Chapter 4 16.college.cengage.com/mathematics/larson/precalculus_limits/1e2/... · Vocabulary Check (page 290) 1. Trignometry 2. angle 3. coterminal 4. radian 5. acute; ... 1 3 tan

Precalculus with Limits, Answers to Section 4.1 1

Chapter 4Section 4.1 (page 290)

Vocabulary Check (page 290)1. Trignometry 2. angle 3. coterminal4. radian 5. acute; obtuse6. complementary; supplementary 7. degree8. linear 9. angular 10.

1. 2 radians 2. 5.5 radians 3. radians

4. radians 5. 1 radian 6. 6.5 radians

7. (a) Quadrant I (b) Quadrant III

8. (a) Quadrant III (b) Quadrant III

9. (a) Quadrant IV (b) Quadrant III

10. (a) Quadrant IV (b) Quadrant II

11. (a) Quadrant III (b) Quadrant II

12. (a) Quadrant IV (b) Quadrant II

13. (a) (b)

14. (a) (b)

15. (a) (b)

16. (a) (b)

17. (a) (b)

18. (a) (b)

19. (a) (b)

20. (a) (b)

21. (a) Complement: Supplement:

(b) Complement: none; Supplement:

22. (a) Complement: Supplement:


23. (a) Complement:

Supplement:


24. (a) Complement: none; Supplement:

(b) Complement:

Supplement:

25. 26. 27. 28.

29. 30.

31. (a) Quadrant II (b) Quadrant IV32. (a) Quadrant I (b) Quadrant III33. (a) Quadrant III (b) Quadrant I34. (a) Quadrant II (b) Quadrant IV35. (a) (b)

150°

x

y

x30°

y

10�165�

�330��60�120�210�

� � 1.5 � 1.64

�

2� 1.5 � 0.07;

� � 3 � 0.14

� � 2 � 1.14

� � 1 � 2.14

�

2� 1 � 0.57;

�

12

11�

125�

12;

�

4

2�

3

�

6;

28�

15, �

32�

15

7�

4, �

�

4

25�

12, �

23�

128�

3, �

4�

3

�

6, �

23�

6

19�

6, �

5�

6

17�

6, �

7�

6

13�

6, �

11�

6

7π

x

y

4

x

y

−3

x

y

116π

x

y

x

y

52π

− 74π

y

x

− 23π

x

y

54π

x

y

�4

�3

A �12r2�

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333202CB04_AN.qxd 1/1/70 09:38 AM Page 1

(Continued)

36. (a) (b)

37. (a) (b)

38. (a) (b)

39. (a) (b)40. (a) (b)41. (a) (b)42. (a) (b)43. (a) Complement: Supplement:

(b) Complement: none; Supplement:44. (a) Complement: Supplement:

(b) Complement: Supplement:45. (a) Complement: ; Supplement:

(b) Complement: none; Supplement:46. (a) Complement: none; Supplement:


47. (a) (b) 48. (a) (b)

49. (a) (b) 50. (a) (b)

51. (a) (b) 52. (a) (b)53. (a) (b) 54. (a) (b)55. 2.007 56. 1.525 57. 58.

59. 9.285 60. 6.021 61. 62. 0.00963. 64. 65.66. 67. 68.69. 70.71. (a) (b)72. (a) (b)73. (a) (b)74. (a) (b)75. (a) (b)76. (a) (b)77. (a) (b)78. (a) (b)

79. radians 80. radians 81. radians

82. radian 83. radian 84. radian

85. radians 86. 2 radians

87. 88.89. 3 meters 90.

91.

92.93. 12.27 square feet 94. 5.64 square miles95. 591.3 miles 96. 686.2 miles97. 98.99. 100.

101. (a) 728.3 revolutions per minute(b) 4576 radians per minute

102. (a) radians per minute; radians per minute(b) 850 revolutions per minute

103. (a)

(b) feet per minute feet per minute104. (a)

(b)105. (a) radians per minute

(b) centimeters per minute106.

107.

108. (a) feet per second; miles per hour

(b) (c)

(d) The functions are both linear.109. False. A measurement of radians corresponds to two

complete revolutions from the initial to the terminal sideof an angle.

4�

d �7�

7920td �

7�

7920n

�1014�

3

A � 476.39� square meters � 1496.62 square meters

140°

35

A � 175� square inches � 549.8 square inches�2400�, 6000��400�, 1000��200� feet per minute � 628.3 feet per minute 8� radians per minute � 25.13 radians per minute

� 9869.849425��3 � 32,672.56 radians per minute

10,400� radians per minute

1700�3400�

275�5

12 radian0.063 radian � 3.59�0.071 radian � 4.04�

18� square millimeters � 56.55 square millimeters

8�

3 square inches � 8.38 square inches

5� centimeters � 15.71 centimeters3� feet � 9.42 feet15� inches � 47.12 inches

5029

47

29�

45

327

2910

65

0� 47� 11.4��0� 21� 18�

�3� 34� 48�2� 30�

0� 27��345� 7� 12�

�145�48�240� 36�

�408.272��135.01�

330.007�85.308�

2.2�245.167�

�128.5�54.75�

�32.659��114.592�

864.000��756.000�1170.000�

337.500�81.818�25.714�

�0.014

�0.842�3.776408�330��66�420�

20��105�210�270�

4�

5�

3�

2�

4�

3�

�

9

2�

3

7�

45�

6�

6

10�

50�

30�

101�11�

116�26�;177�87�;

65�

162�72�;590�, �130�300�, �60�

180�, �540�600�, �120�

300�, �60�480�, �240�

324�, �396�405�, �315�

−600°

x

y

−750°x

y

480°

x

y

405°

x

y

−120°

x

y

−270°

x

y


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333202CB04_AN.qxd 1/1/70 09:38 AM Page 2


(Continued)

110. True. Let and represent coterminal angles, and let represent an integer.

111. False. The terminal side of the angle lies on the -axis.112. (a) The vertex is at the origin and the initial side is on the

positive axis.(b) Clockwise rotation of the terminal side(c) Two angles in standard position where the terminal

sides coincide(d) The magnitude of the angle is between and

113. Increases. The linear velocity is proportional to the radius.114. Radian. 115. The arc length is increasing. If is constant, the length of

the arc is proportional to the radius

116. Answers will vary. 117. 118.

119. 120.

121. 122.

123. 124.

3

2

1

−1

−2

−3

21−3−4−5x

y

y = x5

y = − (x + 3)5

321−2−3

6

5

4

3

1

−1

−2

−3

x

y

y = x5

y = 2 − x5

4

2

−2

−6

32−2−3 1

y = x5

y = x5 − 4

x

y

432−2

3

2

1

−1

−2

−3

x

yy = x5

y = (x − 2)5

4�132�10

5�24

�22

�s � r� �.�

1 radian � 57.3�

180�.90�

x-

x � � n�360�� n�360��

n

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333202CB04_AN.qxd 1/1/70 09:38 AM Page 3

Section 4.2 (page 299)

Vocabulary Check (page 299)1. unit circle 2. periodic3. period 4. odd; even

1.

2.

3.

4.

5. 6. 7.

8. 9. 10.

11. 12.

13. 14.

15. 16.

17. 18.

19. 20.

21.

is undefined.

23.

24.

25.

is undefined.

is undefined.

26.

is undefined.

is undefined.

27.

28.

29. 30.

31.

32.

33.

34. sin 19�

6� sin

7�

6� �

12

cos�15�

2 � cos �

2� 0

sin 9�

4� sin

�

4�

�22

cos 8�

3� cos

2�

3� �

12

cos 5� � cos � � �1sin 5� � sin � � 0

cot 7�

4� �1tan

7�

4� �1

sec 7�

4� �2cos

7�

4�

�22

csc 7�

4� ��2sin

7�

4� �

�22

cot4�

3 ��33

tan4�

3 � �3

sec4�

3 � �2cos4�

3 � �12

csc4�

3 � �2�3

3sin4�

3 � ��32

cot 3�

2� 0tan

3�

2

sec 3�

2cos

3�

2� 0

csc 3�

2� �1sin

3�

2� �1

cot��

2 � 0tan� �

2sec�

�

2cos��

2 � 0

csc��

2 � �1sin��

2 � �1

cot 5�

6� ��3tan

5�

6� �

�33

sec 5�

6� �

2�33

cos 5�

6� �

�32

csc 5�

6� 2sin

5�

6�

12

cot 3�

4� �1tan

3�

4� �1

sec 3�

4� ��2cos

3�

4� �

�22

csc 3�

4� �2sin

3�

4�

�22

tan�3�

2 cos�

3�

2 � 0

sin�3�

2 � 1

tan 5�

3� ��3tan

11�

6� �

�33

cos 5�

3�

12

cos 11�

6�

�32

sin 5�

3� �

�32

sin 11�

6� �

12

tan�4�

3 � ��3tan�7�

4 � 1

cos�4�

3 � �12

cos�7�

4 ��22

sin�4�

3 ��32

sin�7�

4 ��22

tan��

4 � �1tan��

6 � ��33

cos��

4 ��22

cos��

6 ��32

sin��

4 � ��22

sin��

6 � �12

tan �

3� �3tan

�

4� 1

cos �

3�

1

2cos

�

4�

�22

sin �

3�

�3

2sin

�

4�

�22

��1, 0��0, �1�

12

, ��32 �

12

, ��32 �

�22

, ��22

��32

, �121

2, �32 �2

2, �22

cot � �43tan � �

34

sec � � �54cos � � �

45

csc � � �53sin � � �

35

cot � � �125tan � � �

512

sec � �1312cos � �

1213

csc � � �135sin � � �

513

cot � �125tan � �

512

sec � �1312cos � �

1213

csc � �135sin � �

513

cot � � �815tan � � �

158

sec � � �178cos � � �

817

csc � �1715sin � �

1517


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tan��2�� 0cos��2�� 1sin��2�� 0

333202CB04_AN.qxd 1/1/70 09:39 AM Page 4


(Continued)

35.

36.

37. (a) (b) 38. (a) (b)

39. (a) (b) 40. (a) (b)

41. (a) (b) 42. (a) (b)43. 0.7071 44. 1.7321 45. 1.0378 46. 0.642147. 48. 49. 1.394050. 51. 52.53. (a) (b) 54. (a) 0.7 (b)55. (a) 0.25 or 2.89 (b) 1.82 or 4.4656. (a) 4.0 or 5.4 (b) 0.72 or 5.5657. (a)

(b) (c) The displacement decreases.58. (a) 0.25 foot (b) 0.02 foot (c) foot59. False. means that the function is odd, not

that the sine of a negative angle is a negative number.60. True. The tangent function has a period of 61. (a) -axis symmetry (b)

(c)

62. Answers will vary. 63.

64. 65.

66.

67. 68.

69. 70.y

x−4 46 82−2−6

−2

y

x−1−5−6 1 2

−1

−2

−3

−4

1

2

3

4

−2

y

x−4 4 6 8

−2

−4

−6

−8

2

4

6

8

2−2

y

x−2−4−6 2 4 6 8 10

−2

−4

−6

−8

4

6

8

2

f �1�x� �2�2x � 1�

x � 1

f �1�x� � �x2 � 4, x ≥ 0f �1�x� � 3�4�x � 1�f �1�x� �

23 �x � 1�

cos�� t1� � �cos t1

sin t1 � sin�� t1�y�.

sin��t� � �sin t�0.25

t � 5.5

�0.8�0.4�1�0.7833�1.4486�4.4014

�0.8011�0.1288

�45�

45�

45

45

�43�

34�5�

15

�83�

38�3�

13

cos�8�

3 � cos 4�

3� �

12

sin�9�

4 � sin 7�

4� �

�22

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t 0 1

y 0.25 0.0138 0.0883�0.0249�0.1501

34

12

14

333202CB04_AN.qxd 1/1/70 09:39 AM Page 5


Vocabulary Check (page 308)1. (a) v (b) iv (c) vi (d) iii (e) i (f) ii2. opposite; adjacent; hypotenuse3. elevation; depression

1. sin csc

cos sec

tan cot

2. sin csc

cos sec

tan cot

3. sin csc

cos sec

tan cot

4. sin csc

cos sec

tan cot

5. sin csc

cos sec

tan cot

The triangles are similar, and corresponding sides areproportional.

6. sin csc

cos sec

tan cot


7. sin csc

cos sec

tan cot


8.


9. cos

tan cot

csc

10.

11. sin csc

cos cot

tan

12. sin csc

cos sec

tan

13. sin sec

cos cot

csc

14.

tan � � �35

cot � ��3535

cos � �16

csc � �6�35

35sin � �

�356

6

1

θ

35

� ��10

3

� �1

3� �

�10

10

� � �10� �3�10

10

3

1

10

θ

� �1

5

� ��26

5� �

5�26

26

� � �26� ��26

2626

1

5

θ

� � �3

� ��3

3� �

1

2

� �2�3

3� �

�3

2

1

2 3

θ

csc � �7�612

cot � �5�612

tan � �2�6

5

sec � �75

sin � �2�6

7

5

7 2 6

θ

� �4

3

� ��7

3� �

3�7

7

sec � �4�7

7� �

�7

4

θ

7

4 3

cot � � 2tan � �1

2

sec � ��5

2cos � �

2�55

csc � � �5sin � ��5

5

� �43� �

34

� �54� �

45

� �53� �

35

� �158� �

815

� �1715� �

1517

� �178� �

817

� � 2�2� ��2

4

� �3�2

4� �

2�2

3

� � 3� �1

3

� � 1� � 1

� � �2� ��2

2

� � �2� ��2

2

� �409� �

940

� �4140� �

4041

� �419� �

941

� �125� �

512

� �1312� �

1213

� �135� �

513

� �43� �

34

� �54� �

45

� �53� �

35


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333202CB04_AN.qxd 1/1/70 09:39 AM Page 6


(Continued)

15. sin csc

cos sec

tan

16.

17. 18. 19. 20.

21. 22. 23. 24.

25. 26.

27. (a) (b) (c) (d)

28. (a) 2 (b) (c) (d)

29. (a) (b) (c) (d)

30. (a) (b) (c) (d)

31. (a) 3 (b) (c) (d)

32. (a) (b) (c) (d)

33–42. Answers will vary.

43. (a) 0.1736 (b) 0.1736

44. (a) 0.4348 (b) 0.4348

45. (a) 0.2815 (b) 3.5523

46. (a) 0.9598 (b) 0.9609

47. (a) 1.3499 (b) 1.3432

48. (a) 0.9964 (b) 1.0036

49. (a) 5.0273 (b) 0.1989

50. (a) 1.7946 (b) 0.5572

51. (a) 1.8527 (b) 0.9817

52. (a) 2.6695 (b) 0.0699

53. (a) (b)

54. (a) (b)

55. (a) (b)

56. (a) (b)

57. (a) (b)

58. (a) (b)

59. 60. 61.

62. 63. 443.2 meters; 323.3 meters

64. (a) (b)

(c) 270 feet

65. 66. 137.6 feet

67. (a) 371.1 feet (b) 341.6 feet

(c) Moving down line at 61.8 feet per second

Dropping vertically at 24.2 feet per second

68. 1.3 miles 69.

70. 6.57 centimeters

71. (a)

(b) sin (c) 19.9 meters

(d) The side of the triangle labeled will become shorter.(e)

h

85� �h

20

h20

85°

�x2, y2� � �28, 28�3��x1, y1� � �28�3, 28�

30� ��

6

tan � �63

�h

135

h

3

6132

Not drawn to scale

20�2

32�3

39�330�3

45� ��

460� �

�

3

45� ��

460� �

�

3

60� ��

360� �

�

3

45� ��

460� �

�

3

45� ��

445� �

�

4

30� ��

630� �

�

6

�26

5

1

5

�26

26

1

5

1

3

�2

42�2

3

2�6

52�6

�6

12

1

5

�132

23

3�13

132�13

13

�3�3

2

�3

3

�3

3

�3

2

1

2�3

30�; �

645�;

�

4

45�; �22

30�; �32

45�; �

460�;

�

3

45�; �260�; �3�

4; �22

�

6;

12

tan � �4�273

273

cot � ��273

4cos � �

�27317

sec � �17�273

273sin � �

417

417

273

θ

� �2

3

� ��13

3� �

3�13

13

� ��13

2� �

2�13

13

2

3

13

θ

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Angle,

Height 19.7 18.8 17.3 15.3

50�60�70�80��

Angle,

Height 12.9 10.0 6.8 3.5

10�20�30�40��

333202CB04_AN.qxd 1/1/70 09:39 AM Page 7

(Continued)

(f) As

72.

73. True, csc 74. True, sec

75. False,

76. True, for all 77. False,78. False, tan 79. Corresponding sides of similar triangles are proportional.80. Yes, is equal to You can find the value of the

hypotenuse by the Pythagorean Theorem, then you can findwhich is equal to

81. (a)

(b) (c) As approaches 0, sin approaches 0.

82. (a)

(b) Increasing function (c) Decreasing function(d) As the angle increases, the length of the side opposite

the angle increases relative to the length of thehypotenuse and the length of the side adjacent to the angle decreases relative to the length of thehypotenuse. Thus, the sine increases and the cosinedecreases.

83. 84.

85. 86.14

, x � 0, 122�x2 � 5x � 10��x � 2��x � 2�2

2t � 34 � t

, t � ±32

, �4x

x � 2, x � ±6

��

hyp�adj.sec �,

opp�adj.tan �

25� � �tan 5��tan 5��.1.7321 � 0.0349.

�.cot2 � � csc2 � � �1

�2

2�

�2

2� 1.

x � csc�90� � x�.x �1

sin x.

cot 20� � 2.75sec 20� � 1.06csc 20� � 2.92tan 20� � 0.36cos 20� � 0.94sin 20� � 0.34

20 h

θ

� → 0�, h → 0.


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0.1 0.2 0.3 0.4 0.5

sin 0.0998 0.1987 0.2955 0.3894 0.4794�

�

sin 0 0.3090 0.5878 0.8090 0.9511 1

cos 1 0.9511 0.8090 0.5878 0.3090 0�

�

90�72�54�36�18�0��

333202CB04_AN.qxd 1/1/70 09:39 AM Page 8



Vocabulary Check (page 318)

1. 2. 3. 4. 5.

6. 7. reference

1. (a) sin (b) sin

cos cos

tan tan

csc csc

sec sec

cot cot

2. (a) sin

cos

tan

csc

sec

cot

3. (a) sin (b) sin

cos cos

tan tan

csc csc

sec sec

cot

4. (a) sin (b) sin

cos cos

tan tan

csc

sec sec

cot

5. sin csc

cos sec

tan cot

6. sin csc

cos sec

tan cot

7. sin csc

cos sec

tan cot

8. sin csc

cos sec

tan cot

9.

10.

11. Quadrant III 12. Quadrant I

13. Quadrant II 14. Quadrant IV

15. sin csc

cos sec

tan cot

16. sin csc

cos sec

tan cot

17. sin csc

cos sec

tan cot

18. sin csc

cos sec

tan cot

19. sin csc

cos sec

tan cot � � �3� � �1

3

� ��10

3� �

3�10

10

� � ��10� � ��10

10

� � �8

15� � �158

� �178� �

817

� � �1715� � �

1517

� � �8

15� � �158

� �178� �

817

� � �1715� � �

1517

� �43� �

34

� � �54� � �

45

� � �53� � �

35

� � �43� � �

34

� � �54� � �

45

� �53� �

35

cot � � �1431

tan � � �3114

sec � ��1157

14cos � �

14�11571157

csc � � ��1157

31sin � � �

31�11571157

cot � � �3568

tan � � �6835

sec � � ��5849

35cos � � �

35�58495849

csc � ��5849

68sin � �

68�58495849

� �5

2� �

2

5

� � ��29

5� � �

5�29

29

� � ��29

2� � �

2�29

29

� � �2

5� � �

5

2

� � ��29

2� � �

2�29

29

� ��29

5� �

5�29

29

� �815� �

158

� �178� �

817

� �1715� �

1517

� �724� �

247

� �257� �

725

� �2524� �

2425

cot � � �1� � 3

� � �2� ��10

3

csc � � ��2� � �10

� � �1� �1

3

� ��2

2� �

3�10

10

� � ��2

2� �

�10

10

cot � � �4� � �3

� � ��17

4� � �

2�3

3

� � �17� � �2

� � �1

4� �

�3

3

� � �4�17

17� � �

�3

2

� ��17

17� � �

1

2

� �12

5

� � �13

12

� � �13

5

� �5

12

� � �12

13

� � �5

13

� � �8

15� �43

� �178� �

54

� � �1715� �

53

� � �158� �

34

� �817� �

45

� � �1517� �

35

cot �

cos �rx

yx

csc �yr

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(b) sin

cos

tan csc sec cot � � �1

� � ��2� � �2� � �1

� � ��2

2

� ��2

2

333202CB04_AN.qxd 1/1/70 09:39 AM Page 9

(Continued)

20. sin csc

cos sec

tan cot

21. sin csc

cos sec

tan cot

22. is undefined.

is undefined.

23. is undefined.

is undefined.

24. sin csc cos sec is undefined.tan cot

25. sin csc

cos sec

tan cot

26. sin csc

cos sec

tan cot

27. sin csc

cos sec

tan cot

28. sin csc

cos sec

tan cot

29. 0 30. 31. Undefined 32. 33. 1

34. Undefined 35. Undefined 36. 0

37. 38.

39. 40.

41. 42.

43. 44.

45. 46. sin

cos cos

tan tan

47. sin 48.

cos cos��405�� 2

2750� �

�3

2

sin��405�� 2

2750� �

1

2

300� � ��3225� � 1

300� �1

2225� � �

�2

2

300� � ��3

2sin 225� � �

�2

2

′θ

x

y

113π

′θ

3.5

x

y

��

3�� 3.5 � �

′θ

74π

x

y

′θ

23π

x

y

��

4��

�

3

′θ−145°

x

y

′θ

−245°

x

y

�� 35�� 65�

′θ

309°

x

y

′θ

203°

x

y

�� 51�� 23�

�1�1

� � �34� � �

43

� �53� �

35

� � �54� � �

45

� �1

2� � 2

� � ��5� � ��5

5

� � ��5

2� � �

2�5

5

� � 3� �1

3

� � ��10

3� � �

3�10

10

� � ��10� � ��10

10

� � �1� � �1

� � ��2� � ��2

2

� � �2� ��2

2

� � 0� is undefined.�� 0� � �1� � �1

cot �tan � � 0

sec � � �1cos � � �1

csc �sin � � 0

cot �tan � � 0

sec � � �1cos � � �1

csc �sin � � 0

� � ��3

3� � ��3

� � �2� � �1

2

� �2�3

3� �

�3

2

� � ��15� � ��15

15

� � �4�15

15� � �

�15

4

� � 4� �1

4


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333202CB04_AN.qxd 1/1/70 09:39 AM Page 10


(Continued)

tan

49. 50.

51. 52. sin

cos

tan

53. 54.

55. 56. sin

cos

tan

57. 58.

is undefined.

59. 60. 61. 62. 63.

64. 65. 0.1736 66. 67.

68. 2.0000 69. 70.71. 3.2361 72. 73. 4.6373 74. 0.224575. 0.3640 76. 77.78. 1.0436 79. 80. 4.4940

81. (a) (b)

82. (a) (b)

83. (a) (b)

84. (a) (b)

85. (a) (b)

86. (a) (b)

87. (a)

(b) February:March:May:June:August:September:November:

(c) Answers will vary.

88. (a) 26,134 units (b) 31,438 units (c) 21,452 units(d) 26,756 units

89. (a) 2 centimeters (b) 0.14 centimeter(c) centimeters

90. (a) 2 centimeters (b) 0.11 centimeter(c) centimeters

91. 0.79 ampere

92. (a) 12 miles (b) 6 miles (c) 6.9 miles

93. False. In each of the four quadrants, the signs of the secantfunction and cosine function will be the same, becausethese functions are reciprocals of each other.

94. False. For in Quadrant II, For inQuadrant III, For in Quadrant IV,

95. As increases from to , decreases from 12 cm to 0 cm and increases from 0 cm to 12 cm. Therefore,

increases from 0 to 1 and decreases from 1 to 0. Thus, and increaseswithout bound. When , the tangent is undefined.

96. Determine the trigonometric function of the referenceangle and prefix the appropriate sign.

97. -intercepts:

-intercept:Domain: all real numbers

98. Intercept:-intercept:

Domain: all real numbers x�5

2, 0�x�0, 0�

−1−2−3 1 2 3 4 5−1

−2

−3

−4

1

2

y

x(0, 0) , 05

2( (

x�0, �4�y

�1, 0�, ��4, 0�x

−2−6−8 2 4 6 8−2

−4

−8

2

4

6

8

y

x(1, 0)(−4, 0)

(0, −4)

� � 90�tan � � y�x

cos � � x�12sin � � y�12y

x90�0��

�� 360� � �.�� 180�.

�� 180� � �.�

�1.2

�1.98

N � 46.8�, F � 6.5�

N � 68.6�, F � 41.7�

N � 75.5�, F � 55.6�

N � 72.5�, F � 59.5�

N � 63.4�, F � 48.6�

N � 41.6�, F � 13.9�

N � 34.6�, F � �1.4�

F � 36.641 sin�0.502t � 1.831� � 25.610N � 22.099 sin�0.522t � 2.219� � 55.008

240� �4�

3, 300� �

5�

360� �

�

3, 120� �

2�

3

150� �5�

6, 330� �

11�

645� �

�

4, 225� �

5�

4

120� �2�

3, 240� �

4�

360� �

�

3, 300� �

5�

3

135� �3�

4, 315� �

7�

460� �

�

3, 120� �

2�

3

135� �3�

4, 225� �

5�

445� �

�

4, 315� �

7�

4

210� �7�

6, 330� �

11�

630� �

�

6, 150� �

5�

6

�0.4142�0.6052�0.3640

�0.1405�28.6363�1.4826

�0.3420�1.4142�65

4

8

5��3�

�13

2

�10

10

4

5

tan�25�

4 � �1tan�3�

2 cos�

25�

4 ��22

cos�3�

2 � 0

sin�25�

4 � ��22

sin�3�

2 � 1

10�

3� �3tan

11�

4� �1

10�

3� �

1

2cos

11�

4� �

�22

10�

3� �

�3

2sin

11�

4�

�22

tan��

2 is undefined.tan��

6 � ��33

cos��

2 � 0cos��

6 ��32

sin��

2 � �1sin��

6 � �12

�

4� 1tan

4�

3� �3

�

4�

�2

2cos

4�

3� �

12

�

4�

�2

2sin

4�

3� �

�32

tan��840�� 3tan��150�� 33

cos��840�� 12

cos��150�� 32

sin��840�� 32

sin��150�� 12

tan��405�� 1750� �

�3

3

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333202CB04_AN.qxd 1/1/70 09:39 AM Page 11

(Continued)

99. -intercept:-intercept:

Domain: all real numbers

100. -intercepts:

-intercept:Domain: all real numbers


Vertical asymptote:

Horizontal asymptote:

Domain: all real numbersexcept

102. -intercepts:

-intercept:Vertical asymptote:

Slant asymptote:

Domain: all real numbersexcept

103. -intercept:Horizontal asymptote:


104. -intercept:Horizontal asymptote:


105. -intercepts:Vertical asymptote:Domain: all real numbers

except


Vertical asymptote:

Domain: all real numbers such that x > �2

xx � �2

�0, 0.301�y��1, 0�x

3

2

−1

−2

−3

321−1−3

(−1, 0)(0, 0.301)

x

y

x � 0x

x � 0�±1, 0�x

−12 −9 −6 −3 3 6 9 12

6

9

12

(−1, 0) (1, 0)x

y

xy � 2

�0, 5�y

x321−1−2−3−4−5

7

6

5

3

1

2

(0, 5)

y

xy � 0

�0, 12�y

−2 −1 1 2 3 4−1

2

3

4

5

10, 12 ))

x

y

x � �5x

y � x � 5

x � �5

�0, �15�y

�1, 0�, ��1, 0�x

−8−12 4

−8

−16

−24

8

y

x(−1, 0)

(1, 0)

0, 15

−( (

x � �2x

y � 0

x � �2

�0, �74�y

�7, 0�x

−8 −2 2 4 6 8

2

4

y

x

74

0, ( (−

(7, 0)

x�0, �3�y

�1, 0�, ��1, 0�x

−2−3−4 2 3 4

−3

−4

1

2

3

4

y

x(−1, 0)

(0, −3)

(1, 0)

x�0, 8�y��2, 0�x

−6 −4−8 2 4 6 8

−4

10

12

y

x

(0, 8)

(−2, 0)


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333202CB04_AN.qxd 1/1/70 09:39 AM Page 12



Vocabulary Check (page 328)

1. cycle 2. amplitude 3.

4. phase shift 5. vertical shift

1. Period: 2. Period: 3. Period:

Amplitude: 3 Amplitude: 2 Amplitude:4. Period: 5. Period: 6 6. Period: 4

Amplitude: 3 Amplitude: Amplitude:


Amplitude: 3 Amplitude: 1 Amplitude: 3


Amplitude: Amplitude: Amplitude:

13. Period: 1 14. Period: 20Amplitude: Amplitude:

15. is a shift of units to the right.16. is a shift of units to the left.17. is a reflection of in the axis.18. is a reflection of in the axis.19. The period of is twice the period of 20. The period of is one-third the period of 21. is a shift of three units upward.22. is a shift of two units downward.23. The graph of has twice the amplitude of the graph of 24. The period of is the period of 25. The graph of is a horizontal shift of the graph of

units to the right.26. is a shift of two units upward.

27. 28.

29. 30.

31. 32.

33. 34.

35. 36.

37. 38.

39. 40. y

x

−2

1

2

π4

y

xπ4π2π−2

−1

−2

2

y

xππ 2π−2 −π

−2

−4

4

y

xπ π2

π2

1

−1

23

43

1323

43

−

−

−

y

xππ 2π−2 −π

−1

−2

1

2

y

xπ π32 2

−− π2

π32

−4

1

2

3

4

−2

2

ππ 2

f

g

x

y

−3

3

f

g

xππ 2

y

−8

2

4f

g

1 x

y

−1

1

2

3

4

5

f

g

x−π π3

y

−2

2

π

f

g

x

y

−1

g

f

xππ 2

y

− 2

2

π6

fg

x

y

x

−π π32 2

5

43

−5

f

g

y

fg�

fgf.1

3gf.g

fgfg

f.gg.f

x-fgx-f g

f �gf �g

23

14

52

12

13

8�3��

4

�

53�2�

32

12

6�

52

4�2�

3�

2�

b

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333202CB04_AN.qxd 1/1/70 09:39 AM Page 13

(Continued)

41. 42.

43. 44.

45. 46.

47. 48.

49. 50.

51. 52.

53. 54.

55. 56.

57. 58.

59. 60.

61. 62.

63. 64.

65. 66.

67. 68.

69. 70. a � 2, b ��

2, c � �

�

2a � 2, b � 1, c � �

�

4

a � 2, b �12, c � 0a � �3, b � 2, c � 0

a � �1, d � �3a � �4, d � 4

a � 2, d � �1a � 2, d � 1

−0.03

−0 02

0.03

0.02

−20

−0.12

20

0.12

−6

−6

6

2

−3

−1

3

3

−12

−8

12

8

−6 6

−4

4

2

3

x

y

π

−4

−3

−2

−1

1

2

3

4

π π4x

y

−4

2

4

6

10

x

y

π3π2π−2 − ππ

−8

2

4

ππ 2x

y

−7

−6

−5

−4

1

−x

y

πππ 2

0.20.1−0.1 0

1.8

2.2

x

y

−12 4 12

−24−20−16−12−8

48

1216

t

y

–3 –2 –1 1 2 3−1

1

2

4

5

x

y

−6

−4

−2−

2

6

x

y

πππ 2

−6

−4

2

4

6

− ππx

y

2

−2

−1

x

y

−π π32 2

−3

−2

1

2

3

ππ−x

y

1284−4−12

12

8

4

−12

x

y

−1 2 3

−3

−2

2

3

x

y

2

1

−2

62−2−6x

y

1 2

−2

1

2

x

y


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333202CB04_AN.qxd 1/1/70 09:39 AM Page 14


(Continued)

71. 72.

73. (a) 6 seconds (b) 10 cycles per minute(c)

74. (a) 4 seconds (b) 15 cycles per minute(c)

75. (a)

(b) (c)

The model is a good fit. The model is a good fit.

(d) Tallahassee: Chicago:

The constant term gives the annual average tempera-ture.

(e) 12; yes; one full period is one year.

(f ) Chicago; amplitude; the greater the amplitude, thegreater the variability in temperature.

76. (a) seconds (b) 50 heartbeats per minute

77. (a) second (b) 440 cycles per second

78. (a)–(c)

(b)The model is a good fit.

(d) 29 days(e) 0.44

79. (a) 365; answers will vary.(b) 30.3 gallons; the constant term(c)

80. (a) 20 seconds; it takes 20 seconds to complete one revolution on the Ferris wheel.

(b) 50 feet; the diameter of the Ferris wheel is 100 feet.(c)

81. False. The graph of translates thegraph of exactly one period to the left so thatthe two graphs look identical.

82. False. The function has an amplitude that isone-half that of For the amplitudeis

83. True. Because is a

reflection in the -axis of

84. Answers will vary.85. 86.

Conjecture: Conjecture:

sin x � �cosx ��

2sin x � cosx ��

2

2

1

−2

f = g

x

y

π π32 2

π32

−

2

1

−2

f = g

xπ π32 2

− π32

y

y � sinx ��

2.x

y � �cos xcos x � sinx ��

2,

�a�.y � a cos bx,y � cos x.

y �12 cos 2x

f �x� � sin xf �x� � sin�x � 2��

00 20

110

124 < t < 252

00

365

60

y �12 �

12 sin�0.21x � 0.92�

y

x10 20 30 40

0.2

0.4

0.6

0.8

1.0

Perc

ent o

f m

oon’

sfa

ce il

lum

inat

ed

Day of the year

1440

65

56.55�77.90�;

00

12

100

00

12

100

C�t� � 56.55 � 26.95 cos�

6t � 3.67

1 3 5 7

−2

−3

1

2

3

t

v

t2 4 8 10

0.25

0.50

0.75

1.00

−0.25

−1.00

v

x � �, ��x � �

�

6, �

5�

6,

7�

6,

11�

6

−2

2

2

�−2�

−2

2

2

�−2�

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333202CB04_AN.qxd 1/1/70 09:39 AM Page 15

(Continued)

87. (a)

The graphs appear to coincide from to

(b)

The graphs appear to coincide from to

(c)

The interval of accuracy increased.88. (a) 0.4794, 0.4794 (b) 0.8417, 0.8415 (c) 0.5, 0.5

(d) 0.8776, 0.8776 (e) 0.5417, 0.5403(f) 0.7074, 0.7071The error increases as moves farther away from 0.

89. 90.

91. 92.

93. 94.

95. 96. 97. Answers will vary.ln�x2�2x�ln 3xy4

log2�x3y�log10 �xy

12 ln z �

12 ln�z2 � 1�3 ln t � ln�t � 1�

2 log2 x � log2�x � 3�12 log10�x � 2�

x

−2

2�−2�

2

−2

−2 2� �

2

�x7

7!, �

x 6

6!

�

2.�

�

2

−2

−2 2� �

2

�

2.�

�

2

−2

−2 2� �

2


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333202CB04_AN.qxd 1/1/70 09:39 AM Page 16



Vocabulary Check (page 339)1. vertical 2. reciprocal 3. damping4. 5. 6.7.

1. e, 2. c, 3. a, 1 4. d,5. f, 4 6. b, 4

7. 8.

9. 10.

11. 12.

13. 14.

15. 16.

17. 18.

19. 20.

21. 22.

23. 24.

25. 26. y

x

−1

−2

π32

π2

2ππ

1

2

3

4

x

− ππ322

y

4

3

2

1

πx

y

−4 4

2

4

6

x

y

2

3

ππ−

y

x

3

xπ π

y

−

2

4

6

2−2x

y

1

2

3

x

y

π−2 2π

2

4

6

ππ 2x

y

2

4

6

xπ

y

2

4

6

x

y

π4

π4

π2

−

1−1−2−3 2 3−1

x

y

8

6

4

2

−2− π4

π4

x

y

x

4

3

2

1

−3

−4

21−1−2

y

π

3

2x

y

1

2

3

xπ

y

− π

2

−8

−4

x

y

x

4

3

2

1

− π3

π3

y

−3

1

2

3

ππ−x

y

1

2

3

xπ π

y

−

2�2��

2�

�� , �1� � �1, �x � n��

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333202CB04_AN.qxd 1/1/70 09:39 AM Page 17

(Continued)

27. 28.

29. 30.

31. 32.

33. 34.

35. 36.

37. 38.

39. 40.

41. 42.

43. 44.

45. 46.

47. 48.

49. Even 50. Odd

51. (a) (b)

(c) approaches 0 and approaches because thecosecant is the reciprocal of the sine.

52. (a)

(b) (c) The intervals are the same.

53.

The expressions are equivalent except that when is undefined.

54. 55.

The expressions are The expressions areequivalent. equivalent.

56.

The expressions are equivalent.

57. d, 58. a,

59. b, 60. c, g → 0 as x → 0.g → 0 as x → 0.

f → 0 as x → 0.f → 0 as x → 0.

−1

−

3

�2

3�2

3

−4

2�−2�

4

−4

−2 2

4

� �

y1

sin x � 0,

−2

3−3

2

��1, 13�;��1, 13�−3

g

f

3

1−1

� gf

�

6< x <

5�

6

1

−1

2

3

f

g

xπ ππ32 4

π4

y

�2�

3, �

�

3,

4�

3,

5�

3�

7�

4, �

5�

4,

�

4,

3�

4

�5�

3, �

�

3,

�

3,

5�

3�

4�

3, �

2�

3,

2�

3,

4�

3

�7�

4, �

3�

4,

�

4,

5�

4�

4�

3, �

�

3,

2�

3,

5�

3

�5�

3, �

2�

3,

�

3,

4�

3�

7�

4, �

3�

4,

�

4,

5�

4

−6

−2

6

2

−6

−0.6

6

0.6

−4

−� �

4

−3

− �2

�2

3

−3

�2

3�2

3−

3

−3

− �2

3�2

3

3

−3

−2

3

2

−4

�2

�2

−

4

−3

3

�4

3�4

3−

−5

5�−5�

5

−2

xπ32

− π32

y

xπ2

1

2

y

y

x1 2 3 4

1

2

3

y

x

−1

1

2

3

4

π 2π−π 3π


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333202CB04_AN.qxd 1/1/70 09:39 AM Page 18


(Continued)

61. 62.

The functions are equal. The functions are equal.

63. 64.

The functions are equal. The functions are equal.

65. 66.

As As

67. 68.

As As

69. 70.

As As

71. 72.

As As

73. 74.

As oscillates As oscillates.between 1 and

75. 76.

77. (a)

(b) As the predator population increases, the number ofprey decreases. When the number of prey is small, thenumber of predators decreases.

(c) 24 months; 24 months

78.

79. (a) 12 months; 12 months(b) Summer; winter (c) 1 month

80. (a)

(b) approaches 0 as increases.

81. True. For a given value of the coordinate of is thereciprocal of the coordinate of sin x.y-

csc xy-x,

ty

0 4�

−0.6

0.6

L:H:

Month (1 ↔ January)

Law

n m

ower

sal

es(i

n th

ousa

nds

of u

nits

) 150135120105907560453015

2 4 6 8 10 12t

S

R :C :

00 100

50,000

R

C

x

20

40

60

80

Angle of camera

Dis

tanc

e

d

0 π2

π4

π4

π2

− −

Gro

und

dist

ance

x

14

10

6

2

−2

−6

−10

−14Angle of elevation

d

π π π32 4

π4

d � 27 sec xd � 7 cot x

�1.x → 0, h�x�x → 0, f �x�

−1

−� �

2

−2

−� �

2

x → 0, y → .x → 0, g�x� → 1.

−1

−6� 6�

1

−1

6�−6�

2

x → 0, y → .x → 0, y → .

0

−2

6�

6

0

−2

8�

6

x → , h�x� → 0.x → , f �x� → 0.

−8

−1

8

1

−9 9

−6

6

x → , f �x� → 0.x → , g�x� → 0.

−3 6

−3

3

−1

8−8

1

−1

−3 3 6−6

2

3

x

y

–1

2

3

y

xππ−

−4

2

4

xππ−

y

−3 −2 −1 1 2 3

−3

−2

−1

1

2

3

x

y

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333202CB04_AN.qxd 1/1/70 09:39 AM Page 19

(Continued)

82. True. is equal to and if the recip-rocal of is translated units to the left, then

83. As approaches from the left, approaches As approaches from the right, approaches

84. As approaches from the left, approaches As approaches from the right, approaches

85. (a)

0.7391(b) 1, 0.5403, 0.8576, 0.6543, 0.7935, 0.7014, 0.7640,

0.7221, 0.7504, 0.7314, . . . ; 0.7391

86. 87.

The graphs appear to The graphs appear tocoincide on the interval coincide on the interval

88. (a)

(b)

(c)

89. 90.

91.

92.

93.

94.

95. 96.97. 2 98. �65 � 8.062

e10 � 4 � 22,022.466±�e3.2 � 1 � ±4.851

14 � e68

2� �1.702 � 1029

2 � e73

3� 1.684 � 1031

1365

log10 5log10 1.00041096 � 10.732

�ln 2 � �0.693

13

log10 98log10 8

� 0.735ln 54

2� 1.994

�17

sin�7�x� �19

sin�9�x�� y4 �

4

� sin��x� �1

3 sin�3�x� �

1

5 sin�5�x�

−3 3

−2

2

�17

sin�7�x�� y3 �

4

� sin��x� �1

3 sin�3�x� �

1

5 sin�5�x�

−3 3

−2

2

y2

−3 3

−2

2

y1

�1.1 ≤ x ≤ 1.1.�1.1 ≤ x ≤ 1.1.

−6

−

6

�2

3�2

3

−6

− �2

3�2

3

6

3

−2

−3

2

� .f�x .f�x

� .f��2x .f��2x

1

sinx ��

2�

1cos x

� sec x.

��2y � sin xy � 1�cos x,y � sec x


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333202CB04_AN.qxd 1/1/70 09:39 AM Page 20



Vocabulary Check (page 349)1.2.

3.

1. 2. 0 3. 4. 5. 6.

7. 8. 9. 10. 11.

12. 13. 14. 15. 0 16. 0

17. 18.

19. 1.29 20. 0.47 21. 22. 2.3523. 24. 1.50 25. 0.32 26. 1.3127. 1.99 28. 29. 0.74 30. 1.2331. 0.85 32. 1.91 33. 1.29 34.

35. 36.

37. 38.

39. 40.

41. 42.

43. 0.3 44. 25 45. 46. 47. 0

48. 49. 50. 51. 52.

53. 54. 55. 56.

57. 58. 59. 60.

61. 62. 63.

64. 65. 66.

67. 68.

69. 70.

Asymptotes: Asymptote:

71.

72. 73. 74.

75. 76.

The graph of is a The graph of is ahorizontal shift one unit horizontal stretch of theto the right of graph of

77. 78.

79. 80.

81. 82.

83. 84.

−0.5 0.5

2�

−2�0

2�

−1

−4 −2 2 4x

π

y

−2 1 2

π

y

v

−4 −2 2 4x

π

y

−4 −2 2 4

π

π

y

x

−

−4 −3 −2 −1t

π

y

−2 −1 1 2

π

π2

x

y

f.f.

gg

1 2−2x

π

π

y

−

−1 1 2 3

π

π

y

x

−

�4x � x2

x � 2�x � 1�

�x2 � 2x � 10

x6

�9

�x2 � 81, x < 0

9

�x2 � 81, x > 0;

x � 0y � ±1

−3

−2

3

2

−3 3

−2

2

�r2 � �x � h�2

r

�x 2 � 2

x

x�9 � x 2

x

1

�2x � x2

�1 � x 2�9x2 � 1�1 � 4x 2

x

�x2 � 1

1

x

8

5

�5

3

�3�7

7

�34

5�

13

5

12

13

2�5

5

�5

553

3

5�

2

�0.2�0.1

� � arctan 1

x � 1, x � 1� � arccos

x � 32x

� � arctan x � 1

10� � arcsin

x � 2

5

� � arccos 4

x� � arctan

x

4

�, 2�

3, �3

2�

�

3, �

�3

3, 1

�1.50�0.13

�1.25�0.85

−2

g

f

2

�2

�2

−

−1

1.5−1.5

fg

1

��

6

�

3

�

4

2�

3

�

3�

�

3�

�

4

5�

6

��

4

�

6

�

2

�

3

�

6

y � tan�1 x; � < x < ; ��

2< y <

�

2

y � arccos x; 0 ≤ y ≤ �

y � sin�1 x; �1 ≤ x ≤ 1

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333202CB04_AN.qxd 1/1/70 09:39 AM Page 21

(Continued)

85. 86.

87. 88.

89.

The graph implies that the identity is true.

90.

The graph implies that the identity is true.

91. (a) (b) 0.13, 0.25

92. (a) (b)

93. (a)

(b) 2 feet (c) As increases, approaches 0.94. (a) (b) 12.94 feet95. (a) (b) 24.4 feet

96. (a) (b)

97. (a) (b)

98. False. is not in the range of the arcsine.

99. False. is not in the range of the arctangent.

100. False. The graphs are not the same.

101. Domain:Range:

102. Domain:Range:

103. Domain:Range:

104. (a) (b) 0 (c) (d)

105. (a) (b) (c) 1.25 (d) 2.03

106.

As increases to infinity, approaches , but has nomaximum.a � 87.54

f3�gx

00

6

g

f

12

�

2

�

4

�

6

5�

6

�

4

x−2 −1 21

−

y

π2

π2

��2, 0� � �0, ��2�� , �1� � �1, �

x−2 −1 1 2

y

π

π2

�0, ��2� � �� 2, �� , �1� � �1, �

x−1−2 21

π2

y

π

�0, �� , �

5�

4

5�

6

14.0�, 31.0�� arctan x

20

40.6�, 80.5�� arctan 6x

� � 26.0�

� � 32.9�

x � 0;

0

−0.5

6

1.5

21.8�, 58.0�� arctan s

750

� � arcsin 5

s

−6

6−6

6

5 sin�� t � arctan 43�

−6

−2

6

� 2�

3�2 sin2t ��

4

5

−2

−4

4

5

−2

−4

4

−4 4

−2�

�2

−2 4

�−

�


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333202CB04_AN.qxd 1/1/70 09:39 AM Page 22


(Continued)

107. (a)

(b) The domains and ranges of the functions are restricted.The graphs of and differ because of thedomains and ranges of and

108. (a)–(e) Answers will vary. 109. 1279.284110. 0.051 111. 117.391 112.

113.

114.

115.

116.

117. Eight people 118. 3 miles per hour

119. (a) $21,253.63 (b) $21,275.17

(c) $21,285.66 (d) $21,286.01

120. 2008: $458,504

2�2

tan � �2�2

1

cot � ��24

cos � �13

csc � �3�2

4sin � �

2�23

3

θ

�11

csc � �6�11

11

cot � �5�11

11tan � �

�115

sec � �65

sin � ��11

66

5

θ

�5

csc � ��52

cot � �12

cos � ��55

sec � � �5sin � �2�5

5

2

θ

�7

csc � �43

cot � ��73

tan � �3�7

7

sec � �4�7

7cos � �

�74

4 3

θ

2.718 � 10�8

f �1.ff �1

� ff � f �1

−2

−� �

2

−2

−� �

2

f �1� ff � f �1

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333202CB04_AN.qxd 1/1/70 09:39 AM Page 23


Vocabulary Check (page 359)1. elevation; depression 2. bearing3. harmonic motion

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 2.56 inches 12. 1.62 meters

13. 19.99 inches 14. 2.80 feet 15. 107.2 feet16. 1648.5 feet 17. 19.7 feet 18. 81.2 feet19. (a)

(b) (c) 19.9 feet20. 123.5 feet 21. 2236.8 feet 22.

23. (a) (b) (c)

24. 25.26. (a) 5099 feet (b) 117.7 seconds 27. 0.73 mile28. 29. 554 miles north; 709 miles east30. (a) 429.26 miles north; 2434.44 miles east

(b)31. (a) 58.18 nautical miles west; 104.95 nautical miles south

(b) S W; 32. (a) 21.4 hours

(b) 5.86 nautical miles east; 239.93 nautical miles south(c)

33. (a) (b) 68.82 meters34. 35.36. 37. 1933.3 feet 38. 11.8 kilometers39.

40.41. 42. 43. 44.45. 29.4 inches 46. 25 inches 47.48. 9.06 centimeters 49.50. feet, feet, feet

51. 52.

53. 54.

55. (a) 4 (b) 4 (c) 4 (d)

56. (a) (b) 10 (c) (d)

57. (a) (b) 60 (c) 0 (d)

58. (a) (b) 396 (c) 0 (d)

59. 60.

61. (a) (b) (c)

62. (a)

(b)

7.0 (minimum length)

�

32

�

8

t

1

−1

y

π π38 8

π4

π2

d �7

4 cos

� t

5� � 528�

1792

164

1120

116

140

12

12

116

d � 2 cos� t

5 d � 3 cos4� t

3 d � 3 sin� t

3 d � 4 sin�� t�

c � 13b � 7.2a � 21.6a � 12.2, b � 7

y � �3r54.7�35.3�52.1�78.7�

� 1.025 miles or � 5412 feet

� 3.23 miles or � 17,054 feet208�

N 56.31� Wd � 5.46 kilometersN 58� E178.6�

distance � 130.9 nautical miles36.7�

280�

6.8�; 2516.3 feet

2.06�71.34�

35.8�tan � �121

2

1713

θ

17 ft

12 ft

1

1

3

2

56.3�

h � 50�tan 47�40� � tan 35��

50 ft

47° 40′

35°

h

x

y

A � 24�48�

c � 33.85b � 30.73

B � 77�45�B � 8.03�B � 17.92�

b � 420.70A � 81.97�A � 72.08�

a � 91.34a � 9.36a � 49.48B � 44.42�B � 59.04�B � 81.6�

A � 45.58�A � 30.96�c � 277.24b � 24.49c � 11.66b � 274.27A � 19�A � 36�B � 70�

c � 25.38b � 12.14c � 10.64a � 8.26a � 8.82a � 3.64


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0.1 23.0

0.2 13.1

0.3 9.9

0.4 8.43

cos 0.42

sin 0.4

3cos 0.3

2sin 0.3

3cos 0.2

2sin 0.2

3cos 0.1

2sin 0.1

L1 � L2L2L1�

0.5 7.6

0.6 7.2

0.7 7.0

0.8 7.13

cos 0.82

sin 0.8

3cos 0.7

2sin 0.7

3cos 0.6

2sin 0.6

3cos 0.5

2sin 0.5

L1 � L2L2L1�

333202CB04_AN.qxd 1/1/70 09:39 AM Page 24


(Continued)

(c)

(d) 7.0 (minimum length)

63. (a)

(b)

83.14 square feet(c)(d)

square feet when The answers are the same.

64. (a)

(b) or

The model is a good fit.(c) 12. Yes, sales of outerwear are seasonal.(d) Maximum displacement from average sales of $8

million65. False. The tower is leaning, so it is not perfectly vertical

and does not form a right angle with the ground.66. False. One period is the time for one complete cycle of the

motion.67. No. means 24 degrees east of north.68. Air navigation is always measured clockwise from the

north. Nautical navigation measures the acute angle a pathmakes with a fixed north-south line.

69. 70.

71. 72.y

x−1−2−3 2 3

−1

−2

−3

1

2

3

y

x−1−2 1 2 3 4 5

−1

1

2

3

4

6

7

y � �43x �

13y � �

45x �

225

y

x−1−2−3 2 3

−1

−2

−3

1

2

3

y

x−1−2−3−4 1 2 3 4

−1

1

2

3

5

6

7

y � �12x �

16y � 4x � 6

N 24� E

S � 8 � 6.3 sin�

6t �

�

2S � 8 � 6.3 cos�

6t

t2 4 121086

3

6

9

12

15

Ave

rage

sal

es(i

n m

illio

ns o

f do

llars

)

Month (1 ↔ January)

S

� � 60�� 83.1

0900

100

A � 64�1 � cos ��sin ��

−12

−2� 2�

12

L � L1 � L2 �2

sin ��

3

cos �

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Base 1 Base 2 Altitude Area

8 59.7

8 72.7

8 80.5

8 83.1

8 80.7

8 74.0

8 64.08 sin 90�8 � 16 cos 90�

8 sin 80�8 � 16 cos 80�

8 sin 70�8 � 16 cos 70�

8 sin 60�8 � 16 cos 60�

8 sin 50�8 � 16 cos 50�

8 sin 40�8 � 16 cos 40�

8 sin 30�8 � 16 cos 30�

Base 1 Base 2 Altitude Area

8 82.73

8 83.04

8 83.11

8 83.14

8 83.11

8 83.048 sin 62�8 � 16 cos 62�

8 sin 61�8 � 16 cos 61�

8 sin 60�8 � 16 cos 60�

8 sin 59�8 � 16 cos 59�

8 sin 58�8 � 16 cos 58�

8 sin 56�8 � 16 cos 56�

333202CB04_AN.qxd 1/1/70 09:39 AM Page 25

Review Exercises (page 365)

1. 0.5 radian 2. 4.5 radians3. (a) 4. (a)

(b) Quadrant II (b) Quadrant I

(c) (c)

5. (a) 6. (a)

(b) Quadrant II (b) Quadrant I

(c) (c)

7. (a) 8. (a)

(b) Quadrant I (b) Quadrant IV(c) (c)

9. (a) 10. (a)

(b) Quadrant III (b) Quadrant IV(c) (c)

11. 8.378 12. 13. 14. 3.44315. 16. 17.18. 19. 48.17 inches 20. 11.52 meters21. (a) radians per minute

(b) inches per minute22. 212.1 inches per second23.

24. 25.

26. 27. 28.

29.

30.

31.

32. is undefined.

is undefined.

33.

34. 35.

36. 37.

38. 39. 3.2361 40.

41.

cot � �54

sec � ��41

5

csc � ��41

4

tan � �45

cos � �5�41

41

sin � �4�41

41

�0.3420�1.1368

�75.3130cos�13�

3 � cos 5�

3�

12

sin�17�

6 � sin 7�

6� �

12

cos 4� � cos 0 � 1

sin 11�

4� sin

3�

4�

�22

cot 2�tan 2� � 0

sec 2� � 1cos 2� � 1

csc 2�sin 2� � 0

cot�2�

3 ��33

tan�2�

3 � �3

sec�2�

3 � �2cos�2�

3 � �12

csc�2�

3 � �2�3

3sin�

2�

3 � ��32

cot �

4� 1tan

�

4� 1

sec �

4� �2cos

�

4�

�22

csc �

4� �2sin

�

4�

�22

cot 7�

6� �3tan

7�

6�

�33

sec 7�

6� �

2�33

cos 7�

6� �

�32

csc 7�

6� �2sin

7�

6� �

12

�12

, �32 �

�32

, 12�

�22

, �22

�12

, �32 Area � 55.31 square millimeters

Area � 339.29 square inches

400�662

3�

326.586�

�200.535��330.000�128.571�

�0.589�2.225

315�, �45�250�, �470�

x

−405°

y

−110°

x

y

640�, �80�430�, �290�

x

280°

y

70°

y

x

�

3, �

17�

32�

3, �

10�

3

x

−

y

233π

− 43π

x

y

20�

9, �

16�

93�

4, �

5�

4

x

y

29π

114π

x

y

Precalculus with Limits, Answers to Review Exercises 26

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42.

cot � � 1

sec � � �2

csc � � �2

tan � � 1

cos � ��22

sin � ��22

333202CB04_AN.qxd 1/1/70 09:39 AM Page 26


(Continued)

43. 44.

45. (a) 3 (b) (c) (d)

46. (a) (b) (c) (d)

47. (a) (b) (c) (d)

48. (a) (b) (c) (d) 5

49. 0.6494 50. 5.2408 51. 0.5621 52. 5.3860

53. 3.6722 54. 0.2045 55. 71.3 meters 56. 19.5 feet

57.

58.

59.

60.

61.

62.

63.

64.

65. 66.

67. 68.

69. 70.

cot � � ��3cot � � �2�21

21

sec � �2�3

3sec � � �

5

2

csc � � �2csc � �5�21

21

tan � � ��33

tan � � ��21

2

cos � ��32

sin � ��21

5

cot � �45

cot � � ��55

3

sec � � ��41

4sec � � �

8�55

55

csc � � ��41

5csc � �

8

3

cos � � �4�41

41tan � � �

3�55

55

sin � � �5�41

41cos � � �

�55

8

cot � � ��52

cot � � �5�11

11

sec � � �3�5

5csc � � �

6�11

11

tan � � �2�5

5tan � � �

�11

5

cos � � ��53

cos � �5

6

sin � �23

sin � � ��11

6

cot � �23

tan � �32

sec � � ��13

2cos � � �

2�1313

csc � � ��13

3sin � � �

3�1313

cot � �1

4tan � � 4

sec � � �17cos � ��17

17

csc � ��17

4sin � �

4�17

17

cot � �34tan � �

43

sec � �53cos � �

35

csc � �54sin � �

45

cot � � �19

tan � � �9

sec � � ��82cos � ��82

82

csc � ��82

9sin � �

9�8282

cot � � 5tan � �15

sec � � ��26

5cos � � �

5�2626

csc � � ��26sin � � ��2626

cot � �4

15tan � �

15

4

sec � ��241

4cos � �

4�241

241

csc � ��241

15sin � �

15�241

241

cot � � �34tan � � �

43

sec � �53cos � �

35

csc � � �54sin � � �

45

cot � �34tan � �

43

sec � �53cos � �

35

csc � �54sin � �

45

�612

2�615

�1515

4�1515

�154

14

�174

�1717

�1714

�24

3�24

2�23

cot � �2�14

5cot � �

�33

sec � �9�14

28sec � � 2

csc � �95

csc � �2�3

3

tan � �5�14

28tan � � �3

cos � �2�14

9cos � �

12

sin � �59

sin � ��32

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333202CB04_AN.qxd 1/1/70 09:39 AM Page 27

(Continued)

71. 72.

73. 74.

75.

76.

77.

78.

79.

80.

81.

82.

83. 84. 85. 0.0584

86. 0.0878 87. 3.2361 88. 4.3813

89. 90.

91. 92.

93. 94.

95. 96.

97. (a) (b) 264 cycles per second

98. (a)

(b) 12. Yes. One period is one year.(c) 1.41. 1.41 represents the maximum change in time

from the average time of sunset.�d � 18.09�

014

12

22

y � 2 sin 528�x

−4

−3

1

2

3

4

tπ

y

−4

−3

−2

−1

1

3

4

tπ

y

321−2−3x

−1

−2

−3

−5

−6

−1

y

4

3

2

−1

−2

xπππ 2

y

−

−8

−6

−4

8

x

y

π8π4

−6

−2

2

4

6

xπ6

y

x

2

−1

−2

2πππ−

y

2

1

−2

x

− ππ322

y

�0.1425�0.7568

tan 315� � �1sin 315� � ��22

; cos 315� ��22

;

tan��240�� 3

sin��240�� 32

; cos��240�� 12

;

tan��150�� 33

cos��150�� 32

;sin��150�� 12

;

tan 495� � �1cos 495� � ��22

;sin 495� ��22

;

tan�5�

4 � �1

cos�5�

4 � ��22

;sin�5�

4 ��22

;

tan�7�

3 � ��3

cos�7�

3 �12

;sin�7�

3 � ��32

;

tan �

4� 1cos

�

4�

�22

;sin �

4�

�22

;

tan �

3� �3cos

�

3�

12

;sin �

3�

�32

;

′θ

x

y

173π

′θx

y

65π−

��

3��

�

5

′θ

635°

x

y

′θ

264°

x

y

�� 85�� 84�


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333202CB04_AN.qxd 1/1/70 09:39 AM Page 28


(Continued)

99. 100.

101. 102.

103. 104.

105. 106.

107. 108.

As As

109. 110. 111. 0.41 112. 0.21

113. 114. 1.10 115. 116.

117. 118. 119. 1.24 120. 2.66

121. 122. 1.45123. 124.

125. 126.

127. 128. 129. 130.

131. 132. 133.

134. 9.6 feet 135. 1221 miles,

136.

137. False. The sine or cosine function is often useful for modeling simple harmonic motion.

138. True. The inverse sine, is defined byif and only if where

and

139. False. For each there corresponds exactly one value of

140. False. is not in the range of the arctangent function.

141. d; The period is and the amplitude is 3.

142. a; The period is and, because the graph isreflected in the -axis.

143. b; The period is 2 and the amplitude is 2.

144. c; The period is and the amplitude is 2.

145. The function is undefined because 146. (a)

(b) tan� ��

2 � �cot �

sec � � 1�cos �.

4�

xa < 0,2�

2�

3��4

y.�

��2 ≤ y ≤ ��2.�1 ≤ x ≤ 1sin y � xy � arcsin x

y � arcsin x,

d � 0.75 cos2�t3

85.6�

66.8�1

�1 � �x � 1�2

�4 � x2

x

�512

135

43

45

−1.5 1.5

�2

−�2

−4 4

�2

−�2

−1.5 1.5

�3

0

−1.5 1.5

�

−�

�0.98

�

6�

�

4�

6�0.46

��

2�

�

6

x → � , f �x� → � x → � , f �x� → �

−2

−300

300

� 2�−9

−6

9

6

tπ

2

y

4

3

2

1

−3

−4

x

− ππ322

y

tπ

1

y

ππ−−1

−2

−3

−4

x

y

t

3

2

1

y

ππ−x

4

3

2

1

−3

−4

ππ−

y

t

1

2

3

y

π2

π2

−

4

3

2

1

πx

y

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0.1 0.4 0.7

tan 9.9666 2.3652 1.1872

9.9666 2.3652 1.1872��cot �

��

2�

1.0 1.3

tan 0.6421 0.2776

0.6421 0.2776��cot �

��

2�

333202CB04_AN.qxd 1/1/70 09:39 AM Page 29

(Continued)

147. The ranges of the other four trigonometric functions areor

148. (a) The displacement is increased.(b) The friction damps the oscillations more quickly.(c) The frequency of the oscillations increases.

149. (a) (b)The area function increases more rapidly.

150. Answers will vary.

03

30

A s

00

0 6

4

A s

s � 10�, � > 0s � 0.8r, r > 0A � 50�, � > 0;A � 0.4r2, r > 0;

�� , �1� � �1, �.�� , �


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333202CB04_AN.qxd 1/1/70 09:39 AM Page 30

Precalculus with Limits, Answers to Chapter Test 31

Chapter Test (page 369)

1. (a) (b)

(c)

2. 3000 radians per minute 3. square feet

4. sin csc

cos sec

tan cot

5. For For

sin

cos

csc

sec

cot

6.

7. Quadrant III 8. 9. 1.33, 1.81

10. 11.

12. 13.

14. 15.

Period: 2 Not periodic

16. 17.

18. 19.

20. d � �6 cos �t

310.1�

1−2 2

−

x

π

π

y

�5

2a � �2, b �

12

, c � ��

4

0

−2

32

6

−6

−4

6

4

α1

2

3

4

− π π2

π−π2

y

−4

−3

−2

−1

1

3

4

π2x

y

cot � � �8

15cot � � �34

csc � �1715sec � �

53

tan � � �158csc � � �

54

cos � � �817tan � � �

43

sin � �1517sin � � �

45

150�, 210�

′θ

290°

x

y

�� 70�

cot � �23

� �2

3

sec � � ��13

2� �

�13

2

csc � � ��13

3� �

�13

3

cos � � �2�13

13� �

2�13

13

sin � � �3�13

13� �

3�13

13

� ≤ � <3�

2:0 ≤ � <

�

2:

� � �1

3� � �3

� � ��10� � ��10

10

� ��10

3� �

3�10

10

� 709.04

225�

13�

4, �

3�

4

54π

x

y

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333202CB04_AN.qxd 1/1/70 09:39 AM Page 31

Problem Solving (page 371)

1. (a) radians or (b) feet

2. Gear 1:

Gear 2:

Gear 3:

Gear 4:

Gear 5:

3. (a) 4767 feet (b) 3705 feet(c) feet,

4. (a) Answers will vary. (b) The ratios are equal.(c) No; No(d) Yes, because all six trigonometric functions are ratios

of right triangles.5. (a) (b)

Even Even

6. (a) is even. (b) is even.

7.

8. (a)

(b) answers will vary.(c) 20 millimeters; answers will vary.

(d) 80 beats per minute (e)

9. (a)

(b) (c)

10. (a) (b) Period of

Period of

(c) Yes, because the sine and cosine functions are periodic.

11. (a) 3.35, 7.35 (b)(c) Yes. There is a difference of nine periods between the

values.

12. (a) Equal; two-period shift(b) Not equal; is a horizontal translation and

is a period change.(c) Not equal; For example,

13. (a) (b)(c)(d) As you move closer to the rock, must get smaller and

smaller. The angles and will decrease along withthe distance so will decrease.

14. (a) The approximation isaccurate over the interval

(b) The accuracy improved.

−2

2

�2

−�2

x9

9

�1 ≤ x ≤ 1.

−2

2

�2

−�2

dy,�2�1

d� 1.75 feet

x � 1.71 feet; y � 3.46 feet40.5�

sin�12�� 2�� sin�1

2��.f �1

2t�f �t �

12c�

�0.65

g : �

f : 2� ;

−6

−

6

� �

g

f

I�7369� � 0.945E�7369� � 0.901P�7369� � 0.631

−2

7348 7377

2

P

I

E

−2

7300 7380

2

P IE

Period �1516

seconds; 32�

15

Period �34 seconds;

070

5

130

h � 51 � 50 sin8�t ��

2h�x�h�x�

−1

−2

3

2� �

−1

−2

3

2� �

tan 63� �w � 3705

3000

w � 2183

48�

19 radians� 454.7�,

5�

2 radians450�,

24�

11 radians� 392.7�,

24�

13 radians� 332.3�,

3�

2 radians270�,

� 816.42990�11�

2

Precalculus with Limits, Answers to Problem Solving 32

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Documents

Chapter 4 16.college.cengage.com/mathematics/larson/precalculus_limits/1e2/... · Vocabulary Check (page 290) 1. Trignometry 2. angle 3. coterminal 4. radian 5. acute; ... 1 3 tan