CHAPTER 4 Trigonometric Functions · 274 Chapter 4 Trigonometric Functions 12. (a) Coterminal...

107
CHAPTER 4 Trigonometric 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 © Houghton Mifflin Company. All rights reserved.

Transcript of CHAPTER 4 Trigonometric Functions · 274 Chapter 4 Trigonometric Functions 12. (a) Coterminal...

Page 1: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 2: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 3: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 4: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 5: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 6: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 7: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 8: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 9: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 10: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 11: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 13: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 14: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 15: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 16: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 17: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 18: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 19: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 20: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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

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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rved

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Page 21: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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.

©H

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ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 22: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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.

©H

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ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 23: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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

©H

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ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 24: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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

©H

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ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 25: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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 �

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 26: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 27: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 28: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 29: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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

©H

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ton

Miff

lin C

ompa

ny. A

ll rig

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rved

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Page 30: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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�

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

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Page 31: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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�

©H

ough

ton

Miff

lin C

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ny. A

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Page 32: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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

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Page 33: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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

.

Page 34: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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

.

Page 35: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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

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hts

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rved

.

Page 36: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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

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ny. A

ll rig

hts

rese

rved

.

Page 37: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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

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

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ny. A

ll rig

hts

rese

rved

.

Page 38: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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, �� �

Hou

ghto

n M

ifflin

Com

pany

. All

right

s re

serv

ed.

Page 39: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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

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Page 40: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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 ⇒ �

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Page 41: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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 ⇒

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Page 42: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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

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Page 43: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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

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Page 44: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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

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Page 45: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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

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Page 46: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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

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Page 47: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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

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Page 48: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Miff

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Page 49: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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

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

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Page 51: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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

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Page 52: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 53: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Miff

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Page 54: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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.

Page 55: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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

ough

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Miff

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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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Page 57: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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!

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Page 59: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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

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Page 60: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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-

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Page 61: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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

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Page 62: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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�

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Page 63: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 65: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 66: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 68: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 69: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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 �

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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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Page 71: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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

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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Page 72: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 73: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 74: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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.

Page 75: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 76: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 77: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 78: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 79: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 80: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 81: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 82: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 83: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 84: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 85: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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

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Page 86: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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

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Page 87: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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

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Page 88: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Miff

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Page 89: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 90: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 91: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 92: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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

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Page 93: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 94: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 95: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 96: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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,

©H

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Page 97: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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

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 ⇒

©H

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Page 98: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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

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�

©H

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Page 99: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 100: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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

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

yf �x� � 3 cos�x � ��

0

3 0 0 3�3f �x�

��

2�

2��x

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Page 101: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 102: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 103: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 104: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 105: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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Page 106: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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

Hou

ghto

n M

ifflin

Com

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. All

right

s re

serv

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Page 107: CHAPTER 4 Trigonometric Functions · 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 8 4: 7 7 6 2 4 5 9 6 7 6

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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