrant I or Quadrant II.
Values in degrees:
Values in radian:
(b) reference angle is or
and is in Quadrant III or Quadrant IV.
Values in degrees:
Values in radians:7�
6,
11�
6
210�, 330�
��
6
30�sin � � �1
2 ⇒
�
6,
5�
6
30�, 150�
��
6
30�sin � �1
2 ⇒ 90. (a) cos reference angle is 45º or
and is in Quadrant I or IV.
Values in degrees:
Values in radians:
(b) cos reference angle is or
and is in Quadrant II or III.
Values in degrees:
Values in radians:3�
4,
5�
4
135�, 225�
��
4
45�� � ��2
2 ⇒
�
4,
7�
4
45�, 315�
��
4
� ��2
2 ⇒
©H
ough
ton
Miff
lin C
ompa
ny. A
ll rig
hts
rese
rved
.
312 Chapter 4 Trigonometric Functions
92. (a) csc
Reference angle is
Values in degrees:
Values in radians:
(b) csc
Reference angle is
Values in degrees:
Values in radians:�
6,
5�
6
30�, 150�
�
6 or 30�.
sin � �12
� � 2 ⇒
5�
4,
7�
4
225�, 315�
45� or �
4.
sin � ��1�2
� � ��2 ⇒ 91. (a) reference angle is or
and is in Quadrant I or Quadrant II.
Values in degrees:
Values in radians:
(b) reference angle is or
and is in Quadrant II or Quadrant IV.
Values in degrees:
Values in radians:3�
4,
7�
4
135�, 315�
��
4
45�cot � � �1 ⇒
�
3,
2�
3
60�, 120�
��
3
60�csc � �2�3
3 ⇒
93. (a) reference angle is or
and is in Quadrant II or Quadrant III.
Values in degrees:
Values in radians:
(b) reference angle is or
and is in Quadrant II or Quadrant III.
Values in degrees:
Values in radians:2�
3,
4�
3
120�, 240�
�
60�,�
3cos � � �
12
⇒
5�
6,
7�
6
150�, 210�
�30�,
�
6sec � � �
2�33
⇒ 94. (a)
Reference angle is
Values in degrees:
Values in radians:
(b) Values in degrees:
Values in radians:�
4 or
7�
4
45� or 315�
5�
6,
11�
6
150�, 330�
�
6 or 30�.
cot � � ��3 ⇒ cos �sin �
� ��3
95. (a)
(b)
(c) cos 30�2 � ��32 �2
�34
cos 30� � sin 30� ��3 � 1
2
f��� � g��� � sin 30� � cos 30� �12
��32
�1 � �3
2
96. (a)
(b)
(c) cos 60�2 � �12�
2
�14
cos 60� � sin 60� �12
��32
�1 � �3
2
f��� � g��� � sin 60� � cos 60� ��32
�12
��3 � 1
2
(d)
(e)
(f) cos��30�� � cos 30� ��32
2 sin 30� � 2�12� � 1
sin 30� cos 30� � �12���3
2 � ��34
(d)
(e)
(f) cos��60�� � cos 60� �12
2 sin 60� � 2�32
� �3
sin 60� cos 60� � ��32
��12� �
�34
©H
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.
Section 4.4 Trigonometric Functions of Any Angle 313
97. (a)
(b)
(c) cos 315�2 � ��22 �2
�12
cos 315� � sin 315� ��22
� ���22 � � �2
f��� � g��� � sin 315� � cos 315� � ��22
��22
� 0
98. (a)
(b)
(c)
(d) sin 225� cos 225� � ���22 ����2
2 � �12
cos 225�2 � ���22 �2
�12
cos 225� � sin 225� ���2
2� ���2
2 � � 0
f��� � g��� � sin 225� � cos 225� � ��22
��22
� ��2
(d)
(e)
(f) cos��315�� � cos�315�� ��22
2 sin 315� � 2���22 � � ��2
sin 315� cos 315� � ���22 ���2
2 � � �12
(e)
(f) cos��225�� � cos�225�� � ��22
2 sin 225� � 2���22 � � ��2
99. (a)
(b)
(c)
(d) sin 150� cos 150� �12
��3
2�
��34
cos 150�2 � ���32 �2
�34
cos 150� � sin 150� ���3
2�
12
��1 � �3
2
f��� � g��� � sin 150� � cos 150� �12
���3
2�
1 � �32
(e)
(f ) cos��150�� � cos�150�� ���3
2
2 sin 150� � 2�12� � 1
100. (a)
(b)
(c)
(d) sin 300� cos 300� � ���32 ��1
2� ���3
4
cos 300�2 � �12�
2
�14
cos 300� � sin 300� �12
� ���32 � �
1 � �32
f��� � g��� � sin 300� � cos 300� ���3
2�
12
�1 � �3
2
(e)
(f) cos��300�� � cos�300�� �12
2 sin 300� � 2���32 � � ��3
101. (a)
(b)
(c)
(d) sin 7�
6 cos
7�
6� ��
12���
�32 � �
�34
�cos 7�
6 �2� ���3
2 �2
�34
cos 7�
6� sin
7�
6�
��32
� ��12� �
1 � �32
f��� � g��� � sin 7�
6� cos
7�
6� �
12
��32
��1 � �3
2
(e)
(f) cos��7�
6 � � cos�7�
6 � ���3
2
2 sin 7�
6� 2��
12� � �1
©H
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314 Chapter 4 Trigonometric Functions
102. (a)
(b)
(c)
(d) sin 5�
6 cos
5�
6� �1
2����32 � �
��34
�cos 5�
6 �2� ���3
2 �2
�34
cos 5�
6� sin
5�
6�
��32
�12
��1 � �3
2
f��� � g��� � sin 5�
6� cos
5�
6�
12
���3
2�
1 � �32
(e)
(f) cos��5�
6 � � cos�5�
6 � ���3
2
2 sin 5�
6� 2�1
2� � 1
103. (a)
(b)
(c)
(d) sin 4�
3 cos
4�
3� ���3
2 ���12� �
�34
�cos 4�
3 �2� ��
12�
2
�14
cos 4�
3� sin
4�
3� �
12
� ���32 � �
�3 � 12
f��� � g��� � sin 4�
3� cos
4�
3�
��32
�12
��1 � �3
2
(e)
(f) cos��4�
3 � � cos�4�
3 � � �12
2 sin 4�
3� 2���3
2 � � ��3
104. (a)
(b)
(c)
(d) sin 5�
3 cos
5�
3� ���3
2 ��12� �
��34
�cos 5�
3 �2� �1
2�2
�14
cos 5�
3� sin
5�
3�
12
� ���32 � �
1 � �32
f��� � g��� � sin 5�
3� cos
5�
3�
��32
�12
�1 � �3
2
(e)
(f) cos��5�
3 � � cos�5�
3 � �12
2 sin 5�
3� 2���3
2 � � ��3
105. (a)
(b)
(c) cos 270�2 � 02 � 0
cos 270� � sin 270� � 0 � ��1� � 1
f��� � g��� � sin 270� � cos 270� � �1 � 0 � �1 (d)
(e)
(f) cos��270�� � cos�270�� � 0
2 sin 270� � 2��1� � �2
sin 270� cos 270� � ��1��0� � 0
106. (a)
(b)
(c) cos 180�2 � ��1�2 � 1
cos 180� � sin 180� � �1 � 0 � �1
f��� � g��� � sin 180� � cos 180� � 0 � 1 � �1 (d)
(e)
(f) cos��180�� � cos�180�� � �1
2 sin 180� � 2�0� � 0
sin 180� cos 180� � 0��1� � 0
107. (a)
(b)
(c) �cos 7�
2 �2� 02 � 0
cos 7�
2� sin
7�
2� 0 � ��1� � 1
f��� � g��� � sin 7�
2� cos
7�
2� �1 � 0 � �1 (d)
(e)
(f) cos��7�
2 � � cos�7�
2 � � 0
2 sin 7�
2� 2��1� � �2
sin 7�
2 cos
7�
2� ��1��0� � 0
©H
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Section 4.4 Trigonometric Functions of Any Angle 315
108. (a)
(b)
(c) �cos 5�
2 �2� 02 � 0
cos 5�
2� sin
5�
2� 0 � 1 � �1
f��� � g��� � sin 5�
2� cos
5�
2� 1 � 0 � 1 (d)
(e)
(f) cos��5�
2 � � cos�5�
2 � � 0
2 sin 5�
2� 2�1� � 2
sin 5�
2 cos
5�
2 � �1��0� � 0
109.
(a) January:
(b) July:
(c) December: t � 12 ⇒ T � 31.75�
t � 7 ⇒ T � 70�
t � 1 ⇒ T � 49.5 � 20.5 cos�� �1�6
�7�
6 � � 29�
T � 49.5 � 20.5 cos�� t6
�7�
6 �
110.
(a) thousand
(b) thousand
(c) thousand
(d) thousand
Answers will vary.
S�6� � 25.8
S�5� � 27.5
S�14� � 33.0
S�1� � 23.1 � 0.442�1� � 4.3 sin��
6� � 25.7
S � 23.1 � 0.442t � 4.3 sin��t6 �, t � 1 ↔ Jan. 2004
111. sin
(a)
miles
(b)
miles
(c)
milesd �6
sin 120�� 6.9
� � 120�
d �6
sin 90��
6
1� 6
� � 90�
d �6
sin 30��
6
�1�2�� 12
� � 30�
� �6
d ⇒ d �
6
sin �
113. True. The reference angle for isand sine is positive
in Quadrants I and II.�� � 180� � 151� � 29�,
� � 151�
112. As increases from to x decreases from 12 cm to 0 cm and y increases from 0 cm to 12 cm. Therefore, increases from 0to 1, and decreases from 1 to 0.Thus, begins at 0 and increases without bound. When the tangent is undefined.
� � 90�,tan � � y�x
cos � � x�12sin � � y�12
90�,0��
114. False. and
cot���
4� � �1
�cot�3�
4 � � ���1� � 1
©H
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316 Chapter 4 Trigonometric Functions
115. (a)
(b) It appears that sin � � sin�180� � ��.
0 0.3420 0.6428 0.8660 0.9848
0 0.3420 0.6428 0.8660 0.9848sin�180� � ��
sin �
80�60�40�20�0��
116.
Patterns and conclusions may vary.
Function
Domain All reals except
Range
Evenness No Yes No
Oddness Yes No Yes
Period
Zeros n��
2� n�n�
�2�2�
���, ���1, 1�1, 1
�
2� n����, �����, ��
tan xcos xsin x
Function
Domain All reals except All reals except All reals except
Range
Evenness No Yes No
Oddness Yes No Yes
Period
Zeros None None�
2� n�
�2�2�
���, �����, �1 � 1, �����, �1 � 1, ��
n��
2� n�n�
cot xsec xcsc x
118.
x � �179 � �1.889
9x � �17
44 � 9x � 61
120.
x � 1.186, �1.686
x ��1 ±�1 � 4�4��2�
4�
�1 ±�334
2x2 � x � 4 � 0
117.
x � 7
3x � 21
3x � 7 � 14
119.
x � 3.449, x � �1.449
x �2 ±�4 � 20
2� 1 ±�6
x2 � 2x � 5 � 0
©H
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Section 4.5 Graphs of Sine and Cosine Functions
Section 4.5 Graphs of Sine and Cosine Functions 317
121.
x � �5.908, 4.908
x ��1 ±�1 � 4�29�
2�
�1 ±�1172
x2 � x � 29 � 0
27 � �x � 1��x � 2�
3
x � 1�
x � 29
123.
x � 3 � log4 726 � 3 �ln 726ln 4
� �1.752
3 � x � log4 726
43�x � 726
125.
x � e�6 � 0.002479 � 0.002
ln x � �6
122.
( extraneous)x � 0x � 6
x�x � 6� � 0
x2 � 6x � 0
10x � x2 � 4x
5x
�x � 4
2x
124.
x �12
ln 86 � 2.227
2x � ln 86
86 � e2x
90 � 4 � e2x
4500
4 � e2x � 50
126. ⇒ x � 10 � e2 ⇒ x � e2 � 10 � �2.611ln �x � 10 �12 ln�x � 10� � 1 ⇒ ln�x � 10� � 2
■ You should be able to graph and
■ Amplitude:
■ Period:
■ Shift: Solve and
■ Key increments: (period)1
4
bx � c � 2�.bx � c � 0
2�
�b�
�a�y � a cos�bx � c�.y � a sin�bx � c�
Vocabulary Check
1. amplitude 2. one cycle 3. 4. phase shift2�
b
©H
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318 Chapter 4 Trigonometric Functions
1.
(a) -intercepts:
(b) -intercept:
(c) Increasing on:
Decreasing on:
(d) Relative maxima:
Relative minima: ���
2, �1�, �3�
2, �1�
��3�
2, 1�, ��
2, 1�
��3�
2, �
�
2�, ��
2,
3�
2 �
��2�, �3�
2 �, ���
2,
�
2�, �3�
2, 2��
�0, 0�y
��2�, 0�, ���, 0�, �0, 0�, ��, 0�, �2�, 0�x
f�x� � sin x
2.
(a) -intercepts:
(b) -intercept:
(c) Increasing on:
Decreasing on:
(d) Relative maxima:
Relative minima: ���, �1�, ��, �1�
��2�, 1�, �0, 1�, �2�, 1�
��2�, ���, �0, ��
���, 0�, ��, 2��
�0, 1�y
��3�
2, 0�, ��
�
2, 0�, ��
2, 0�, �3�
2, 0�x
f�x� � cos x
7.
Period:
Amplitude: �23� �2
3
2�
�� 2
y �2
3 sin �x
3.
Period:
Amplitude: �3� � 3
2�
2� �
y � 3 sin 2x
5.
Period:
Amplitude: �52� �5
2
2�
1�2� 4�
y �5
2 cos
x
2
Xmin = -2Xmax = 2Xscl = Ymin = -4Ymax = 4Yscl = 1
��2��
Xmin = -4Xmax = 4Xscl = Ymin = -3Ymax = 3Yscl = 1
���
4.
Period:
Amplitude: �a� � 2
2�
b�
2�
3
y � 2 cos 3x
6.
Period:
Amplitude: �a� � ��3� � 3
2�
b�
2�
�1�3�� 6�
y � �3 sin x
3
8.
Period:
Amplitude: �a� �3
2
2�
b�
2�
���2�� 4
y �3
2 cos
�x
2
Xmin = -Xmax = Xscl = Ymin = -3Ymax = 3Yscl = 1
��4��
Xmin = -Xmax = Xscl = Ymin = -4Ymax = 4Yscl = 1
�6�6�
©H
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.
Section 4.5 Graphs of Sine and Cosine Functions 319
9.
Period:
Amplitude: ��2� � 2
2�
1� 2�
y � �2 sin x 10.
Period:
Amplitude: �a� � ��1� � 1
2�
b�
2�
2�5� 5�
y � �cos 2x5
11.
Period:
Amplitude: �14� �1
4
2�
2�3� 3�
y �1
4 cos
2x
3
13.
Period:
Amplitude: �13� �13
2�
4��
12
y �13
sin 4�x
15.
The graph of is a horizontal shift to the right units of the graph of (a phase shift).f�
g
g�x� � sin�x � ��
f �x� � sin x
17.
The graph of is a reflection in the axis of thegraph of f.
x-g
g�x� � �cos 2x
f �x� � cos 2x
19.
The graph of has five times the amplitude of and reflected in the axis.x-
f,g
g�x� � �5 cos x
f�x� � cos x
21.
The graph of is a vertical shift upward of fiveunits of the graph of f.
g
g�x� � 5 � sin 2x
f �x� � sin 2x
12.
Period:
Amplitude: �a� �5
2
2�
b�
2�
1�4� 8�
y �5
2 cos
x
414.
Amplitude: �a� �34
Period: 2�
b�
2�
��12� 24
y �34
cos � x12
16.
g is a horizontal shift of f units to the left.�
f�x� � cos x, g�x� � cos�x � ��
18.
g is a reflection of f about the y-axis.(or, about the -axis)x
f �x� � sin 3x, g�x� � sin��3x�
20.
The amplitude of g is one-half that of f. g is areflection of f in the x-axis.
f�x� � sin x, g�x� � �12 sin x
22.
g is a vertical shift of f six units downward.
f �x� � cos 4x, g�x� � �6 � cos 4x
24. The period of g is one-half the period of f.23. The graph of has twice the amplitude as the graphof The period is the same.f.
g
25. The graph of is a horizontal shift units to theright of the graph of f.
�g 26. Shift the graph of f two units upward to obtain thegraph of g.
©H
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320 Chapter 4 Trigonometric Functions
27.
Period:
Amplitude: 1
Period:
Amplitude:
−2
−3
−4
1
4
x
y
π2
π23−
f
g
��4� � 4
2�
g�x� � �4 sin x
2�
f �x� � sin x
29.
Period:
Amplitude: 1
is a vertical shift of the graph offour units upward.
f
g
ππ−−1
−2
22
2
3
4
6
x
y
f�x�g�x� � 4 � cos x
2�
f�x� � cos x
28.
–2
2
xπ6
fg
y
g�x� � sin x
3
f�x� � sin x
30.
–2
2
xπ
f
g
y
g�x� � �cos 4x
f�x� � 2 cos 2x
x 0
0 1 0 0
0 1�32
�32
12
sin x3
�1sin x
2�3�
2�
�
2
x 0
2 0 0 2
1 1 �1�1�1�cos 4x
�22 cos 2x
�3�
4�
2�
4
31.
Period:
Amplitude:
is the graph of
shifted vertically three units upward.
f �x�g�x� � 3 �1
2 sin
x
2
1
2
4�
x
y
π3π2 π4ππ4 −−
f
g
−2
−1
−3
−4
1
2
3
4
f�x� � �1
2 sin
x
2
©H
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.
Section 4.5 Graphs of Sine and Cosine Functions 321
33.
Period:
Amplitude: 2
is the graph of shifted units to the left.
–3
3
xπ π2
f
g
y
�f�x�g�x� � 2 cos�x � ��
2�
f�x� � 2 cos x
36. f�x� � sin x, g�x� � �cos�x ��
2�
x 0
0 1 0 0
0 1 0 0�1�cos�x ��
2��1sin x
2�3�
2�
�
2
32.
x
y
f
g−1
−2
−2 2−1 1
2
3
4
1
g�x� � 4 sin �x � 2
f�x� � 4 sin �x
34.
f g
ππ
ππ
−
−
1
−1
22
x
y
g�x� � �cos�x ��
2�f�x� � �cos x
x 0 1 2
0 4 0 0
2 �2�6�2�2g�x�
�4f �x�
32
12
x 0
0 1 0
0 0 1 0�1�cos�x ��
2��1�1�cos x
2�3�
2�
�
2
35.
Period:
Amplitude: 1
−2
−2
2
2
f = g
��
2�
sin x � cos�x ��
2�
f �x� � sin x, g�x� � cos�x ��
2�
Conjecture: sin x � �cos�x ��
2�
−2
−2
2
2
��
f = g
©H
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322 Chapter 4 Trigonometric Functions
38.
Conjecture: cos x � �cos�x � ��
−2
−2
2
2
��
f = g
f�x� � cos x, g�x� � �cos�x � ��37.
Thus,
−2
−2
2
2
f = g
��
f �x� � g�x�.
g�x� � �sin�x ��
2� � sin��
2� x� � cos x
f�x� � cos x
x 0
1 0 0 1
1 0 0 1�1�cos�x � ��
�1cos x
2�3�
2�
�
2
39.
Period:
Amplitude: 3
Key points:
x
y
π2
π2
π23 π
23− −
−4
1
2
3
4
�2�, 0��3�
2, �3�,��, 0�,��
2, 3�,�0, 0�,
2�
y � 3 sin x
41.
Period:
Amplitude: 1
Key points:
x
y
−1
−2
−3
2
3
π2 π4
�2�, �1�, �3�, 0�, �4�, 1��0, 1�, ��, 0�,
4�
y � cos x2
40.
Period:
Amplitude:
x
y
−0.5
−1
0.5
1
π π2
π2
π2−
14
2�
y �14
cos x
42.
Period:
Amplitude:
x
y
π8
π8
−2
1
2
3π83 π
85−
1
2�
4�
�
2
y � sin 4x
©H
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.
Section 4.5 Graphs of Sine and Cosine Functions 323
43.
Period:
Amplitude: 1
Shift: Set and
Key points: ��
4, 0�, �3�
4, 1�, �5�
4, 0�, �7�
4, �1�, �9�
4, 0�
x �9�
4x �
�
4
x ��
4� 2�x �
�
4� 0
2�
–3
–2
1
2
3
xπ−π
yy � sin�x ��
4�; a � 1, b � 1, c ��
4
44.
Horizontal shift units to the right�
x
y
π2
π2
−−1
−2
1
2
π23
y � sin�x � ��
45.
Period:
Amplitude: 8
Key points: ���, �8�, ���
2, 0�, �0, 8�, ��
2, 0�, ��, �8�
2�
x
y
−2
−4
−6
−8
−10
2
8
10
πππ2 π2− −
y � �8 cos�x � ��
46.
Amplitude: 3
Horizontal shift units to the left
Note: 3 cos�x ��
2� � �3 sin x
1
3
4
−2
−3
−4
x
y
π2
π23π
2−
�
2
y � 3 cos�x ��
2� 47.
Amplitude: 2
Period:
6−6
−4
4
2�
2��3� 3
y � �2 sin 2�x
3
©H
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324 Chapter 4 Trigonometric Functions
48.
Amplitude: 10
Period:
18−18
−12
12
2�
��6� 12
y � �10 cos �x6
49.
Amplitude: 5
Period:
30−30
−20
20
2�
��12� 24
y � �4 � 5 cos � t12
50.
Amplitude: 2
Period:
6−6
−2
6
2�
2��3� 3
y � 2 � 2 sin 2�x
3
51.
Amplitude:
Period:
−2
2
4π 5π−
2�
1�2� 4�
23
y �23
cos�x2
��
4� 52.
Amplitude: 3
Period:
−6
6
π2
π2
−
2�
6�
�
3
y � �3 cos�6x � �� 53.
Amplitude: 2
Period:
−
−4
4
��
�
2
y � �2 sin�4x � ��
55.
Amplitude: 1
Period: 1
−3
−1
3
3
y � cos�2�x ��
2� � 154.
Amplitude: 4
Period:
−8
8
12−12
3�
y � �4 sin�2
3x �
�
3� 56.
Amplitude: 3
Period: 4
−7
1
6−6
y � 3 cos��x
2�
�
2� � 3
58.
Amplitude: 5
Period:
−0
12
��
�
y � 5 cos�� � 2x� � 657.
Amplitude: 5
Period:
−
−4
20
��
�
y � 5 sin�� � 2x� � 10 59.
Amplitude:
Period:
−
−0.02
0.02
�180
�180
160
1100
y �1
100 sin 120� t©
Hou
ghto
n M
ifflin
Com
pany
. All
right
s re
serv
ed.
Section 4.5 Graphs of Sine and Cosine Functions 325
60.
Amplitude:
Period:
−0.04
0.04
0.06−0.06
125
1100
y ��1100
cos�50� t� 61.
Amplitude:
Since is the graph ofreflected about the
axis and shifted vertically fourunits upward, we have and Thus,
� 4 � 4 cos x.
f�x� � �4 cos x � 4
d � 4.a � �4
x-g�x� � 4 cos x
f �x�
1
28 � 0 � 4
f�x� � a cos x � d
63.
Amplitude:
Graph of is the graph ofreflected about the
-axis and shifted vertically oneunit upward. Thus,f �x� � �6 cos x � 1.
xg�x� � 6 cos x
f
12
7 � ��5� � 6
f �x� � a cos x � d 65.
Amplitude:
Since the graph is reflectedabout the axis, we have
Period:
Phase shift:
Thus, f �x� � �3 sin 2x.
c � 0
2�
b� � ⇒ b � 2
a � �3.x-
�a� � 3
f �x� � a sin�bx � c�
67.
Amplitude:
Period:
Phase shift: when
Thus, f �x� � sin�x ��
4�.
�1���
4� � c � 0 ⇒ c ��
4
x ��
4.bx � c � 0
2� ⇒ b � 1
a � 1
f �x� � a sin�bx � c�
62.
Amplitude:
Reflected in the x-axis:
y � �3 � cos x
d � �3
�4 � �1 cos 0 � d
a � �1
�2 � ��4�2
� 1
f�x� � a cos x � d
64.
Reflected in -axis,
y � �4 �12
cos x
d � �4
a � �12
x
Period: 2�
Amplitude: 12
y � a cos x � d
66.
Amplitude:
Period:
Phase shift:
y � 2 sin�x
2�c � 0
2�
b� 4� ⇒ b �
1
2
4�
2 ⇒ a � 2
y � a sin�bx � c�
68.
Amplitude:
Period:
Phase shift:
y � 2 sin��x
2�
�
2�
c
b� �1 ⇒ c � �
�
2
2�
b� 4 ⇒ b �
�
2
4
2 ⇒ a � 2
y � a sin�bx � c� 69.
In the interval when
−2
−2
2
2
��
y1
y2
x � �5�
6, �
�
6,
7�
6,
11�
6.
sin x � �12�2�, 2�,
y2 � �1
2
y1 � sin x
©H
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.
326 Chapter 4 Trigonometric Functions
70.
.
−2
−2
2
2
��
y1
y2
y1 � y2 when x � �, ��
y2 � �1
y1 � cos x
72.
(a)
(b) Maximum sales: December
Minimum sales: June �t � 6�
�t � 12�
10
12
120
S � 74.50 � 43.75 cos � t
6
71.
(a)
(b)
(c) Cycles per min cycles per min
(d) The period would change.
�60
6� 10
Time for one cycle � one period �2�
��3� 6 sec
0
−2
4
2
�
v � 0.85 sin �t
3
73.
(a)
(b) Minimum:
Maximum: 30 � 25 � 55 feet
30 � 25 � 5 feet
00
135
60
h � 25 sin �
15�t � 75� � 30
74.
1 heartbeat�3�4� ⇒ 4
3 heartbeats�second � 80 heartbeats�minute
Period: 2�
�8��3� �34
060
1.5
130
P � 100 � 20 cos 8�
3t
75.
(a)
This is to be expected:
(b) The constant 30.3 gallons is the average dailyfuel consumption.
365 days � 1 year
Period: 2�
b�
2�
�2��365�� 365 days
C � 30.3 � 21.6 sin�2�t
365� 10.9�
(c)
Consumption exceeds 40 when (Graph together with
(Beginning of May through part of September)y � 40.)C124 ≤ x ≤ 252.
gallons�day
00
365
60
©H
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Section 4.5 Graphs of Sine and Cosine Functions 327
76. (a) Yes, y is a function of t because for eachvalue of t there corresponds one and onlyone value of y.
(b) The period is approximately
The amplitude is approximately12�2.35 � 1.65� � 0.35 centimeters.
2�0.375 � 0.125� � 0.5 seconds.
(c) One model is
(d)
00
0.9
4
y � 0.35 sin 4� t � 2.
77. (a)
(b) y � 0.506 sin�0.209x � 1.336� � 0.526
−10 10
1.0
2.0
0.5
1.5
20 30 40 50 60 70
−0.5
x
y (c)
The model is a good fit.
(d) The period is
(e) June 29, 2007 is day 545. Using the model,or 27.09%.y � 0.2709
2�
0.209� 30.06.
−10 10
1.0
2.0
1.5
20 30 40 50 60 70
−0.5
x
y
78. (a)
(b)
The model somewhat fits the data.
(c)
The model is a good fit.
045
12
95
00
12
80
A�t� � 19.73 sin�0.472t � 1.74� � 70.2 (d) Nantucket:
Athens:
The constant term (d) gives the average daily hightemperature.
(e) Period for
Period for
You would expect the period to be 12 (1 year).
(f ) Athens has greater variability. This is given by theamplitude.
A�t� �2�
0.472� 13
N�t� �2�
�2��11� � 11
70.2�
58�
79. True. The period is 2�
3�10�
20�
3.
81. True
80. False. The amplitude is that of y � cos x.12
82. Answers will vary.
83. The graph passes through and has period
Matches (e).
�.�0, 0� 84. The amplitude is 4 and the period Sinceis on the graph, matches (a).�0, �4�
2�.
85. The period is and the amplitude is 1. Sinceand are on the graph, matches (c).��, 0��0, 1�
4� 86. The period is Since is on the graph,matches (d).
�0, �1��.
©H
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328 Chapter 4 Trigonometric Functions
87. (a) is even.
−
−2
2
��
h(x) = cos2 x
h�x� � cos2 x (b) is even.
−
−2
2
��
h(x) = sin2 x
h�x� � sin2 x (c) is odd.
−
−2
2
��
h(x) = sin x cos x
h�x� � sin x cos x
88. (a) In Exercise 87, is even and we saw that is even. Therefore, for even and we make the conjecture that is even.
(b) In Exercise 87, is odd and we saw that is even. Therefore, for odd and we make the conjecture that is even.
(c) From part (c) of 87, we conjecture that the product of an even function and an odd function is odd.
h�x�h�x� � g�x�2,g�x�h�x� � sin2xg�x� � sin x
h�x�h�x� � f�x�2,f�x�h�x� � cos2 xf�x� � cos x
89. (a)
0.8415 0.9983 1.0 1.0sin x
x
�0.001�0.01�0.1�1x
0 0.001 0.01 0.1 1
Undef. 1.0 1.0 0.9983 0.8415sin x
x
x
(b)
As approaches 1.
(c) As approaches 0, approaches 1.sin x
xx
x → 0, f�x� �sin x
x
−1 10.8
1.1
90. (a) (b)
As approaches 0.
(c) As approaches 0, approaches 0.1 � cos x
xx
x → 0, 1 � cos x
x
1−1
0.5
−0.5
�0.0005�0.005�0.05�0.45971 � cos x
x
�0.001�0.01�0.1�1x
0 0.001 0.01 0.1 1
Undef. 0.0005 0.005 0.05 0.45971 � cos x
x
x
91. (a)
(b)
−2π π
−2
2
2
−2π π
−2
2
2
(c) Next term for sine approximation:
Next term for cosine approximation:
−2π π
−2
2
2−2π π
−2
2
2
�x6
6!
�x7
7!
©H
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Section 4.6 Graphs of Other Trigonometric Functions
Section 4.6 Graphs of Other Trigonometric Functions 329
92. (a) (d)
(b) (e)
(c) (f )
In all cases, the approximations are very accurate.
cos��12� � 0.8776sin��
8� � 0.3827
cos���
4� � 0.7071sin�12� � 0.4794
cos��1� � 0.5403sin�1� � 0.8415 93.
Slope �7 � 12 � 0
� 3
x
y
−1−2−3−4 1 2 3 4−1
7
6
5
4
3
2
1
(2, 7)
(0, 1)
95. 8.5 � 8.5�180�
� � � 487.014�94.
m ��2 � 43 � 1
��64
��32
−2
−1−1 1 2 3 4−2−3−4
−3
−4
1
2
3
4
x
y
(−1, 4)
(3, −2)
96. �0.48 � �0.48�180�
� � � �27.502� 97. Answers will vary. (Make a Decision)
■ You should be able to graph:
■ When graphing or you should know to first graph or since
(a) The intercepts of sine and cosine are vertical asymptotes of cosecant and secant.
(b) The maximums of sine and cosine are local minimums of cosecant and secant.
(c) The minimums of sine and cosine are local maximums of cosecant and secant.
■ You should be able to graph using a damping factor.
y � a sin�bx � c�y � a cos�bx � c�y � a csc�bx � c�y � a sec�bx � c�
y � a csc�bx � c�y � a sec�bx � c�y � a cot�bx � c�y � a tan�bx � c�
Vocabulary Check
1. vertical 2. reciprocal 3. damping
©H
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330 Chapter 4 Trigonometric Functions
1.
(a) x-intercepts:
(c) Increasing on:
Never decreasing
(e) Vertical asymptotes: x � �3�
2, �
�
2,
�
2,
3�
2
��2�, �3�
2 �, ��3�
2, �
�
2�, ���
2,
�
2�, ��
2,
3�
2 �, �3�
2, 2��
��2�, 0�, ���, 0�, �0, 0�, ��, 0�, �2�, 0�
f�x� � tan x
2.
(a) x-intercepts:
(c) Decreasing on:
Never increasing
(e) Vertical asymptotes: x � �2�, ��, 0, �, 2�
��2�, ���, ���, 0�, �0, ��, ��, 2��
��3�
2, 0�, ��
�
2, 0�, ��
2, 0�, �3�
2, 0�
f�x� � cot x
3.
(a) No x-intercepts
(b) y-intercept:
(c) Increasing on intervals:
Decreasing on intervals:
(d) Relative minima:
Relative maxima:
(e) Vertical asymptotes: x � �3�
2, �
�
2,
�
2,
3�
2
���, �1�, ��, �1�
��2�, 1�, �0, 1�, �2�, 1�,
���, ��
2�, ���
2, 0�, ��,
3�
2 �, �3�
2, 2��
��2�, �3�
2 �, ��3�
2, ���, �0,
�
2�, ��
2, ��
�0, 1�
f�x� � sec x 4.
(a) No x-intercepts
(b) No y-intercepts
(c) Increasing on intervals:
Decreasing on intervals:
(d) Relative minima:
Relative maxima:
(e) Vertical asymptotes: x � ±�, ±2�
���
2, �1�, �3�
2, �1�
��3�
2, 1�, ��
2, 1�
��2�, �3�
2 �, ���
2, 0�, �0,
�
2�, �3�
2, 2��
��3�
2, ���, ���, �
�
2�, ��
2, ��, ��,
3�
2 �
f�x� � csc x
6.
Period:
Asymptotes:
1
2
3
π π−1
−2
−3
x
y
x � ± �
2
�
y �14
tan x5.
Period:
Two consecutive asymptotes: and
x
y
1
2
3
4
π π2
π2
−
x ��
2x � �
�
2
�
y �1
2 tan x
0
012
�12
y
�
4�
�
4x
(b) intercept:
(d) No relative extrema
�0, 0�y-
(b) No intercepts
(d) No relative extrema
y-
©H
ough
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.
Section 4.6 Graphs of Other Trigonometric Functions 331
7.
Period:
Two consecutive asymptotes:
2x ��
2 ⇒ x �
�
4
−
6
−6
−4
π4 4
π3π4
−4π3
x
y2x � ��
2 ⇒ x � �
�
4
�
2
y � �2 tan 2x
0
2 0 �2y
�
8�
�
8x
8.
Period:
Asymptotes:
4x ��
2 ⇒ x �
�
8
4x ���
2 ⇒ x �
��
8
�
4
−4
x
y
π2
−8
y � �3 tan 4x
10.
Period: 2�
x
y
πππ2 π2− −
1
2
3
y �14 sec x9.
Graph first.
Period:
One cycle: 0 to 2�
2�
y � �12 cos x
1
2
3
xπ−π
yy � �12 sec x
11.
Period: 2
Shift graph of down three units.
sec �x
1
1 2−1−2
−3
−4
−5
x
yy � sec �x � 3
13.
Graph first.
Period:
One cycle: 0 to 4�
2�
1�2� 4�
y � 3 sin x
2x
y
−8
−10
2
4
6
8
10
π3π2 π4π π−π3− π2−
y � 3 csc x
2
12.
Period:
Asymptotes:
x � ��
8, x �
�
8
2�
4�
�
2
y � �2 sec 4x � 2
x 0
y 0 �0.828�0.828
�
16�
�
16
xπππ244
−
2
4
6
y
14.
Period:
Asymptotes:
x � 0, x � 3�
2�
�1�3�� 6�
y � �csc x
3
x
y 1.155�1.155�1.155
4�2��
3
π−π−1
−2
−3
π4π4−x
y
©H
ough
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.
332 Chapter 4 Trigonometric Functions
15.
Period:
Two consecutive asymptotes:
x
2� � ⇒ x � 2�
x
2� 0 ⇒ x � 0
�
1�2� 2�
y �1
2 cot
x
2
0 �12
12
y
3�
2�
�
2x
6
−6
−4
ππ−x
y 16.
Period:
Asymptotes:x � 0, x � 1
�
�� 1
x
y
−1−2−3 1 2 3
−3
1
2
3
y � 3 cot �x
18.
Period: 1
Asymptotes:
x � �12
, x �12
3
1−1
−3
−1
−2
2
1
x
yy � �12
tan � x
x 0
y 0 �12
12
14
�14
17.
Period:
Two consecutive asymptotes:
�x
4�
�
2 ⇒ x � 2
�x
4� �
�
2 ⇒ x � �2
�
��4� 4
y � 2 tan �x
4
0 1
0 2�2y
�1x
2
4
4
6
2−2 6−6x
y
19.
Period:
Asymptotes: x � ±�
4
2�
2� �
1
2
3
4
x
y
π2
π2
−
y �12
sec�2x � �� 20.
Period:
Asymptotes: x � ±�
2
2� 6
8
−6
−8
−2
x
y
−4
π2π2−
y � �sec�x � ��
21.
Graph first.
Period:
Asymptotes: Set
x � � x � ��
� � x � 0 and � � x � 2�
2�
y � sin�� � x�
2
4
6
π− ππ−2 π2x
yy � csc�� � x�
©H
ough
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.
Section 4.6 Graphs of Other Trigonometric Functions 333
22.
Period:
x
y
−1
−2
−3
−4
−5
3
4
5
−− ππ2
π π2
2�
2� �
y � csc�2x � ��
24.
Period:
x
y
−2
−3
−4
−5
3
2
1
4
5
π2
π2
π23 π
23− −
�
y �14
cot�x � ��
23.
Period:
Two consecutive asymptotes:
x ��
2� � ⇒ x �
3�
2
x ��
2� 0 ⇒ x �
�
2
�
y � 2 cot�x ��
2�
2 0 �2y
5�
4�
3�
4x
ππ−
−6
−4
−2
x
y
25.
Period:
−
−10
10
��
−
−10
10
��
2�
3
y � 2 csc 3x �2
sin�3x�
27.
−4
4
−�2
�2
−4
4
−�2
�2
y ��2
cos 4x
y � �2 sec 4x
26.
−
−3
3
�2
�2
−
−3
3
�2
�2
y ��1
sin�4x � ��
y � �csc�4x � ��
28.
−2
2
2−2
−2
2
2−2
y �1
4 sec �x �
1
4 cos �x29.
Period: 4
−2
2
6−6
−2
2
6−6
�1
3 cos��x2
��
2�
y �13
sec��x2
��
2�
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334 Chapter 4 Trigonometric Functions
30.
−
−3
3
��−
−3
3
��
y �12
csc�2x � �� 31.
x � �7�
4, �
3�
4,
�
4,
5�
4
tan x � 1
33.
x � ±2�
3, ±
4�
3
sec x � �232.
The solutions appear to be:
(or in decimal form: 2.618, 5.760)�3.665, �0.524,
x � �7�
6, �
�
6,
5�
6,
11�
6
cot x � ��3
34.
The solutions appear to be:
(or in decimal form: 0.785, 2.356)�5.498, �3.927,
�7�
4, �
5�
4,
�
4,
3�
4
csc x � �2
36.
Thus, the function is odd and the graph of is symmetric with the origin.
y � tan x
tan��x� � �tan x
f �x� � tan x
35. The graph of has axis symmetry.Thus, the function is even.
y-f�x� � sec x
37. The function
has origin symmetry. Thus, the function is odd.
f �x� � csc 2x �1
sin 2x
38.
Odd
�cos��2x�sin��2x� � �
cos�2x�sin�2x� � �f �x�
f ��x� � cot��2x�
f �x� � cot 2x 39. and
Not equivalent because is not defined at 0.
sin x csc x � sin x� 1
sin x� � 1, sin x � 0
y1
−2
2
3−3
y2 � 1y1 � sin x csc x
40.
It appears that
sin x sec x � sin x 1
cos x�
sin x
cos x� tan x
y1 � y2.
−2
−4
2
4
��
y1 � sin x sec x, y2 � tan x 41. and
Equivalent
cot x �cos x
sin x−2
−4
2
4
��
y2 � cot x �1
tan xy1 �
cos x
sin x
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Section 4.6 Graphs of Other Trigonometric Functions 335
42.
It appears that
tan2 x � sec2 x � 1
1 � tan2 x � sec2 x
y1 � y2.
−1
3
3�2
3�2
−
y1 � sec2 x � 1, y2 � tan2 x 43.
As
Odd function
Matches graph (d).
f �3�
2 � � 0
f �x� → 0.x → 0,
f�x� � x cos x
47.
The graph is the line y � 0.−2
−2
2
2
��f = g
f�x� � g�x�
f �x� � sin x � cos�x ��
2�, g�x� � 0 48.
It appears that That is,
sin x � cos�x ��
2� � 2 sin x.
f�x� � g�x�.
g�x� � 2 sin x −2
−3
2
3
��
f = g
f�x� � sin x � cos�x ��
2�
49.
−2
−2
2
2
��
f = gf �x� � g�x�
f �x� � sin2 x, g�x� �1
2�1 � cos 2x� 50.
It appears that That is, that
cos2 �x
2�
1
2�1 � cos �x�.
f�x� � g�x�.
g�x� �1
2 �1 � cos �x�
−2
−2
2
2
��
f = gf�x� � cos2
�x
2
44.
Matches graph (a) as and for all x.f �x� ≥ 0f�x� → 0,
x → 0,
f�x� � �x sin x� 45.
As
Odd function
Matches graph (b).
g�2�� � 0
x → 0, g�x� → 0.
g�x� � �x� sin x 46.
Even function
Matches graph (c) asx → 0, g�x� → 0.
g�x� � �x� cos x
51.
Damping factor:
x → �, f �x� → 0
e�x
−3
3
6−3
f�x� � e�x cos x
54.
Damping factor:
x → ±�, g�x� → 0
�e�x 2�2 ≤ g�x� ≤ e�x 2�2
y � e�x 2�2
−1
1
8−8
g�x� � e�x 2�2 sin x
52.
Damping factor:
f �x� → 0 as x → �
e�2x
−3
−2
3
2f�x� � e�2x sin x
53.
Damping factor:
h → 0 as x → �
e�x2�43−3
−2
2h �x� � e�x2�4 cos x
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336 Chapter 4 Trigonometric Functions
55. (a)
(b)
(c)
(d) As x → ���
2, f�x� → �.
As x → ���
2, f�x� → ��.
As x → ��
2, f�x� → �.
As x → ��
2, f�x� → ��. 56.
(a)
(b)
(c)
(d) As x → ���
2, f�x� → ��.
As x → ���
2, f�x� → �.
As x → ��
2, f �x� → �.
As x → ��
2, f�x� → ��.
f�x� � sec x
57.
(a)
(b)
(c)
(d) As x → ��, f�x� → ��.
As x → ��, f�x� → �.
As x → 0�, f�x� → ��.
As x → 0�, f�x� → �.
f�x� � cot x 58.
(a)
(b)
(c)
(d) As x → ��, f�x� → �.
As x → ��, f�x� → ��.
As x → 0�, f�x� → ��.
As x → 0�, f�x� → �.
f�x� � csc x
59. As the predator population increases, the number of prey decreases. When the numberof prey is small, the number of predators decreases.
61.
d �5
tan x� 5 cot x
0
−36
36
�
tan x �5
d62.
d �36
cos x� 36 sec x
−36
72
�2
�2
cos x �36
d
60.
(a)
Period of
Period of sin
Period of
Period of L�t� : 12
H�t� : 12
� t
6:
2�
���6�� 12
cos � t
6:
2�
���6�� 12
00
12
90
H
L
L�t� � 39.36 � 15.70 cos � t
6� 14.16 sin
� t
6
H�t� � 54.33 � 20.38 cos � t
6� 15.69 sin
� t
6
(b) From the graph, it appears that the greatest differencebetween high and low temperatures occurs in summer.The smallest difference occurs in winter.
(c) The highest high and low temperatures appear to occuraround the middle of July, roughly one month after thetime when the sun is northernmost in the sky.
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Section 4.6 Graphs of Other Trigonometric Functions 337
63. (a)
0
−0.6
0.6
�4
64. (a) 850 rev min
(b) The direction of the saw is reversed.
(c)
(d)
L � 60��
2� �� � cot �, 0 < � <
�
2
�1700
2�
(e) Straight line lengths change faster.
(f )
00
500
�2
0.3 0.6 0.9 1.2 1.5
L 306.2 217.9 195.9 189.6 188.5
�
(b) The displacement function is not periodic, but damped. It approaches 0 as increases.t
65. True. and is a vertical
asymptote for the tangent function.
x � ��
2�
3�
2� � � �
�
2 66. True. For x → ��, 2x → 0.
67.
(a)
0
−1
3
�
g
f
f �x� � 2 sin x, g�x� �12
csc x
(b) for �
6< x <
5�
6f �x� > g�x� (c) As and
since is the
reciprocal of f �x�.
g�x�12
csc x → �,
2 sin x → 0x → �,
68. (a)
−2
2
3−3
−2
2
3−3
69. (a)
1.5574 1.0033 1.0 1.0tan x
x
�0.001�0.01�0.1�1x
0 0.001 0.01 0.1 1
Undef. 1.0 1.0 1.0033 1.5574tan x
x
x
(b)
As
(c) The ratio approaches 1 as x approaches 0.
x → 0, f�x� �tan x
x → 1.
1.1−1.10
2
(b)
(c) y4 � y3 �4�
�19
sin 9�x�
−2
2
3−3
y3 �4� �sin �x �
13
sin 3�x �15
sin 5�x �17
sin 7�x�
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338 Chapter 4 Trigonometric Functions
70. (a)
1.0311 1.0003 1.0�0.0475tan 3x
3x
�0.001�0.01�0.1�1x
0 0.001 0.01 0.1 1
Undef. 1.0 1.0003 1.0311 �0.0475tan 3x
3x
x
(b)
As
(c) The ratio approaches 1 as x approaches 0.
x → 0, f �x� �tan 3x
3x → 1.
1
−1.1
−1
1.1
71. Period is and graph is increasing on
Matches (a).
���
4,
�
4�.�
272. Period is and is on the graph.
Matches (b).
��
8, 1��
2
73.
The graphs of and
are similar on the interval ���
2,
�
2�.
y2 � x �2x3
3!�
16x5
5!y1 � tan x
3−3
−2
2
y = tan(x)y = 2x3
3!16x5
5!+ +x
74.
The graphs of and
are similar on the interval ���
2,
�
2�.
y2 � 1 �x2
2!�
5x4
4!y1 � sec x
−3 30
4y = sec(x)
x2
2!5x4
4!+y = 1 +
75. Distributive Property 77. Additive Identity Property
79. Not one-to-one
81.
f �1�x� �13�x2 � 14�, x ≥ 0
y �13�x2 � 14�
x2 � 3y � 14
x � �3y � 14, y ≥ 143 , x ≥ 0
y � �3x � 14, x ≥ 143 , y ≥ 0
76.
Multiplicative Inverse Property
7�17� � 1
78.
Associative Property of Addition
�a � b� � 10 � a � �b � 10� 80. Not one-to-one
82. One-to-one
f �1�x� � x3 � 5
x3 � y � 5
x � �y � 5�1�3
y � �x � 5�1�3
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Section 4.7 Inverse Trigonometric Functions 339
Section 4.7 Inverse Trigonometric Functions
■ You should know the definitions, domains, and ranges of and
Function Domain Range
■ You should know the inverse properties of the inverse trigonometric functions.
and
and
and
■ You should be able to use the triangle technique to convert trigonometric functions or inverse trigonometricfunctions into algebraic expressions.
arctan�tan y� � y, ��
2< y <
�
2tan�arctan x� � x
arccos�cos y� � y, 0 ≤ y ≤ �cos�arccos x� � x, �1 ≤ x ≤ 1
arcsin�sin y� � y, ��
2≤ y ≤
�
2sin�arcsin x� � x, �1 ≤ x ≤ 1
��
2< y <
�
2�� < x < �y � arctan x ⇒ x � tan y
0 ≤ y ≤ ��1 ≤ x ≤ 1y � arccos x ⇒ x � cos y
��
2≤ y ≤
�
2�1 ≤ x ≤ 1y � arcsin x ⇒ x � sin y
y � arctan x.y � arcsin x, y � arccos x,
Vocabulary Check
1. 2.
3. y � tan�1 x, �� < x < �, ��
2< y <
�
2
y � arccos x, 0 ≤ y ≤ �y � sin�1 x, �1 ≤ x ≤ 1
2. (a)
(b) y � arccos 0 ⇒ cos y � 0 for 0 ≤ y ≤ � ⇒ y ��
2
y � arccos 1
2 ⇒ cos y �
1
2 for 0 ≤ y ≤ � ⇒ y �
�
3
1. (a) (b) arcsin 0 � 0arcsin 12
��
6
3. (a) because and
(b) because and 0 ≤ 1 ≤ �.cos 0 � 1arccos 1 � 0
��
2≤
�
2≤
�
2.sin
�
2� 1arcsin 1 �
�
2
4. (a) because and
(b) because and ��
2< 0 <
�
2.tan 0 � 0arctan 0 � 0
��
2<
�
4<
�
2.tan
�
4� 1arctan 1 �
�
4©H
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340 Chapter 4 Trigonometric Functions
5. (a)
(b) arctan��1� � ��
4
arctan �33
��
66. (a)
(b) arcsin���22 � � �
�
4
arccos���22 � �
3�
4
8. (a)
(b) y � arcsin �2
2 ⇒ sin y �
�2
2 for �
�
2≤ y ≤
�
2 ⇒ y �
�
4
y � arccos��1
2� ⇒ cos y � �1
2 for 0 ≤ y ≤ � ⇒ y �
2�
3
7. (a) for
(b) y � arctan �3 ⇒ tan y � �3 ⇒ y ��
3
��
2< y <
�
2 ⇒ y � �
�
3y � arctan���3 � ⇒ tan y � ��3
9. (a) for
(b) ⇒ y � ��
6 y � tan�1���3
3 � ⇒ tan y ���3
3
��
2≤ y ≤
�
2 ⇒ y �
�
3y � sin�1
�3
2 ⇒ sin y �
�3
2
10. (a) (b)
(c)
(d) symmetric to the origin�0, 0�,
−2
2
1−1
–1 1
–2
–1
1
2
x
y
x
y �0.2014�0.4115�0.6435�0.9273�1.5708
�0.2�0.4�0.6�0.8�1.0
x 0 0.2 0.4 0.6 0.8 1
y 0 0.2014 0.4115 0.6435 0.9273 1.5708
11.
(a) (b)
(c)
(d) Intercepts: no symmetry�1, 0�,�0, �
2�,
−10
4
1
–2 –1 1 2
1
3
4
x
y
y � arccos x
3.1416 2.4981 2.2143 1.9823 1.7722y
�0.2�0.4�0.6�0.8�1x
0 0.2 0.4 0.6 0.8 1.0
1.5708 1.3694 1.1593 0.9273 0.6435 0y
x
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Section 4.7 Inverse Trigonometric Functions 341
12. (a) (b)
(c)
(d) Horizontal asymptotes: y � ±�
2
−2
2
10−10
–10 –5 5 10
–2
1
2
x
y
x
y �1.1071�1.3258�1.4056�1.4464�1.4711
�2�4�6�8�10
x 0 2 4 6 8 10
y 0 1.1071 1.3258 1.4056 1.4464 1.4711
13.
���3, ��
3�, ���33
, ��
6�, �1, �
4�y � arctan x ↔ tan y � x
15. cos�1�0.75� � 0.72
14.
�cos �
6�
�32 �y �
�
6 ⇒ x �
�32
,
�cos 2�
3� �
12�x � �
12
⇒ y �2�
3,
�cos � � �1�x � �1 ⇒ y � �,
y � arccos x
17. arcsin��0.75� � �0.85
19. arctan��6� � �1.41
16. sin�1 0.56 � 0.59
18. arccos��0.7� � 2.35 20. tan�1 5.9 � 1.40
21.
� � arctan x
8θ
8
x
tan � �x
8
23.
� � arcsin�x � 2
5 �θ
5x + 2
sin � �x � 2
5
22.
� � arccos 4
x 4
x
θ
cos � �4
x
24.
� � arctan�x � 1
10 �10
θ
x + 1
tan � �x � 1
10
25. Let y be the third side. Then
tan � �x
�4 � x2 ⇒ � � arctan
x�4 � x2
cos � ��4 � x2
2 ⇒ � � arccos
�4 � x2
2
sin � �x2
⇒ � � arcsin x2
y2 � 22 � x2 � 4 � x2 ⇒ y � �4 � x2
26. Let y be the third side. Then
tan � �x
�9 � x2 ⇒ � � arctan
x�9 � x2
cos � ��9 � x2
3 ⇒ � � arccos
�9 � x2
3
sin � �x3
⇒ � � arcsin x3
y2 � 32 � x2 � 9 � x2 ⇒ y � �9 � x2.
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342 Chapter 4 Trigonometric Functions
27. Let y be the hypotenuse. Then
tan � �x � 1
2 ⇒ � � arctan
x � 12
cos � �2
�x2 � 2x � 5 ⇒ � � arccos
2�x2 � 2x � 5
sin � �x � 1
�x2 � 2x � 5 ⇒ � � arcsin
x � 1�x2 � 2x � 5
y2 � �x � 1�2 � 22 � x2 � 2x � 5 ⇒ y � �x2 � 2x � 5.
28. Let y be the hypotenuse. Then
tan � �x � 2
3 ⇒ � � arctan
x � 23
cos � �3
�x2 � 4x � 13 ⇒ � � arccos
3�x2 � 4x � 13
sin � �x � 2
�x2 � 4x � 13 ⇒ � � arcsin
x � 2�x2 � 4x � 13
y2 � �x � 2�2 � 32 � x2 � 4x � 13 ⇒ y � �x2 � 4x � 13.
29. sin�arcsin 0.7� � 0.7 31. cos�arccos��0.3�� � �0.3
33.
Note: is not in the range of the arcsine function.3�
arcsin�sin 3�� � arcsin�0� � 0
35. arctan�tan 11�
6 � � arctan���33 � � �
�
634. arccos�cos
7�
2 � � arccos�0� ��
2
36. arcsin�sin 7�
4 � � arcsin���22 � � �
�
4
30. tan�arctan 35� � 35
32. sin�arcsin��0.1�� � �0.1
37. sin�1�sin 5�
2 � � sin�1 1 ��
2
38. cos�1�cos 3�
2 � � cos�1 0 ��
239. sin�1�tan
5�
4 � � sin�1 1 ��
2
40. cos�1�tan 3�
4 � � cos�1��1� � � 41. tan�arcsin 0� � tan 0 � 0
42. cos�arctan��1�� � cos���
4� ��22
43. sin�arctan 1� � sin��
4� ��22
44. sin�arctan��1�� � sin���
4� � ��22
45. arcsincos���
6� � arcsin��32 � �
�
3
46. arccossin���
6� � arccos��12� �
2�
3 47. Let Then
and
sin y �45
.
tan y �43
, 0 < y <�
2, 4
3
5
y
y � arctan 43
.
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Section 4.7 Inverse Trigonometric Functions 343
49. Let Then
and cos y �7
25.sin y �
2425
,
7
2425
y
y � arcsin 2425
.48. Let
�hyp
adj�
5
4
sec�arcsin 3
5� � sec u
sin u �3
5, 0 < u <
�
2.
5
4
3u
u � arcsin 3
5,
50. Let
cscarctan��12
5 � � csc u �hyp
opp� �
13
12
5
13
u
−12
tan u � �12
5, �
�
2< u < 0.
u � arctan��12
5 �, 51. Let Then,
and
xy
34−3
5
y
sec y ��34
5.tan y � �
3
5, �
�
2< y < 0
y � arctan��3
5�.
53. Let Then,
and
xy
3
−2
5
y
sin y ��5
3.cos y � �
2
3,
�
2< y < �
y � arccos��2
3�.52. Let
tan�arcsin��3
4�� � tan u ��3�7
��3�7
7
4−3
u7
sin u � �3
4, �
�
2< u < 0.
u � arcsin��3
4�,
54. Let
cot�arctan 5
8� � cot u �8
5
tan u �5
8, 0 < u <
�
2.
u � arctan 5
8,
8
5u
89
55. Let Then,
and cot y �1
x.tan y � x
x2 + 1x
y
1
y � arctan x.
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344 Chapter 4 Trigonometric Functions
56. Let
sin�arctan x� � sin u �opp
hyp�
x�x2 � 1
x + 12
1
x
u
tan u � x �x
1.u � arctan x,
58. Let
sec�arcsin �x � 1�� � sec u �hyp
adj�
1�2x � x2
x −1
2x x− 2
1
u
sin u � x � 1 �x � 1
1.u � arcsin�x � 1�,
57. Let
Opposite side:
1 − (x + 2)2
y
x + 2
1
sin y ��1 � �x � 2�2
1� ��x2 � 4x � 3
�1 � �x � 2�2
cos y � x � 2.y � arccos�x � 2�,
59. Let Then and
x
5
y
25 − x2
tan y ��25 � x2
x.
cos y �x5
,y � arccos x5
.
60. Let
cot�arctan 4
x� � cot u �adj
opp�
x
4
4
ux
x + 162
tan u �4
x.u � arctan
4
x,
62. Let
cos�arcsin x � h
r � � cos u ��r2 � �x � h�2
r
rx h−
r x h− −( )22
u
sin u �x � h
r.u � arcsin
x � h
r,
61. Let Then and
x
y
7 + x2
7
csc y ��7 � x2
x.
tan y �x�7
y � arctan x�7
.
63. Let Then and
Thus,
x
y
196 + x2
14
y � arcsin� 14�196 � x2�.sin y �
14�196 � x2
.
tan y �14x
y � arctan 14x
.©
Hou
ghto
n M
ifflin
Com
pany
. All
right
s re
serv
ed.
Section 4.7 Inverse Trigonometric Functions 345
65. Let Then,
and Thus,
y � arcsin �x � 1���x � 1�2 � 9
� arcsin �x � 1��x2 � 2x � 10
.
sin y ��x � 1�
��x � 1�2 � 9.
cos y �3
�x2 � 2x � 10�
3
��x � 1�2 � 9
y � arccos 3
�x2 � 2x � 10.64. If then
arcsin �36 � x2
6� arccos
x
6
6
ux
36 − x 2
sin u ��36 � x2
6.arcsin
�36 � x2
6� u,
66. If then
since 2 < x < 4arccos x � 2
2� arctan
�4x � x2
x � 2,
2
ux − 2
4x x− 2
cos u �x � 2
2.arccos
x � 2
2� u, 2 < x < 4 67.
Domain:
Range:
Vertical stretch of
02−2
2�
f�x� � arccos x
0 ≤ y ≤ 2�
�1 ≤ x ≤ 1
y � 2 arccos x
69. The graph of is a horizontaltranslation of the graph of by two units.
40
−�2
�2
y � arcsin xf �x� � arcsin�x � 2�68.
Domain:
Range:
−
3−3
�2
�2�
�
2≤ y ≤
�
2
�2 ≤ x ≤ 2
y � arcsin x
2
70.
Domain:
This is the graph of shifted two unitsto the left.
0
�
0−4
y � arccos t
�3 ≤ t ≤ �1
g�t� � arccos�t � 2� 71.
Domain: all real numbers
Range:
3−3
−�2
�2
��
2< y <
�
2
f �x� � arctan 2x
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346 Chapter 4 Trigonometric Functions
72.
Domain:
Range: �0, ��
��4, 4�
05−5
�f�x� � arccos x
4
73.
The graphs are the same.
−2
−6
2
6
��
� 3�2 sin�2t ��
4� � 3�2 sin�2t � arctan 1�
� �32 � 32 sin�2t � arctan 3
3� f�t� � 3 cos 2t � 3 sin 2t 74.
The graph suggests that
is true.
−6
6
9−9
A cos �t � B sin �t ��A2 � B2 sin��t � arctan A
B�
� 5 sin�� t � arctan 4
3�
� �42 � 32 sin�� t � arctan 4
3�f�t� � 4 cos � t � 3 sin � t
75. As x → 1�, arcsin x → �
2. 76. As x → 1�, arccos x → 0. 77. As x →�, arctan x → �
2.
78. As x → �1�, arcsin x → ��
2. 79. As x → �1�, arccos x → �. 80. As x → ��, arctan x → �
�
2.
81. (a)
(b)
��22.6��s � 26:� � arcsin�1026� � 0.3948,
��11.1��s � 52: � � arcsin�1052� � 0.1935,
sin � �10s
⇒ � � arcsin�10s �
82. (a)
(b)
(c)
� 12.94 feet
tan�0.5743� �h
20 ⇒ h � 20 tan�0.5743�
tan � �1117
⇒ � � 0.5743 or 32.9�
34 ft
11 ft
83. (a)
(b) When
When ��65��.� � 1.13 radians,s � 1600,
��28��.� � arctan�400750� � 0.49 radian,
s � 400,
� � arctan� s750�
tan � �s
750
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Section 4.7 Inverse Trigonometric Functions 347
86. (a)
(b) When
(0.24 rad).
When
� � arctan 12
20� 31.0�, �0.54 rad�.
x � 12,
� � arctan 5
20� 14.0�,
x � 5,
� � arctan x
20
tan � �x
2085. (a)
(b) When
When
��63��.� � arctan�63� � 1.11 radians,
x � 3,
��31��.� � arctan� 610� � 0.54 radian,
x � 10,
� � arctan�6x�
tan � �6x
87. False. arcsin 12
��
688. False. tan x �
sin xcos x
89. if and only if and
x−2 −1 1 2
π2
y
π
0 < y < �.�� < x < �cot y � x,y � arccot x 90. if and only if where
and andThe domain of
is and the range is
x−2 −1 1 2
π
y
π2
�0, ��2� � ���2, ��.���, �1� � �1, ��
y � arcsec x��2 < y ≤ �.0 ≤ y < ��2x ≤ �1 � x ≥ 1
sec y � xy � arcsec x
91. if and only if
Domain:
Range: ��
2, 0� � �0,
�
2���, �1� � �1, ��
x−2 −1 1 2
π2
π2−
ycsc y � x.y � arccsc x
84.
(a)
−0.5
1.5
60
� arctan 3x
x2 � 4, x > 0
(b) is maximum when feet.
(c) The graph has a horizontal asymptote at As the camera moves further from the picture, theangle subtended by the camera approaches 0.
� 0.
x � 2
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348 Chapter 4 Trigonometric Functions
92. (a) if and only if and
Thus,
Hence, graph
(b) if and only if or and or
Thus, and Hence graph
(c) if and only if and or
Thus, Hence, graph y � arcsin�1x�, x � ���, �1� � �1, ��.1
x� sin y and y � arcsin�1
x�.
0 < y ≤�
2.�
�
2≤ y < 0x ≤ �1 or x ≥ 1,x � csc y,y � arccsc x
x � ���, �1� � �1, ��.y � arccos 1x,y � arccos�1
x�.1x
� cos y
�
2< y < �.0 ≤ y <
�
2x ≥ 1,x � sec y, x ≤ �1y � arcsec x
y � � � arctan�1�x�,��2,arctan�1�x�,
x < 0 x � 0 x > 0
.
1x
� tan y and y � arctan�1x�.
0 < y < �.�� < x < �x � cot y,y � arccot x
93. y � arcsec �2 ⇒ sec y � �2 and 0 ≤ y <�
2�
�
2< y ≤ � ⇒ y �
�
4
94. y � arcsec 1 ⇒ sec y � 1 and 0 ≤ y <�
2�
�
2< y ≤ � ⇒ y � 0
95. y � arccot���3 � ⇒ cot y � ��3 and 0 < y < � ⇒ y �5�
6
96. y � arccsc 2 ⇒ csc y � 2 and ��
2≤ y < 0 � 0 < y ≤
�
2 ⇒ y �
�
6
97. Let Then,
Therefore, arcsin��x� � �arcsin x.
y � �arcsin x.
�y � arcsin x
sin��y� � x
�sin y � x
sin y � �x
y � arcsin��x�. 98.
y � �arctan x
�y � arctan x
arctan�tan��y�� � arctan x
tan��y� � x, ��
2< �y <
�
2
�tan y � x
tan y � �x, ��
2< y <
�
2
y � arctan��x�
99.
Let
Let
Hence, and are complementary angles arcsin x � arccos x ��
2.⇒a � �
�
2⇒sin a � cos ⇒ a
� arccos x ⇒ cos � x.
� arcsin x ⇒ sin � x.
arcsin x � arccos x ��
2
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Section 4.7 Inverse Trigonometric Functions 349
100.
(a)
(c)
� 1.25
� 1.25 � 0 � 1.25
Area � arctan 3 � arctan 0
a � 0, b � 3
��
4� 0 �
�
4� 0.785
Area � arctan 1 � arctan 0
a � 0, b � 1
Area � arctan b � arctan a
(b)
(d)
� 1.25 � ���
4� � 2.03
Area � arctan 3 � arctan��1�
a � �1, b � 3
��
4� ��
�
4� ��
2� 1.571
Area � arctan 1 � arctan��1�
a � �1, b � 1
101.4
4�2�
1�2
��22
103.2�3
6�
�33
105.
Adjacent side:
cot � ��11
5
sec � �6
�11�
6�1111
csc � �65
tan � �5
�11�
5�1111
cos � ��11
6
�62 � 52 � �11
sin � �56
102.2�3
�2�3�3�3
�2�3
3
104.5�5
2�10�
5�5
2�2�5�
5
2�2�
5�24
106.
cot � �12
sec � � �5
csc � ��52
cos � ��55
sin � �2�5
�2�5
5
θ
2
1
5
tan � � 2, 0 < � <�
2
θ
56
11
108.
cot � �1
2�2�
�24
csc � �3
2�2�
3�24
tan � � 2�2
cos � �13
sin � �2�2
3
θ1
32 2
sec � � 3, 0 < � <�
2107.
Adjacent side:
cot � ��73
sec � �4�7
�4�7
7
csc � �43
tan � �3�7
�3�7
7
cos � ��74
�16 � 9 � �7
sin � �34
θ
34
7
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Section 4.8 Applications and Models
350 Chapter 4 Trigonometric Functions
■ You should be able to solve right triangles.
■ You should be able to solve right triangle applications.
■ You should be able to solve applications of simple harmonic motion: or d � a cos wt.d � a sin wt
Vocabulary Check
1. elevation, depression 2. bearing 3. harmonic motion
1. Given:
cos A �bc ⇒ c �
bcos A
�10
cos 30�� 11.55
tan A �ab
⇒ a � b tan A � 10 tan 30� � 5.77
B � 90� � 30� � 60�
A � 30�, b � 10
2. Given:
cos B �ac ⇒ a � c cos B � 15 cos 60� �
152
� 7.50
sin B �bc ⇒ b � c sin B � 15 sin 60� �
15�32
� 12.99
A � 90� � 60� � 30�
B � 60�, c � 15
3. Given:
A � 90� � 71� � 19�
sin B �bc ⇒ c �
bsin B
�14
sin 71�� 14.81
tan B �ba
⇒ a �b
tan B�
14tan 71�
� 4.82
B � 71�, b � 14 4. Given:
sin A �ac ⇒ c �
asin A
�20.5
sin 7.4�� 159.17
tan A �ab
⇒ b �a
tan A�
20.5tan 7.4�
� 157.84
B � 90� � 7.4� � 82.6�
A � 7.4�, a � 20.5
5. Given:
B � 90� � 26.57� � 63.43�
tan A �ab
�612
�12
⇒ A � arctan 12
� 26.57�
c2 � a2 � b2 ⇒ c � �36 � 144 � 13.42
a � 6, b � 12 6. Given:
cos B �ac ⇒ B � arccos�25
45� � 56.25�
sin A �ac ⇒ A � arcsin�25
45� � 33.75�
b � �c2 � a2 � �1400 � 10�14 � 37.42
a � 25, c � 45
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Section 4.8 Applications and Models 351
7. Given:
B � 90� � 72.76� � 17.24�
cos A �bc �
1654
⇒ A � arccos�1654� � 72.76�
a � �c2 � b2 � �2660 � 51.58
b � 16, c � 54 8. Given:
sin B �bc ⇒ B � arcsin�1.32
18.9� � 4.00�
cos A �bc �
1.3218.9
⇒ A � arccos�1.3218.9� � 86.00�
a � �c2 � b2 � �355.4676 � 18.85
b � 1.32, c � 18.9
10. Given:
tan B �ba
⇒ b � a tan B � 145.5 tan�65� 12�� � 314.89
cos B �ac ⇒ c �
acos B
�145.5
cos�65� 12�� � 346.88
A � 90� � 65� 12� � 24� 48�
B � 65� 12�, a � 145.5
9.
b � 430.5 cos 12� 15� � 420.70
cos 12� 15� �b
430.5
a � 430.5 sin 12� 15� � 91.34
sin 12� 15� �a
430.5
B � 90� � 12� 15� � 77� 45�b
c = 430.5a
AC
B
12°15′
A � 12� 15�, c � 430.5
11.
h �12
�8� tan 52� � 5.12 in.
h �12
b tan �
tan � �h
b�212.
h �12
�12� tan 18� � 1.95 meters
h �12
b tan �
tan � �h
b�2
13.
h �12
�18.5� tan 41.6� � 8.21 ft
h �12
b tan �
tan � �h
b�214.
h �12
�3.26� tan 72.94� � 5.31 cm
h �12
b tan �
tan � �h
b�2
15. (a)
θ
60 ft
L
(b)
� 60 cot �
L �60
tan �
tan � �60
L(c)
(d) No, the shadow lengths do not increase inequal increments. The cotangent function is not linear.
340 165 104 72 50L
50�40�30�20�10��
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352 Chapter 4 Trigonometric Functions
17.
� 19.70 feet
h � 20 sin 80�
80°
20 h
sin 80� �h
20
16. (a)
(c)
θL
850 ft
L 4821 2335 1472 1013 713
50�40�30�20�10��
(b)
(d) No, the cotangent function is not a linear function.
� 850 cot �
L �850tan �
tan � �850L
18.
58.2 meters13°
h
h � 58.2 tan�13�� � 13.44 meters
tan�13�� �h
58.2
21. (a)
(b) Let the height of the church and the heightof the church and steeple Then:
and
and
(c) feeth � 19.9
h � y � x � 50�tan 47� 40� � tan 35��
y � 50 tan 47� 40�x � 50 tan 35�
tan 47� 40� �y
50tan 35� �
x
50
� y.� x
50 ft
47 40° ′
35°
h
22.
s
a
100
39.75°
28°
� 30 feet
� 100 tan 39.75� � 100 tan 28�
s � 100 tan 39.75� � a
a � s � 100 tan 39.75�
tan 39.75� �a � s
100
tan 28� �a
100 ⇒ a � 100 tan 28�
19.
� 76.6 feet
h � 100 sin 50�
100h
50°
sin 50� �h
10020.
� 2089.99 feet
x � 4000 sin 31.5�
31.5°
4000x
sin 31.5� �x
4000
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Section 4.8 Applications and Models 353
26. (a)
(b)
(c) � � arctan 121
2
1713
� 35.8�
tan � �121
2
1713
θ
17 ft
12 ft
1
1
3
2
25.
95
75
θ
� � arctan 1519
� 38.29�
tan � �7595
24. (a)
(b)
(c)
is in the interval �45.0, 55.72�.d
55.72 ≥ d ≥ 45.00
htan 25�
≥ d ≥h
tan 30�
tan 25� ≤hd
≤ tan 30�
tan 25� ≤ tan � ≤ tan 30�
25� ≤ � ≤ 30�
tan � �hd
�15�3
d�
25.98d
60°
30 h
d
θ
θ
� 15�3 � 25.98 ft
sin 60� �h30
⇒ h � 30 sin 60�
27.
θα16,500
4000
� � 90� � � � 75.97�
sin � �4000
16,500 ⇒ � � 14.03�
28.
250 feetθ2.5 miles
not drawn to scale
� � arctan� 2502.5�5280�� � 1.09�
tan � �250
2.5�5280� 29. Since the airplane speed is
after one minute its distance travelled is 16,500 feet.
� 5099 ft
a � 16,500 sin 18�18°
16500asin 18� �
a
16,500
�275 ft
sec��60 sec
min� � 16,500 ft
min,
23. (a)
(b)
(c) If then Then
� 53 feet, 14
inch.
h � 53.02 feet
17 � h � 70.02
�17 � h�2 � l2 � 1002 � 4902.906
�17 � h�2 � 1002 � l2
l �100
cos 35�� 122.077.� � 35�,
cos � �100
l ⇒ � � arccos�100
l �100 ft
3 ft 3 ft20 ft
lh
θ
� �h2 � 34h � 10,289
l � �1002 � �17 � h�2
l2 � 1002 � �20 � 3 � h�2
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354 Chapter 4 Trigonometric Functions
30.
If
If h � 16,000, s �16,000
275 sin 18�� 188.3 seconds.
h � 10,000, s �10,000
275 sin 18�� 117.7 seconds.
s �h
275 sin 18�
sin 18� �h
275�s� 31.
� 0.66 mile
x � 4 sin 9.5�
x
9.5°
4 sin 9.5� �x4
32.
⇒ L �1808
sin�25.2�� � 4246.3 feet
1808
25.2°
L
sin�25.2�� �1808
L
33.
cos 61� �b
120 ⇒ b � 120 cos 61� � 58.18 nautical miles
sin 61� �a
120 ⇒ a � 120 sin 61� � 104.95 nautical miles
�20��6� � 120 nautical miles
S
EW
120a
b
61°
N90� � 29� � 61�
34.
miles
milesx � 900 cos 38� � 709.2
y � 900 sin 38� � 554.1
x
yc
38°52°
c � 600�1.5� � 900
275( )sh
18°
35. Note: forms a right triangle.
(a)
Bearing from to N E
(b)
tan C �d
50 ⇒ tan 54� �
d
50 ⇒ d � 68.82 m
C � � � � 54�
� 90� � � 22�
� � � � 32�
58�C:A
� � 90� � 32� � 58�
S
EW
C
B
d
A
θ α
φγ
β
β50
NABC� � 32�, � 68�
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Section 4.8 Applications and Models 355
36.
� 5.46 kilometers d �30
cot 34� � cot 14�
d cot 34� � 30 � d cot 14�
cot 34� �30 � d cot 14�
d
tan 34� �d
y ⇒ d
30 � x�
d
30 � d cot 14�
tan 14� �d
x ⇒ x � d cot 14�
38.
S
EW
Airport
Plane
θ
θ
85
160
N
Bearing: S 27.98� W
tan � �85160
⇒ � � 27.98�
37.
Bearing:
S
EW
45
30θ
Port
Ship
N
N 56.31� W
tan � �4530
�32
⇒ � � 56.31�
39.
Distance between ships:
350 4°
dD
S1 S2
6.5°
D � d � 1933.3 ft
tan 4� �350
D ⇒ D � 5005.23 ft
tan 6.5� �350
d ⇒ d � 3071.91 ft
40.
Distance between towns:
D � d � 18.8 � 7 � 11.8 kilometers
cot 28� �D
10 ⇒ D � 18.8 kilometers
cot 55� �d
10 ⇒ d � 7 kilometers
dD
28°
28°
55°
55°10 km
T2T1
42.
h �18�tan 10���tan 2.5��tan 10� � tan 2.5�
� 1.04 miles � 5515 feet
h tan 10� � h tan 2.5� � 18�tan 10���tan 2.5��
h
tan 2.5��
htan 10�
� 18 �h � 18 tan 10�
tan 10�
x �h
tan 2.5, x �
htan 10�
� 18 x
h
10°
− 18not drawn to scale
x
h2.5°
not drawn to scale
tan 2.5� �hx, tan 10� �
hx � 18
41.
⇒ a � 3.23 miles � 17,054 feet
a cot 16� � a cot 57� �55
6
cot 16� �a cot 57� � �55�6�
a
tan 16� �a
a cot 57� � �55�6�
tan 16� �a
x � �55�6�
tan 57� �a
x ⇒ x � a cot 57�
57°16°
x55060
H
a
P1 P2
©H
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rved
.
49.
� 2��3r
2 � � �3r
y � 2b
b ��3r
2
b � cos 30�rb
r30°
y
r2
x
cos 30� �b
r50.
Distance � 2a � 9.06 centimeters
a � c sin 15� � 17.5 sin 15� � 4.53
sin 15º �a
c
c �35
2� 17.5 c
15°a
356 Chapter 4 Trigonometric Functions
43. (a)
Similarly for the back row,
(b) For you need to be 56 feet away since
arctan 5656
� 45�.
45�,
� � arctan 56150
� 20.5�.
� � arctan 5640
� 54.5�
40
50 + 6 = 56
θ
44. (a) Using the two triangles with acute angle
and
Hence,
(b)
(c) The least value is � � 0.785. �� ��
4�
�2
0
20
0
3
3
L1
L2
β
β
L � L1 � L2 � 3 sec � � 3 csc �.
sin � �3L2
.cos � �3L1
�,
45.
� � arctan 5 � 78.7�
tan � � � �1 � �3�2�1 � ��1��3�2�� � ��5�2
�1�2� � 5
L2: x � y � 1 ⇒ y � �x � 1 ⇒ m2 � �1
L1: 3x � 2y � 5 ⇒ y �3
2 x �
5
2 ⇒ m1 �
3
2
47. The diagonal of the base has a length ofNow, we have:
� � arctan 1
�2� 35.3�
a
2a
θ
tan � �a
�2a�
1
�2
�a2 � a2 � �2a.48.
� � arctan �2 � 54.7º
θa
a 2
tan � �a�2
a� �2
46.
� arctan�323� � 74.7� � arctan� 1
5 � ��2�1 �
15��2��
� � arctan� m2 � m1
1 � m2m1�tan � � � m2 � m1
1 � m2m1�L2 � x � 5y � �4 ⇒ m2 �
1
5
L1 � 2x � y � 8 ⇒ m1 � �2
©H
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rved
.
Section 4.8 Applications and Models 357
51. when
Use since when
Thus, d � 8 sin � t.
2�
�� 2 ⇒ � � �
t � 0.d � 0d � a sin �t
t � 0, a � 8, period � 2d � 0 52. Displacement at is
Amplitude:
Period:
d � 3 sin�� t
3 �
2�
�� 6 ⇒ � �
�
3
�a� � 3
0 ⇒ d � a sin � t.t � 0
54. Displacement at is
Amplitude:
Period:
d � 2 cos�� t
5 �
2�
�� 10 ⇒ � �
�
5
�a� � 2
2 ⇒ d � a cos � tt � 053. when
Use since when
Thus, d � 3 cos�43
� t�.
2�
�� 1.5 ⇒ � �
43
�
t � 0.d � 3d � a cos �t
t � 0, a � 3, period � 1.5d � 3
55.
(a)
(b)
� 4 cycles per unit of time
Frequency ��
2��
8�
2�
Maximum displacement � amplitude � 4
d � 4 cos 8�t
56.
(a) Maximum displacement:
(b) Frequency:
(c) When
(d) Least positive value for t for which :
t ��
2
1
20��
1
40
20� t ��
2
cos 20� t � 0
1
2 cos 20� t � 0
d � 0
t � 5, d �12
cos20� �5� �12
.
�
2��
20�
2�� 10
�a� � �12� �1
2
d �1
2 cos 20� t
(c)
(d) 8�t ��
2 ⇒ t �
1
16
d � 4 cos�8� �5�� � 4
57.
(a)
(b)
(c)
(d) 140�t � � ⇒ t �1
140
d � 0
� 70 cycles per unit of time
Frequency ��
2��
140�
2�
Maximum displacement � amplitude �116
d �1
16 sin 140�t
©H
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.
358 Chapter 4 Trigonometric Functions
58.
(a) Maximum displacement:
(b) Frequency:
(c) When t � 5, d �164
sin792� �5� � 0.
�
2��
792�
2�� 396
�a� � � 1
64� �1
64
d �1
64 sin 792� t
59.
� � 2��264� � 528�
2�
��
1264
Period �2�
��
1
frequency
d � a sin �t 60. At buoy is at its high point
Returns to high point every 10 seconds:
d �7
4 cos
� t
5
Period �2�
�� 10 ⇒ � �
�
5
�a� �7
4
Distance from high to low � 2�a� � 3.5
⇒ d � a cos � t.t � 0,
61.
(a)
0
−0.5
0.5
�
y �1
4 cos 16t, t > 0
(b) Period:2�
16�
�
8 seconds (c)
16t ��
2 ⇒ t �
�
32 seconds.
1
4 cos 16t � 0 when
(d) Least positive value for t for which :
t ��
792��
1
792
792� t � �
sin 792� t � 0
1
64 sin 792� t � 0
d � 0
62. (a)
(c) L � L1 � L2 �2
sin ��
3
cos �
(b)
The minimum length of the elevator is 7.0 meters.
(d) From the graph, it appears that the minimum length is 7.0 meters, which agrees with the estimate of part (b).
00
12
�2
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
©H
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.
Section 4.8 Applications and Models 359
63. (a), (b)
Maximum � 83.1 square feet
Base 1 Base 2 Altitude Area
8 22.1
8 42.5
8 59.7
8 72.7
8 80.5
8 83.1
8 80.78 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�
8 sin 20�8 � 16 cos 20�
8 sin 10�8 � 16 cos 10�
(c)
(d) Maximum area is approximately 83.1 square feet for
00
90
100
� � 60�.
� 64�1 � cos �� sin �
� 128 � 8 � 16 cos �8 sin �
A �12�b1 � b2�h
64. (a)
(c) Period:
This corresponds to the 12 months in a year.Since the sales of outerwear is seasonal, this is reasonable.
(d) The amplitude represents the maximum displacement from the average sale of 8 million dollars. Sales are greatest inDecember and least in June.
(cold weather � holidays)
2�
��6� 12
Month (1 January)↔
Ave
rage
sale
s(i
n m
illio
ns o
f do
llars
)
t2 4 121086
3
6
9
12
15
S (b)
Shift:
The model is a good fit.
S � 8 � 6.3 cos�� t
6 �S � d � a cos bt
d � 14.3 � 6.3 � 8
2�
b� 12 ⇒ b �
�
6
Month (1 January)↔
Ave
rage
sale
s(i
n m
illio
ns o
f do
llars
)
t2 4 121086
3
6
9
12
15
Sa �1
2�14.30 � 1.70� � 6.3
65.
(a)
014
12
22
S�t� � 18.09 � 1.41 sin��t6
� 4.60�(b) The period is months, which is 1 year.
(c) The amplitude is 1.41. This gives the maximum changein time from the average time of sunset.�18.09�
2�
���6� � 12
66. False It means 24 degrees east of north.
67. False. The other acute angle is Then
�a
22.56 ⇒ a � 22.56 tan�41.9��.
tan�41.9�� �oppadj
90� � 48.1� � 41.9�.
©H
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rved
.
360 Chapter 4 Trigonometric Functions
Review Exercises for Chapter 4
2. or 4.4 radians250�1. or 0.7 radian40�
3. (a) (b) Quadrant III
(c)
4�
3� 2� � �
2�
3
4�
3� 2� �
10�
3
x
y
π3
4
4. (a) (b) Quadrant IV
(c)
11�
6� 2� � �
�
6
11�
6� 2� �
23�
6
x
y
π6
11
5. (a) (b) Quadrant III
(c)
�5�
6� 2� � �
17�
6
�5�
6� 2� �
7�
6
x
y
π6
5−
6. (a) (b) Quadrant I
(c)
�7�
4� 2� � �
15�
4
�7�
4� 2� �
�
4
x
y
π4
7−
7. Complement:
Supplement: � ��
8�
7�
8
�
2�
�
8�
3�
8
9. Complement:
Supplement: � �3�
10�
7�
10
�
2�
3�
10�
�
5
8. Complement:
Supplement: � ��
12�
11�
12
�
2�
�
12�
5�
12
10. Complement:
Supplement: � �2�
21�
19�
21
�
2�
2�
21�
17�
42
68. False 70.
3x � 6y � 1 � 0
2y � �x �13
y � 0 � �12�x �
13�69.
4x � y � 6 � 0
y � 2 � 4�x � 1�
71.
4x � 5y � 22 � 0
5y � 10 � �4x � 12
y � 2 � �45
�x � 3�
Slope �6 � 2
�2 � 3� �
45
72.
4x � 3y � 1 � 0
3y � 2 � �4x � 1
y �23
� �43�x �
14�
m �1�3 � 2�3
�1�2 � 1�4�
1�3�4
� �43
73. Domain: ���, �� 74. Domain: ���, �� 75. Domain: ���, �� 76. Domain:or x ≤ 7
7 � x ≥ 0
©H
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.
Review Exercises for Chapter 4 361
11. (a)
(b) Quadrant I
(c)
45� � 360� � �315�
45� � 360� � 405�
x
y
45°
12. (a)
(b) Quadrant III
(c)
210� � 360� � �150�
210� � 360� � 570�
x
y
210°
13. (a)
(b) Quadrant III
(c)
�135� � 360� � �495�
�135� � 360� � 225�
x
y
−135°
14. (a)
(b) Quadrant IV
(c)
�405� � 360� � �45�
�405� � 720� � 315�
x
y
−405°
15. Complement of
Supplement of 180� � 5� � 175�5�:
90� � 5� � 85�5�: 16. Complement of
Supplement of 180� � 84� � 96�84�:
90� � 84� � 6�84�:
17. Complement: not possible
Supplement: 180� � 171� � 9�
18. Complement of not possible
Supplement of 180� � 136� � 44�136�:
136�:
19. 135� 16� 45� � �135 �16
60�
45
3600��
� 135.279� 20. �234� 40� � ��234� �40�
3600� � �234.011�
21. 5� 22� 53� � �5 �22
60�
53
3600��
� 5.381�
23. 135.29� � 135� � �0.29��60�� � 135� 17� 24�
25. �85.36� � ��85 � 0.36�60� �� � �85� 21� 36�
24. 25.8� � 25� 48�
26. �327.93� � �327� 55� 48�
22.
� 280� � 0.13� � 0.01� � 280.147�
280� 8� 50� � 280� �8�
60�
50�
3600
27. 415� � 415� � rad180�
�83�
36 rad � 7.243 rad 28.
� �71�
36 rad � �6.196 rad
�355� � �355� � rad180�
©H
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.
362 Chapter 4 Trigonometric Functions
29. �72� � �72� � rad180�
� �2�
5 rad � �1.257 rad 30. 94� � 94�
� rad180�
�47�
90 rad � 1.641 rad
32. �3�
5� �
3�
5 �180�
� � � �108�
34. 1.55 � 1.55�180�
� � � 88.808�
36. radianss � r ⇒ �sr
�24560
� 4.083 �4912
38. cms � r � �15� �3
� 5� � 15.71
31.5�
7�
5�
7 �180�
� � � 128.571�
33. �3.5 � �3.5�180�
� � � �200.535�
35.
�2512
� 2.083
25 � 12
s � r
37.
s � 48.171 m
s � 20�138�� �
180�
s � r
39. In one revolution, the arc length traveled isThe time
required for one revolution is
Linear speed �st
�12�
3�25� 100� cm�sec
t �1
500 minutes �
1500
�60� �325
seconds.
s � 2�r � 2� �6� � 12� cm.40. (a) 28 miles per hour ft/min
Circumference of wheel:
Number of revolutions per minute:
(b) rad/min
t�
73927�
�2�� �14,724
7� 2112
2464�7�3�� �
73927�
� 336.1 rev�min
C � ��213� �
73
�
�28�5280�
60� 2464
41. corresponds to �22
, �22 �.t �
7�
442.
�x, y� � ��22
, 22 �
cos 3�
4� �
22
, sin 3�
4�
22
43.
�x, y� � ��32
, 12�
sin 5�
6�
12
cos 5�
6� �
32
44. corresponds to ��12
, �32 �.t �
4�
3
45. corresponds to ��12
, �32 �.t � �
2�
346. corresponds to ��3
2,
12�.t � �
7�
6
©H
ough
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Miff
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ny. A
ll rig
hts
rese
rved
.
Review Exercises for Chapter 4 363
47. corresponds to ��22
, 22 �.t � �
5�
448. corresponds to ��3
2, �
12�.t � �
5�
6
49.
csc 7�
6� �2tan
7�
6�
13
�33
sec 7�
6� �
23
� �23
3cos
7�
6� �
32
cot 7�
6� 3sin
7�
6� �
12
50.
sec �
4� 2 � csc
�
4
tan �
4� 1 � cot
�
4
sin �
4�
22
� cos �
4
51. corresponds to
undefined
undefinedcsc 2� �1y,tan 2� �
yx
� 0
sec 2� �1x
� 1cos 2� � x � 1
cot 2� �xy,sin 2� � y � 0
�1, 0�.t � 2� 52. corresponds to
undefined
undefinedcsc���� �1y,tan���� �
yx
� 0
sec���� �1x
� �1cos���� � x � �1
cot���� �xy,sin���� � y � 0
��1, 0�.t � ��
53. corresponds to
csc��11�
6 � �1y
� 2
sec��11�
6 � �1x
�23
3
cot��11�
6 � �xy
� 3
tan��11�
6 � �yx
�33
cos��11�
6 � � x �32
sin��11�
6 � � y �12
�32
, 12�.t � �
11�
654. corresponds to
csc��5�
6 � �1y
� �2
sec��5�
6 � �1x
� �23
3
cot��5�
6 � �xy
� 3
tan��5�
6 � �yx
�33
cos��5�
6 � � x � �32
sin��5�
6 � � y � �12
��32
, �12�.t � �
5�
6
55. corresponds to
undefined
undefined csc���
2� �1y
� �1tan���
2� �yx,
sec���
2� �1x,cos��
�
2� � x � 0
cot���
2� �xy
� 0sin���
2� � y � �1
�0, �1�.t � ��
2
©H
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Miff
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rved
.
364 Chapter 4 Trigonometric Functions
58. cos 4� � cos 0 � 1
56. corresponds to
csc���
4� �1y
� �2tan���
4� �yx
� �1
sec���
4� �1x
� 2cos���
4� � x �22
cot���
4� �xy
� �1sin���
4� � y � �22
�22
, �22 �.t � �
�
4
57. sin�11�
4 � � sin�3�
4 � �22
59. sin��17�
6 � � sin�7�
6 � � �12
60. cos��13�
3 � � cos 5�
3�
12
61.
(a)
(b) csc��t� �1
sin��t� � �53
sin��t� � �sin t � �35
sin t �35
62.
(a)
(b) sec��t� �1
cos��t� �135
cos��t� � cos t �513
cos t �5
13
65. cot 2.3 �1
tan 2.3� �0.8935
67. cos 5�
3�
12
66. sec 4.5 �1
cos 4.5� �4.7439
68. tan��11�
6 � � 0.5774
63.
(a)
(b) csc t �1
sin t�
32
sin t � �sin��t� � ���23� �
23
sin��t� � �23
64.
(a)
(b) sec��t� �1
cos��t� �85
cos t � cos��t� �58
cos��t� �58
69. The opposite side is
cot �1
tan �
465
�465
65tan �
oppadj
�65
4
sec �1
cos �
94
cos �adjhyp
�49
csc �1
sin �
965
�965
65sin �
opphyp
�65
9
92 � 42 � 81 � 16 � 65.
©H
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Miff
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ny. A
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hts
rese
rved
.
Review Exercises for Chapter 4 365
70.
cot �4
5
sec �41
441�
41
4
csc �41
541�
41
5
tan �15
12�
5
4
cos �12
341�
441
4115
12
θ
3 41
sin �15
341�
541
41
72.
102
θ96 = 4 6
cot �12
6� 26tan �
2
46�
6
12
sec �5
26�
56
12cos �
46
10�
26
5
csc � 5sin �2
10�
1
5
71. The hypotenuse is .
cot �1
tan �
6
5
sec �1
cos �
61
6
csc �1
sin �
61
5
tan �opp
adj�
10
12�
5
6
cos �adj
hyp�
12
261�
6
61�
661
61
sin �opp
hyp�
10
261�
5
61�
561
61
122 � 102 � 244 � 261
73. csc tan �1
sin
sin cos
�1
cos � sec
75. (a)
(b) sin 6� � 0.1045
cos 84� � 0.104574.
cot � tan cot
� 1 �tan cot
� 1 � tan2 � sec2
76. (a)
(b) �1
cos�54.1167�� � 1.7061 sec�54� 7� � �1
cos�54� 7� �
�1
sin 52.2�� 1.2656 csc�52� 12� � �
1sin�52� 12� �
77. (a)
(b) sec �
4� 1.4142
cos �
4� 0.7071 79.
� 235 feet
x � 125 tan 62�
tan 62� ��
12578. (a)
(b)
� 1.9626
cot�3�
20� �1
tan�3��20�
tan�3�
20� � 0.5095
80. (a) (b)
(c) feeth � 152�12� � 76
sin 30� � h
152 ⇒ h � 152 sin 30�
152 ft
30�
h
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.
366 Chapter 4 Trigonometric Functions
81.
cot �x
y�
3
4tan �
y
x�
4
3
sec �r
x�
5
3cos �
x
r�
3
5
csc �r
y�
5
4sin �
y
r�
4
5
x � 12, y � 16, r � 144 � 256 � 400 � 20
83.
cot � �7
2
sec � �53
7
csc �53
2
tan �y
x� �
2
7
cos �x
r� �
7
53� �
753
53
sin �y
r�
2
53�
253
53
x � �7, y � 2, r � 49 � 4 � 5382.
cot �xy
�15
sec �rx
� 26
csc �ry
�26
5
tan �yx
�102
� 5
cos �xr
�2
226 �
2626
sin �yr �
10
226�
52626
r � 4 � 100 � 104 � 226x � 2, y � 10,
85.
cot �x
y�
16
15
sec �r
x�
481
16
csc �r
y�
481
15
tan �y
x�
5�8
2�3�
15
16
cos �x
r�
2�3481�24
�16481
481
sin �y
r�
5�8481�24
�15481
481
r ��23�
2� �5
8�2
�481
24x �
2
3 and y �
5
8,84.
cot �ry
� �34
sec �rx
�53
csc �xy
� �54
tan �yx
� �43
cos �xr
� 35
sin �yr � �
45
r � 32 � ��4�2 � 5x � 3, y � �4,
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.
Review Exercises for Chapter 4 367
86.
cot �xy
� 5 tan �yx
��2�3
�10�3�
15
sec �rx
� �26
5cos �
xr
� �10�3
226�3�
�526
��526
26
csc �ry
� �26 sin �yr �
�2�3
226�3�
�126
��26
26
r ���103 �2
� ��23�
2�
2263
x � �103
, y � �23
,
87. is in Quadrant IV.
cot � �511
11tan �
y
x� �
11
5
sec �6
5cos �
x
r�
5
6
csc � �611
11sin �
y
r� �
11
6
r � 6, x � 5, y � �36 � 25 � �11 x
6− 11
5θ
ysec �6
5, tan < 0 ⇒
88.
cot �x
y� �
5
12
sec �r
x�
13
�5� �
13
5cos �
x
r� �
5
13
csc �r
y�
13
12sin �
y
r�
12
13
sin > 0 ⇒ y � 12, x � �5tan �y
x� �
12
5 ⇒ r � 13,
89. is in Quadrant II.
cot � �55
3tan �
y
x� �
3
55� �
355
55
sec � �8
55� �
855
55cos �
x
r� �
55
8
csc �8
3sin �
y
r�
3
8
y � 3, r � 8, x � �55
x− 55
38 θ
ysin �3
8, cos < 0 ⇒
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.
368 Chapter 4 Trigonometric Functions
90.
csc �r
y�
5
21�
521
21
cot �x
y�
�2
21� �
221
21tan �
y
x� �
21
2
sec �r
x�
5
�2� �
5
2sin �
y
r�
21
5
sin > 0 ⇒ y � 21cos �x
r�
�2
5 ⇒ y � ±21,
91. Reference angle:
264� � 180� � 84�
x
y
264�=θ
84�=θ ′
93. Coterminal angle:
Reference angle:
� �4�
5�
�
5
2� �6�
5�
4�
5
x
y
θ = π5
6−
= π5θ ′
92. is coterminal with
Reference angle:
360� � 275� � 85�
x
y
= 635°θ
85�=θ ′
275�.635�
94. is coterminal with
Reference angle:
2� �5�
3�
�
3x
y
π3
17
= π3
θ
=θ
′
5�
3.
17�
3
95. is in Quadrant III with reference angle
tan 240� ��3�2
�1�2� 3
cos 240� � �cos 60� � �12
sin 240� � �sin 60� � �32
60�.240� 96. is in Quadrant IV with reference angle
tan 315� � �tan 45� � �1
cos 315� � cos 45� �22
sin 315� � �sin 45� � �22
45�.315�
97. is coterminal with in Quadrant II withreference angle
tan��210�� �1�2
�3�2� �
13
� �33
cos��210�� � �cos�30�� � �32
sin��210�� � sin�30�� �12
30�.150��210� 98. is coterminal with in Quadrant I.
tan��315�� � 1
cos��315�� �22
sin��315�� �22
45��315�
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.
Review Exercises for Chapter 4 369
99. is coterminal with in Quadrant IVwith reference angle
tan��9�
4 � ��2�22�2
� �1
cos��9�
4 � � cos��
4� �22
sin��9�
4 � � �sin��
4� � �22
��4.7��4�9��4
101.
tan�4�� � 0
cos�4�� � 1
sin�4�� � sin�0� � 0
100. is in Quadrant IV.
tan�11�
6 � � �33
cos�11�
6 � �32
sin�11�
6 � � �12
11��6
102.
tan�7�
3 � � 3
cos�7�
3 � �12
sin�7�
3 � � sin��
3� �32
103. tan 33� � 0.6494
105. sec 12�
5�
1cos�12��5� � 3.2361104. csc 105� �
1sin 105�
� 1.0353
107.
Amplitude: 3
1
−4
π
2
3
4
π2x
yf �x� � 3 sin x 108.
Amplitude: 23
π−π−1
−2
−3
π2π2−
2
x
yf �x� � 2 cos x
109.
Amplitude: 14
1
−1
π2π2−
21
21−
x
yf �x� �14 cos x 110.
Amplitude: 72
1
−4
π
2
3
4
π2x
yf �x� �72 sin x
106. sin���
9� � �0.3420
111. Period:
Amplitude: 5
2�
�� 2 113. Period:
Amplitude: 3.4
2�
2� �112. Period:
Amplitude:3
2
2�
�1�2�� 4� 114. Period:
Amplitude: 4
2�
���2�� 4
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.
370 Chapter 4 Trigonometric Functions
115.
Amplitude: 3
Period:2�
2�� 1
−3
3
1
−2
x
y
12
112
−
y � 3 cos 2�x
117.
Amplitude: 5
Period:2�
2�5� 5�
–6
–2
2
4
6
xπ6
yf�x� � 5 sin 2x
5
116.
Amplitude:
Reflected in x-axis
�2 � 2
Period: 2�
�� 2
–3
–2
–1
1
3
x1
yy � �2 sin �x
118.
Amplitude: 8
Reflected in y-axis
Period: 2�
�1�4�� 8�
–8
–6
–4
8
xππ 84
y
f �x� � 8 cos��x
4�
x 0
y 2 0 �2
12�
12
x 0
y 0 8 0 �8�8
4�2��2��4�
119.
Amplitude:
Period:2�
1�4� 8�
52 1
−3
2
3
5
π4 π8π4−
4
−4
−5
x
yf �x� � �
52
cos�x4�
121.
Amplitude:
Period:
Shift:
x � � x � 3�
x � � � 0 and x � � � 2�
2�
5
2
f �x� �5
2 sin�x � ��
–4
–3
–2
–1
1
3
4
xπ
y
120.
Period: 8
Amplitude: 12
1
−1
21−
2 6−2−6x
y
f �x� � �12
sin � x4
122.
Amplitude: 3
This is the graph ofshifted
to the left units.�y � 3 cos x
Period: 2�
–4
–3
1
2
3
4
xπ
yf �x� � 3 cos�x � ��
0
3 0 0 3�3f �x�
��
2�
�
2��x
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.
Review Exercises for Chapter 4 371
123.
x
y
−1−2−3−4 1 2 3 4−1
−2
−3
−4
2
3
4
f �x� � 2 � cos �x2
125.
x
y
−2
−3
−4
2
3
4
π3π2π π−
f�x� � �3 cos�x2
��
4�
124.
Vertical shift downward three units
Period: 2
Amplitude: 12
−4
21−2 −1
−3
−2
−1
x
y
f �x� �12
sin � x � 3
126.
Period: �
24
2
8
− π2
π2
π4
− π4
6
−2
x
yAmplitude: 2
f �x� � 4 � 2 cos�4x � ��
130.
Amplitude:
Period: 2 ⇒ f �x� �1
2 cos �x
1
2
f �x� � a cos�bx � c� 131.
Maximum sales:(June)
Minimum sales:(December)
t � 12
t � 6
040
12
56S � 48.4 � 6.1 cos �t6
132.
Maximum sales:(March)
Minimum sales:(September)
t � 9
t � 3
045
12
70S � 56.25 � 9.50 sin � t6
127. f�x� � �2 cos�x ��
4� 128.
Amplitude: 3
Period: � ⇒ f �x� � 3 cos�2x�
f�x� � a cos�bx � c� 129. f�x� � �4 cos�2x ��
2�
133.
Asymptotes:
Reflected in axisx-
x � �2, x � 2
Period: �
���4�� 4
−4
4
2
6
−2
−4
−6
4−6 −2 2 6x
yf�x� � �tan �x
4
0 1
1 0 �1y
�1x
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.
372 Chapter 4 Trigonometric Functions
137.
Period:
Two consecutive asymptotes:
x2
� � ⇒ x � 2�
x2
� 0 ⇒ x � 0
�
1�2� 2�
2
4
6
π4π2−2π−4πx
yf �x� � 3 cot x2
141.
Period:
x
y
π2
−1
−2
1
2
2�
f �x� �14
sec x 142.
x
y
−2
1
2
π π2 π3π−
f �x� �12
csc x �1
2 sin x
138.
1−1
1
−2 2
2
x
y
f �x� �12
cot � x2
�1
2 tan � x2
139.
Period:
Two consecutive asymptotes:
x ��
2� � ⇒ x �
3�
2
x ��
2� 0 ⇒ x �
�
2
�
π
3
4
π π2π2 −−
5
−1
x
yf �x� �12
cot�x ��
2�
134.
2
4
6
1x
y
f �x� � 4 tan � x 135.
1
2
3
4
−2
−3
−4
−1
x
y
π2
π2
3π2
3−
f �x� �14
tan�x ��
2� 136.
2
4
6
π−2π
−4
−6
π3 π4−3πx
y
f �x� � 2 � 2 tan x3
140.
π−ππ−2 π2
4
x
y
f �x� � 4 cot�x ��
4� �4
tan�x ��
4�
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rved
.
Review Exercises for Chapter 4 373
143.
Period:
3
2
1
x
y
π
�
f �x� �14
csc 2x 144.
1−1
−1
x
y
f �x� �12
sec 2�x �1
2 cos 2�x
145.
Secant function shifted
to right�
4 π π2π−
1
3
2
x
yf �x� � sec�x ��
4� 146.
−1
π π− π2
π2
−x
y
f �x� �12
csc�2x � �� �1
2 sin�2x � ��
147.
−1
ππ−
1
f�x� �14
tan �x2
148.
4
−4
−2π 2π
f�x� � tan�x � �
4�
149.
−10
ππ−
10
f�x� � 4 cot�2x � �� �4
tan�2x � ��150.
−
−
π2
π2
4
4
f�x� � �2 cot�4x � �� � �2
tan�4x � ��
151.
−10
ππ−
10
22
f �x� � 2 sec�x � �� �2
cos�x � �� 152.
−2π π2
8−
8
f�x� � �2 csc�x � �� ��2
sin�x � ��
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.
374 Chapter 4 Trigonometric Functions
153.
−4
4
ππ−
f�x� � csc�3x ��
2� �1
sin�3x � ���2��
161. (a) because
(b) because cos 5�
6� �
32
.arccos��32 � �
5�
6
cos �
4�
22
.arccos�22 � �
�
4162. (a)
(b) arctan�1� ��
4
arctan��3� � ��
3
154.
π π
8−
−
8
f�x� � 3 csc�2x ��
4� �3
sin�2x � ���4��
155.
Damping factor:
−20
20
4−4
y � ex
f �x� � ex sin 2x 156.
Damping factor:
−5
5
4−4
ex
f �x� � ex cos x
157.
Damping factor:
As oscillates between and �2x.2xx →�, f
−14
14
10−10
g�x� � 2x
f �x� � 2x cos x 158.
As oscillatesbetween and x.� x
x →�, f
−6
6
9−9
f �x� � x sin� x
159. (a) because
(b) does not exist because the domain ofarcsin is ��1, 1�.arcsin 4
sin���
2� � �1.arcsin��1� � ��
2160. (a) because
(b) because
sin���
3� � �32
.
arcsin��32 � � �
�
3
sin���
6� � �12
.arcsin��12� � �
�
6
164. arcsin 0.63 � 0.68163. arccos�0.42� � 1.14 165. sin�1��0.94� � �1.22
167. arctan��12� � �1.49
169. tan�1�0.81� � 0.68
166. cos�1��0.12� � 1.69 168. arctan 21 � 1.52
170. tan�1 6.4 � 1.42
171. sin �x � 3
16 ⇒ � arcsin�x � 3
16 � 172. tan �x � 1
20 ⇒ � arctan�x � 1
20 �
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.
Review Exercises for Chapter 4 375
173. Let Then,
and
��x2 � 2x
�x2 � 2x.
1 − (x − 1)2
x − 1
y
1
sec y �1
�x2 � 2x
sin y � �x � 1� �x � 1
1
y � arcsin�x � 1�.
175. Let Then and
�24 � 2x2
4 � x2.
(4 − x2)2 − (x2)24 − x2
x2
y
�16 � 8x2
4 � x2
sin y ��4 � x2�2 � �x2�2
4 � x2.
cos y �x2
4 � x2y � arccos
x2
4 � x2.
174. Let
�4 � x2
x 4 − x2
x
2
u
tan�arccos x
2� � tan u
cos u �x
2.u � arccos
x
2,
176. Let
�hyp
opp�
1
10x1 10x
u
1 100− x 2
csc�arcsin 10x� � csc u
sin u � 10x.u � arcsin 10x,
177.
or 71 meters a � 3.5 sin�1� 10� � � 3.5 sin�7�
6 � � 0.0713not drawn to scale
3.5a
1° 10'
sin�1� 10� � �a
3.5
179.
The towns are approximately 50,000 feet apart or 9.47 miles.
x � 59,212.4 � 9225.1 � 49,987.2 feet
tan 58� �x � y
37,000 ⇒ x � y � 37,000 tan 58� � 59,212.4 feet
37,000
x y
32°
14°
76°58°
tan 14� �y
37,000 ⇒ y � 37,000 tan 14� � 9225.1 feet
178.
(answer depends on angle accuracy) h � 4 sin�0.1194� � 0.48 miles or 2534 feet
sin�� �h4
h4
θ
� arctan� 12100� � 0.1194 or 6.84�
tan �12100
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.
376 Chapter 4 Trigonometric Functions
180.
The distance is 1221 miles and the bearing is N 85.6 E.
sec 4.4� �D
1217 ⇒ D � 1217 sec 4.4� � 1221
tan �93
1217 ⇒ � 4.4�
sin 25� �d4
810 ⇒ d4 � 342
cos 48� �d3
650 ⇒ d3 � 435
cos 25� �d2
810 ⇒ d2 � 734
48°
48°
65°25°
d3d4
d d1 2
810650
D
B
CA θ
N
S
W E
sin 48� �d1
650 ⇒ d1 � 483
��
d1 � d2 � 1217
d3 � d4 � 93
181. Use cosine model with amplitude 3 feet.
Period: 15 seconds
y � 3 cos�2�
15t�
183. False. is a function, but it is not one-to-one.
y � sin
182. Use a cosine model with amplitude
Period: 3 seconds
y � 0.75 cos�2�
3t�
1.52
� 0.75.
184. False. The sine and cosine functions are useful formodeling simple harmonic motion.
185.
(a)
(b) increases at an increasing rate. The function is not linear.
tan �0.672s2
3000
s 10 20 30 40 50 60
38.88�29.25�19.72�11.40�5.12�1.28�
186. (a)
−
−10
10
��
(b) Next term:
The accuracy of the approximation increases as more terms are added.
arctan x � x �x3
3�
x5
5�
x7
7�
x9
9−
−10
10
��
x9
9©
Hou
ghto
n M
ifflin
Com
pany
. All
right
s re
serv
ed.
Practice Test for Chapter 4 377
Chapter 4 Practice Test
1. Express 350 in radian measure.� 2. Express in degree measure.�5���9
3. Convert to decimal form.135� 14� 12� 4. Convert to form.D� M� S��22.569�
5. If use the trigonometric identities to find tan �.
cos � �23, 6. Find given sin � � 0.9063.�
7. Solve for in the figure below.
20°
35
x
x 8. Find the magnitude of the reference angle for� � �6���5.
9. Evaluate csc 3.92. 10. Find given that lies in Quadrant III andtan � � 6.
�sec �
11. Graph y � 3 sin x
2. 12. Graph y � �2 cos�x � ��.
13. Graph y � tan 2x. 14. Graph y � �csc�x ��
4�.
15. Graph using a graphing calculator.y � 2x � sin x, 16. Graph using a graphing calculator.y � 3x cos x,
17. Evaluate arcsin 1. 18. Evaluate arctan��3�.
19. Evaluate sin�arccos 4
�35�. 20. Write an algebraic expression for cos�arcsin x
4�.
For Exercises 21–23, solve the right triangle.
21. A � 40�, c � 12 22.
A C
B
ca
b
B � 6.84�, a � 21.3 23. a � 5, b � 9
24. A 20-foot ladder leans against the side of a barn. Find the height of the top of the ladder if the angle of elevation of the ladder is 67�.
25. An observer in a lighthouse 250 feet above sea level spots a ship off the shore. If the angle of depression to the ship is how far out is the ship?5�,
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