+ All Categories
Home > Documents > Chapter 5 x Section 5.1 (page 379) - Higher Ed eBooks...

Chapter 5 x Section 5.1 (page 379) - Higher Ed eBooks...

Date post: 14-Mar-2018
Category:
Upload: phunghanh
View: 231 times
Download: 0 times
Share this document with a friend
17
Copyright © Houghton Mifflin Company. All rights reserved. Precalculus with Limits, Answers to Section 5.1 1 Chapter 5 Section 5.1 (page 379) Vocabulary Check (page 379) 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 1. 2. sin csc cot 3. 4. sin cos sec cot 5. 6. cos tan csc sec 7. 8. sin tan csc sec cot 9. 10. 11. sin 12. cos csc sec cot 13. 14. sin cos is undefined. is undefined. csc sec is undefined. is undefined. cot 15. d 16. a 17. b 18. f 19. e 20. c 21. b 22. c 23. f 24. a 25. e 26. d 27. 28. 29. 30. 1 31. 32. 33. 34. 35. 1 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 1 49. 50. 51. 52. sin 4 x sec 4 x cos x 2 sec x 1 sin 2 x tan 2 x cos 2 x sin 2 x csc sec cos u sin u 2 sec sec sec t 1 sin y sin x tan x sin 2 cos 2 x sin 2 x cot cos x cos 2 sin csc 0 sec csc 1 1 cot 0 tan tan 0 cos 0 1 sin 1 cot 26 1 2 csc 5 5 sec 56 12 5 2 tan 6 12 tan 2 cos 26 5 5 5 sin 1 5 25 5 cot x 15 15 cot x 22 csc x 415 15 sec x 32 4 sec x 4 csc x 3 tan x 15 tan x 2 4 cos x 1 4 cos x 22 3 sin x 15 4 sin x 1 3 x 4 3 cot 25 5 x 5 4 csc 35 5 x 5 3 sec 3 2 x 3 4 tan 5 2 cos x 4 5 cos 2 3 x 3 5 sin 5 3 10 3 cot x 12 5 10 csc x 13 5 cot 3 sec x 13 12 1 3 tan x 5 12 310 10 cos x 12 13 sin 10 10 sin x 5 13 4 3 cot 1 csc 5 3 csc 2 5 4 sec 2 tan 3 4 tan 1 4 5 cos 2 2 3 5 sin 2 2 x 3 cot x 3 3 sec x 23 3 sec x 2 x 2 csc x 23 3 tan x 3 3 tan x 3 cos x 3 2 cos x 1 2 x 1 2 sin x 3 2 tan u cos u csc u cos u sec 2 u cot 2 u csc u cot u cos u tan u 333202CB05_AN.qxd 1/1/70 09:40 AM Page 1
Transcript
Page 1: Chapter 5 x Section 5.1 (page 379) - Higher Ed eBooks ...college.cengage.com/mathematics/larson/precalculus_limits/1e2/... · 4 5 cos 2 2 3 5 sin 2 2 cot x x 3 3 3 sec x 2 3 3 sec

Cop

yrig

ht ©

Hou

ghto

n M

iffl

in C

ompa

ny. A

ll ri

ghts

res

erve

d.Precalculus with Limits, Answers to Section 5.1 1

Chapter 5Section 5.1 (page 379)

Vocabulary Check (page 379)1. 2. 3. 4.5. 6. 7. 8.9. 10.

1. 2. sin

csc

cot

3. 4. sin

cos

sec

cot

5. 6.

cos

tan

csc

sec

7. 8. sin

tan

csc

sec

cot

9. 10.

11. sin 12.

cos

csc

sec

cot

13. 14. sin

cos

is undefined. is undefined.

csc

sec is undefined.

is undefined. cot

15. d 16. a 17. b 18. f

19. e 20. c 21. b 22. c

23. f 24. a 25. e

26. d 27. 28.

29. 30. 1 31.

32. 33. 34.

35. 1 36. 37.

38. 39. 40.

41. 42. 43.

44. 45. 46.

47. 48. 1 49.

50. 51. 52. sin4 xsec4 xcos x � 2

sec x � 1sin2 x tan2 x

cos2 xsin2 xcsc � sec �

cos u � sin u2 sec �sec �

sec t1 � sin ysin x

tan xsin2 �

cos 2 xsin2 xcot �

cos xcos2 �

sin �csc �

� � 0sec �

�csc � � �1

� � 1cot � � 0

tan �tan �

� � 0cos � � 0

� � 1sin � � �1

cot � � 2�6� �1

2

csc � � �5� � ��5

sec � � �5�612

� � ��5

2

tan � ��612

tan � � 2

cos � � �2�6

5� � �

�5

5

sin � � �15

� � �2�5

5

cot x ��1515

cot x � �2�2

csc x �4�15

15sec x � �

3�24

sec x � 4csc x � 3

tan x � �15tan x � ��24

cos x �14

cos x � �2�2

3

sin x ��15

4sin x �

13

x �4

3cot � � �

2�55

x �5

4csc � � �

3�55

x �5

3sec � �

32

x �3

4tan � � �

�52

cos x �45

cos � �23

x �3

5sin � � �

�53

� � ��10

3cot x �

12

5

� � �10csc x � �13

5

cot � � �3sec x � �1312

� � �1

3tan x �

512

� � �3�10

10cos x � �

12

13

sin � ��1010

sin x � �5

13

� �4

3cot � � �1

csc � �53

csc � � ��2

� �5

4sec � � �2

tan � �34

tan � � �1

� �4

5cos � �

�22

� �3

5sin � � �

�22

x � �3cot x � ��33

sec x � �2�3

3sec x � �2

x � �2csc x �2�3

3

tan x ��33

tan x � ��3

cos x � ��32

cos x � �12

x � �1

2sin x �

�32

�tan ucos ucsc ucos usec2 ucot2 ucsc ucot ucos utan u

333202CB05_AN.qxd 1/1/70 09:40 AM Page 1

Page 2: Chapter 5 x Section 5.1 (page 379) - Higher Ed eBooks ...college.cengage.com/mathematics/larson/precalculus_limits/1e2/... · 4 5 cos 2 2 3 5 sin 2 2 cot x x 3 3 3 sec x 2 3 3 sec

Cop

yrig

ht ©

Hou

ghto

n M

iffl

in C

ompa

ny. A

ll ri

ghts

res

erve

d.

Precalculus with Limits, Answers to Section 5.1 2

(Continued)

53. 54.

55. 56.

57. 58.

59. 60. 61.

62. 63. 64.

65. 66.

67. 68.

69.

70.

71.

72.

73. 74. 75.

76. 77. 78.

79. 80. 81.

82. 83.

84.

85.

86.

87. 88.

89. 90.

91. 92.

93. 94. 0

95. (a)

(b) csc22�

7� cot2

2�

7� 1.6360 � 0.6360 � 1

csc2 132 � cot 2 132 � 1.8107 � 0.8107 � 1

ln�csc t sec t�ln�tan x�ln�cot x�0 < � < �0 ≤ � <

2,

3�

2< � < 2�

2≤ � ≤

3�

20 ≤ � ≤ �

10 sin � � �5�3; sin � � ��32

; cos � �12

4 sin � � 2�2; sin � ��22

; cos � ��22

6 cos � � 3; sin � � ±�32

; cos � �12

3 cos � � 3; sin � � 0; cos � � 110 sec �

5 sec �2 tan �3 tan �

8 sin �3 sin �sec �

tan xcot xcsc x

y1� y2

00 �

2

1200

y1 � y2

10 �

2

12

y1 � y2

00 �

2

6

y1 � y2

00 �

2

1

tan4 x �csc x � 1�3�sec x � tan x�

5�sec x � tan x�1 � cos y

�cot x2 sec x�2 cot 2 x

2 csc2 x9 cos2 x4 cot2 x

�11 � 2 sin x cos x

tan2 x�sec x � 1�cot2 x�csc x � 1�sec2 x � tan2 xsin2 x � cos2 x

0.2 0.4 0.6 0.8 1.0

0.0428 0.2107 0.6871 2.1841 8.3087

0.0428 0.2107 0.6871 2.1841 8.3087y2

y1

x

1.2 1.4

50.3869 1163.6143

50.3869 1163.6143y2

y1

x

0.2 0.4 0.6 0.8 1.0

1.2230 1.5085 1.8958 2.4650 3.4082

1.2230 1.5085 1.8958 2.4650 3.4082y2

y1

x

1.2 1.4

5.3319 11.6814

5.3319 11.6814y2

y1

x

0.2 0.4 0.6 0.8 1.0

0.0403 0.1646 0.3863 0.7386 1.3105

0.0403 0.1646 0.3863 0.7386 1.3105y2

y1

x

1.2 1.4

2.3973 5.7135

2.3973 5.7135y2

y1

x

0.2 0.4 0.6 0.8 1.0

0.1987 0.3894 0.5646 0.7174 0.8415

0.1987 0.3894 0.5646 0.7174 0.8415y2

y1

x

1.2 1.4

0.9320 0.9854

0.9320 0.9854y2

y1

x

333202CB05_AN.qxd 1/1/70 09:40 AM Page 2

Page 3: Chapter 5 x Section 5.1 (page 379) - Higher Ed eBooks ...college.cengage.com/mathematics/larson/precalculus_limits/1e2/... · 4 5 cos 2 2 3 5 sin 2 2 cot x x 3 3 3 sec x 2 3 3 sec

Cop

yrig

ht ©

Hou

ghto

n M

iffl

in C

ompa

ny. A

ll ri

ghts

res

erve

d.Precalculus with Limits, Answers to Section 5.1 3

(Continued)

96. (a)

(b)

97. (a)

(b)

98. (a)

(b)

99.

100. Answers will vary.

101. True. For example,

102. False. A cofunction identity can be used to transform atangent function so that it can be represented by a cotan-gent function.

103. 1, 1 104. 1, 1 105. 106.

107. Not an identity because

108. Not an identity because

109. Not an identity because

110. Not an identity because

111. An identity because sin

112. Not an identity because

113–114. Answers will vary.

115. 116.

117. 118.

119. 120.

121. 122.

123. 124.y

x

−3

−2

−1

3

4

1

5

πππ 2−

y

x

−2

−4

−3

2

3

1

4

π3 π22

π2

y

x

−1

−2

−3

3

1 3−1−3

y

x1 3

−1

−2

1

2

x�x2 � 5x � 1�x2 � 25

�5x2 � 8x � 28�x2 � 4��x � 4�

3�2x � 1�x � 4

x2 � 6x � 8�x � 5��x � 8�

4z � 12�z � 9x � 25

1sin �

1

� �1

sin �� 1

5

cos �

1

5 cos �

sin k�

cos k�� tan k�

cot � � ±�csc2 � � 1

cos � � ±�1 � sin2 �

0, ���, 0

sin��x� � �sin x.

� tan �

sin��12� � �sin�1

2� � �0.4794

sin��250� � �sin 250 � 0.9397

cos��

2� 0.8� � sin 0.8 � 0.7174

cos�90 � 80� � sin 80 � 0.9848

�tan 3.1�2 � 1 � �sec 3.1�2 � 1.0017

�tan 346�2 � 1 � �sec 346�2 � 1.0622

333202CB05_AN.qxd 1/1/70 09:40 AM Page 3

Page 4: Chapter 5 x Section 5.1 (page 379) - Higher Ed eBooks ...college.cengage.com/mathematics/larson/precalculus_limits/1e2/... · 4 5 cos 2 2 3 5 sin 2 2 cot x x 3 3 3 sec x 2 3 3 sec

Cop

yrig

ht ©

Hou

ghto

n M

iffl

in C

ompa

ny. A

ll ri

ghts

res

erve

d.

Precalculus with Limits, Answers to Section 5.2 4

Section 5.2 (page 387)

Vocabulary Check (page 387)1. identity 2. conditional equation 3.4. 5. 6.7. 8.

1–38. Answers will vary.39. (a) (b)

Identity

(c) Answers will vary.

40. (a) (b)

Identity

(c) Answers will vary.

41. (a) (b)

Not an identity

(c) Answers will vary.

42. (a) (b)

Not an identity

(c) Answers will vary.

43. (a) (b)

Identity

(c) Answers will vary.

44. (a) (b)

Identity

(c) Answers will vary.45. (a) (b)

Not an identity

(c) Answers will vary.46. (a) (b)

Not an identity

(c) Answers will vary.47–50. Answers will vary. 51. 152. 1 53. 2 54. 2 55. Answers will vary.56. (a) Answers will vary.

(b)

(c) Greatest: Least: (d) Noon57. False. An identity is an equation that is true for all real

values of 58. True. An identity is an equation that is true for all real

values in the domain of the variable.59. The equation is not an identity because

Possible answer:

60. The equation is not an identity because

Possible answer:

61. 62. 63.

64. 65. 66.

67. 68. 14�1 ± �7�1 ± �5

�5 ± �532

�3 ± �21�9 � 46i

�8 � 4i�21 � 20i2 � �3 � �26 �i

3�

4

±�sec2 � � 1.tan � �

7�

4

±�1 � cos2 � .sin � �

�.

9010;

−5

3

−2� 2�

−3

−2� 2�

3

y2 y1

−1

−2� 2�

1

−1

−2� 2�

5

−5

−� �

5

−1

−2� 2�

5

y2

y1

−1

−2� 2�

3

−5

−5

5

5

sec u�csc usin ucos2 ucot u

tan u

28.36 13.74 8.66 5.96 4.20s

5040302010�

2.89 1.82 0.88 0s

90807060�

333202CB05_AN.qxd 1/1/70 09:40 AM Page 4

Page 5: Chapter 5 x Section 5.1 (page 379) - Higher Ed eBooks ...college.cengage.com/mathematics/larson/precalculus_limits/1e2/... · 4 5 cos 2 2 3 5 sin 2 2 cot x x 3 3 3 sec x 2 3 3 sec

Cop

yrig

ht ©

Hou

ghto

n M

iffl

in C

ompa

ny. A

ll ri

ghts

res

erve

d.Precalculus with Limits, Answers to Section 5.3 5

Section 5.3 (page 396)

Vocabulary Check (page 396)1. general 2. quadratic 3. extraneous

1–6. Answers will vary. 7.

8. 9.

10. 11.

12. 13.

14.

15. 16.

17. 18.

19.

20.

21. 22.

23. 24.

25. 26.

27. No solution 28. 0

29. 30.

31. 32.

33. 34.

35. 36.

37. 38.

39. 40.

41. 42.

43. 44.

45. 2.678, 5.820

46. 0.785, 2.356, 3.665, 3.927, 5.498, 5.760

47. 1.047, 5.236 48. 0.524, 2.618

49. 0.860, 3.426 50. 4.917

51. 0, 2.678, 3.142, 5.820

52. 0.515, 2.726, 3.657, 5.868

53. 0.983, 1.768, 4.124, 4.910

54. 0.524, 0.730, 2.412, 2.618

55. 0.3398, 0.8481, 2.2935, 2.8018

56. 0.5880, 2.0344, 3.7296, 5.1760

57. 1.9357, 2.7767, 5.0773, 5.9183

58. 1.7794, 4.5038

59.

60.

61. 62.

63. (a) (b)

Maximum:Minimum:

64. (a)

Maximum:

Maximum:

Minimum:

Minimum:65. 1 66. 0.739

67. (a) All real numbers except

(b) -axis symmetry; Horizontal asymptote:

(c) Oscillates (d) Infinitely many solutions

(e) Yes, 0.6366

68. (a) All real numbers except

(b) -axis symmetry (c) approaches 1.

(d) Four solutions: ±�, ±2�

yy

x � 0x

y � 1y

x � 0x

�4.7124, �3��1.5708, 1��2.6180, 1.5��0.5236, 1.5�

0

−3

2�

3

�3.9270, �1.4142��0.7854, 1.4142�

5�

4� 3.9270

4� 0.7854

−3

0 2�

3

6,

5�

6�

3,

5�

3

arctan��2� � �, arctan��2� � 2�, �

4,

5�

4

4,

5�

4, arctan 5, arctan 5 � �

�2 � 8n, 2 � 8n�2 � 6n, 2 � 6n

34 � n�1 � 4n

8�

3� 4n�,

10�

3� 4n�

2� 4n�,

7�

2� 4n�

12�

n�

2,

5�

12�

n�

2�

12�

n�

3

2�

3� n�,

5�

6� n�

6� n�,

5�

6� n�

4,

5�

4�

2

3,

5�

3

6,

5�

6,

7�

6,

11�

6

7�

6,

3�

2,

11�

6�,

3,

5�

3

3,

5�

3

3,

5�

3, �

2,

3�

2,

2�

3,

4�

30, �,

6,

5�

6,

7�

6,

11�

6

0, �0, �

2, �,

3�

2

4�

n�

2,

2�

3� 2n�,

4�

3� 2n�

n�

3,

4� n�

9�

n�

3,

2�

9�

n�

3�

8�

n�

2,

3�

8�

n�

2

3� n�,

2�

3� n�

3� n�,

2�

3� n�

6� n�,

5�

6� n�,

3� n�,

2�

3� n�

n�, 3�

2� 2n�

3� n�,

2�

3� n�

6� n�,

5�

6� n�

2�

3� n�

3� 2n� ,

2�

3� 2n�

7�

6� 2n�,

11�

6� 2n�

2�

3� 2n�,

4�

3� 2n�

(b)

3�

2� 4.7124

5�

6� 2.618

2� 1.5708

6� 0.5236

333202CB05_AN.qxd 1/1/70 09:40 AM Page 5

Page 6: Chapter 5 x Section 5.1 (page 379) - Higher Ed eBooks ...college.cengage.com/mathematics/larson/precalculus_limits/1e2/... · 4 5 cos 2 2 3 5 sin 2 2 cot x x 3 3 3 sec x 2 3 3 sec

Cop

yrig

ht ©

Hou

ghto

n M

iffl

in C

ompa

ny. A

ll ri

ghts

res

erve

d.

Precalculus with Limits, Answers to Section 5.3 6

(Continued)

69. 0.04 second, 0.43 second, 0.83 second

70. 1.96 seconds

71. February, March, and April

72. January, October, November, December

73. 74.

75. (a) Between seconds and seconds

(b) 5 times: seconds

76. (a)

(b) 1

(c) The constant term, 5.45

(d) years

(e) 2010

77. (a) (b)

78. (a)

(b)

For the approximation appears to be good.

(c) 3.46, 8.81

3.46 is close to the zero of in the interval

79. True. The first equation has a smaller period than the sec-ond equation, so it will have more solutions in the interval

80. False. There is no value of for which

81. 1 82. 3

83. 84.

85. 86.

87. 88.

89. 90. 30 feet

91. Answers will vary.

1.36

tan��1410� ��33

tan��1845� � �1

cos��1410� ��32

cos��1845� ��22

sin��1410� �12

sin��1845� � ��22

tan�600� � �3tan 390 ��33

cos�600� � �12

cos 390 ��32

sin�600� � ��32

sin 390 �12

c � 5.0b � 50.1

a � 15.4a � 54.8

C � 19C � 24

sin x � 3.4.x

0, 2��.

0, 6.f

3.5 ≤ x ≤ 6,

−4

0 6

fg

4

103

A � 1.12

0.6 < x < 1.1

−2

0 �2

2

�13.37

y

t

8

6

4

2

2 4 6 8 10 12 14

Une

mpl

oym

ent r

ate

Year (0 ↔ 1990)

t � 16, 48, 80, 112, 144

t � 24t � 8

1.936.9, 53.1

−4

0 10

4

333202CB05_AN.qxd 1/1/70 09:40 AM Page 6

Page 7: Chapter 5 x Section 5.1 (page 379) - Higher Ed eBooks ...college.cengage.com/mathematics/larson/precalculus_limits/1e2/... · 4 5 cos 2 2 3 5 sin 2 2 cot x x 3 3 3 sec x 2 3 3 sec

Cop

yrig

ht ©

Hou

ghto

n M

iffl

in C

ompa

ny. A

ll ri

ghts

res

erve

d.Precalculus with Limits, Answers to Section 5.4 7

Section 5.4 (page 404)

Vocabulary Check (page 404)1.

2. 3.

4.

5. 6.

1. (a) (b)

2. (a) (b)

3. (a) (b)

4. (a) (b)

5. (a) (b)

6. (a) (b)

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

tan��165� � 2 � �3

cos��165� � ��24

�1 � �3 �

sin��165� � ��24

��3 � 1�

tan��105� � 2 ��3

cos��105� ��2

4�1 � �3 �

sin��105� � ��2

4��3 � 1�

tan 285 � ��2 � �3 �

cos 285 ��2

4��3 � 1�

sin 285 � ��2

4��3 � 1�

tan�� �

12� � �2 � �3

cos�� �

12� ��2

4��3 � 1�

sin�� �

12� ��2

4�1 � �3 �

tan 17�

12� 2 � �3

cos 17�

12�

�2

4�1 � �3 �

sin 17�

12� �

�2

4��3 � 1�

tan 7�

12� �2 ��3

cos 7�

12�

�2

4�1 � �3 �

sin 7�

12�

�2

4��3 � 1�

tan 11�

12� �2 � �3

cos 11�

12� �

�2

4��3 � 1�

sin 11�

12�

�2

4��3 � 1�

tan 255 � 2 ��3

cos 255 ��2

4�1 � �3 �

sin 255 � ��2

4��3 � 1�

tan 195 � 2 � �3

cos 195 � ��2

4��3 � 1�

sin 195 ��2

4�1 � �3 �

tan 165 � �2 ��3

cos 165 � ��2

4��3 � 1�

sin 165 ��2

4��3 � 1�

tan 105 � �2 � �3

cos 105 ��2

4�1 � �3 �

sin 105 ��2

4��3 � 1�

��2 � �32

�2 � �64

��3 � 1

2

1

2

�2 � 1

2��6 � �2

4

�2 � 1

2

�2 � �6

4

�2 � �32

�6 � �24

�1 � �22

��2 � �64

tan u � tan v1 � tan u tan v

cos u cos v � sin u sin v

sin u cos v � cos u sin v

tan u � tan v1 � tan u tan v

cos u cos v � sin u sin v

sin u cos v � cos u sin v

333202CB05_AN.qxd 1/1/70 09:40 AM Page 7

Page 8: Chapter 5 x Section 5.1 (page 379) - Higher Ed eBooks ...college.cengage.com/mathematics/larson/precalculus_limits/1e2/... · 4 5 cos 2 2 3 5 sin 2 2 cot x x 3 3 3 sec x 2 3 3 sec

Cop

yrig

ht ©

Hou

ghto

n M

iffl

in C

ompa

ny. A

ll ri

ghts

res

erve

d.

Precalculus with Limits, Answers to Section 5.4 8

(Continued)

18.

19.

20.

21.

22.

23. 24. 25.

26. 27. 28.

29. 30. 31.

32. 33. 34.

35. 36. 37.

38. 39. 40.

41. 42. 43.

44. 45. 46.

47. 48. 49. 50.

51. 1 52. 53. 0

54. 55–64. Answers will vary.

65. 66. 67.

68. 69. 70. 71.

72. 73. 74.

75. (a)

(b) feet (c) cycle per second

76. Answers will vary.

77. False.

78. False.

79. False.

80. True.

81– 84. Answers will vary.

85. (a) (b)

86. (a) (b)

87. (a) (b)

88. (a) (b)

89. 90. 91. Proof

92. (a) All real numbers except

(b)

x � 0x

�5�2

2 sin � �

5�2

2 cos �2 cos �

�2 cos�2� ��

4��2 sin�2� ��

4�13 cos�3� � 1.1760�13 sin�3� � 0.3948�

5 cos�2� � 0.6435�5 sin�2� � 0.9273�

�2 cos�� ��

4��2 sin�� ��

4�

sin�x ��

2� � sin x cos �

2� cos x sin

2� �cos x

cos�x ��

2� � cos x cos �

2� sin x sin

2� sin x

cos�u ± v� � cos u cos v � sin u sin v

sin�u ± v� � sin u cos v ± cos u sin v

1

5

12

y �5

12 sin�2t � 0.6435�

0, ��

4,

7�

40,

3, �,

5�

3

5�

4,

7�

4�

3,

5�

3�

2tan �

�cos ��cos x�sin x

x � x�1 � x2

�x2 � 1

2x2 � �1 � x2

�4x2 � 1

117125

53

11744�

44117

45

35�

1663

6556

6533�

6316

�3365

1665

5665

�6365

�33

�1

�22

�32

�22

��32

cos�3x � 2y�tan 3x

cos 12�

35sin 1.8tan 80

tan 239sin 190cos 40

tan 5�

12� �3 � 2

cos 5�

12�

�2

4��3 � 1�

sin 5�

12�

�2

4�1 � �3 �

tan��13�

12 � � �2 � �3

cos��13�

12 � � ��2

4��3 � 1�

sin��13�

12 � ��2

4��3 � 1�

tan��7�

12� � 2 � �3

cos��7�

12� ��24

�1 � �3 �

sin��7�

12� � ��24

��3 � 1 �

tan 13�

12� 2 � �3

cos 13�

12� �

�24

�1 � �3 �

sin 13�

12�

�24

�1 � �3 �

tan 15 � 2 � �3

cos 15 ��24

�1 � �3 �

sin 15 ��24

��3 � 1�

0.01 0.02 0.05

�0.521�0.509�0.504g�h�

�0.521�0.509�0.504f �h�

h

0.1 0.2 0.5

�0.691�0.583�0.542g�h�

�0.691�0.583�0.542f �h�

h

333202CB05_AN.qxd 1/1/70 09:40 AM Page 8

Page 9: Chapter 5 x Section 5.1 (page 379) - Higher Ed eBooks ...college.cengage.com/mathematics/larson/precalculus_limits/1e2/... · 4 5 cos 2 2 3 5 sin 2 2 cot x x 3 3 3 sec x 2 3 3 sec

Cop

yrig

ht ©

Hou

ghto

n M

iffl

in C

ompa

ny. A

ll ri

ghts

res

erve

d.Precalculus with Limits, Answers to Section 5.4 9

(Continued)

(c)

(d) As the function approaches and thefunction approaches

93. 94.95.

96. (a) and (b) Answers will vary.

97.

98.

99. Because is not one-to-one, does not exist.

100. 101.

102. 103.

104. x2 � 10x

6x � 33x2

4x � 3f �1�x� � x2 � 16

f �1f

f �1�x� � �8x � 7

f �1�x� �x � 15

5

sin2�� ��

4� � sin2�� ��

4� � 1

−3

−2� 2�

3

1515

�0.5.g�h��0.5f �h�h → 0,

−2

−3 3

2

333202CB05_AN.qxd 1/1/70 09:40 AM Page 9

Page 10: Chapter 5 x Section 5.1 (page 379) - Higher Ed eBooks ...college.cengage.com/mathematics/larson/precalculus_limits/1e2/... · 4 5 cos 2 2 3 5 sin 2 2 cot x x 3 3 3 sec x 2 3 3 sec

Cop

yrig

ht ©

Hou

ghto

n M

iffl

in C

ompa

ny. A

ll ri

ghts

res

erve

d.

Precalculus with Limits, Answers to Section 5.5 10

Section 5.5 (page 415)

Vocabulary Check (page 415)1. 2.3.

4. 5.

6.

7.

8.

9.

10.

1. 2. 3. 4. 5.

6. 7. 8. 9.

10. 11.

12. 13.

14. 15.

16. 17.

18. 19.

20. 21. 22.

23. sin 24. sin

cos cos

tan tan

25. sin 26. sin

cos cos

tan tan

27. sin 28. sin

cos cos

tan tan

29.

30.

31.

32.

33.

34.

35. 36.

37. 38.

39. 40. 4

41. 42. sin

cos

tan

43. sin

cos

tan

44. sin 45. sin

cos cos

tan tan

46. sin 47.

cos

tan

48. 49.

50. 51. sin

cos

tan u

2�

8 ��89

5tan

u2

�12

u

2� ��89 � 8�89

178cos

u2

�2�5

5

u

2��89 � 8�89

178sin

u2

��55

tan u2

� 5tan 7�

12� �2 � �3

cos u2

��2626

cos 7�

12� �

12�2 � �3

sin u2

�5�26

26sin

7�

12�

12�2 � �3

tan 3�

8� �2 � 1

12� 2 � �3

cos 3�

8�

12�2 � �2

12�

1

2�2 � �3

sin 3�

8�

12�2 � �2

12�

1

2�2 � �3

8� �2 � 167 30� � 1 ��2

8�

1

2�2 � �267 30� �

1

2�2 � �2

8�

1

2�2 � �267 30� �

1

2�2 � �2

112 30� � �1 � �2

112 30� � �12�2 � �2

112 30� �12�2 � �2

165 � �3 � 2 tan 75 � 2 ��3

165 � �12�2 ��3 cos 75 �

12�2 � �3

165 �12�2 � �3 sin 75 �

12�2 � �3

�17

�174

1

4

�1717

4�1717

116�1 � cos 2x � cos 4x � cos 2x cos 4x�

116�1 � cos 2x � cos 4x � cos 2x cos 4x�

1128�3 � 4 cos 4x � cos 8x�

18�1 � cos 4x�

1128�35 � 48 cos 2x � 28 cos 4x �16 cos 2x cos 4x � cos 8x�

18�3 � 4 cos 2x � cos 4x�

2u � �4�2

7 2u �

4�21

17

2u �7

9 2u � �

17

25

2u � �4�2

3 2u � �

4�21

25

2u � �8152u �

247

2u �15172u �

725

2u � �8172u �

2425

2u � 4�5 2u � �24

7

2u � �1

9 2u � �

7

25

2u � �4�5

9 2u �

24

25

cos 2x4 cos 2x3 cos 2x

3 sin 2x0, �

4,

2,

3�

4, �,

5�

4,

3�

2,

7�

4

0, �

2, �,

3�

2

6,

2,

5�

6,

3�

2

2,

6,

5�

6,

7�

6,

3�

2,

11�

6

2,

7�

6,

11�

6

0, 2�

3,

4�

3

2,

3�

2,

4,

3�

4,

5�

4,

7�

4

12,

5�

12,

13�

12,

17�

12

2,

7�

6,

3�

2,

11�

6

0, �

3, �,

5�

3158

178

1715

815

817

1517

14

�1717

�2 sin�u � v2 � sin�u � v

2 �2 sin�u � v

2 � cos�u � v2 �

12sin�u � v� � sin� u � v�

12cos�u � v� � cos�u � v�

1 � cos usin u

�sin u

1 � cos u

±�1 � cos u2

tan2 u

cos2 u � sin2 u � 2 cos2 u � 1 � 1 � 2 sin2 ucos2 u2 sin u cos u

333202CB05_AN.qxd 1/1/70 09:40 AM Page 10

Page 11: Chapter 5 x Section 5.1 (page 379) - Higher Ed eBooks ...college.cengage.com/mathematics/larson/precalculus_limits/1e2/... · 4 5 cos 2 2 3 5 sin 2 2 cot x x 3 3 3 sec x 2 3 3 sec

Cop

yrig

ht ©

Hou

ghto

n M

iffl

in C

ompa

ny. A

ll ri

ghts

res

erve

d.Precalculus with Limits, Answers to Section 5.5 11

(Continued)

52. sin 53. sin

cos cos

tan tan

54. sin 55.

cos

tan

56. 57. 58.

59. 60.

61. 62.

63. 64.

65. 66.

67. 68.

69. 70.

71. 72.

73.

74. 75.

76. 77.

78. 79.

80.

81. 82.

83. 84.

85. 86.

87.

88.

89. 90.

91. 92. 93. 94.95–110. Answers will vary.

111. 112.

113. 114.

115. 116.

x

2

1

−1

−2

ππ 2

y

x

2

1

−1

−2

ππ 2

y

−3

−� �

3

−3

−2� 2�

3

−3

−2� 2�

3

−3

−2� 2�

3

3665

413

144169

25169

0

−1

2�

1

0

−2

2�

2

0, �

2, �,

3�

2,

4,

3�

4,

5�

4,

7�

4

6,

5�

6

0

−2

2�

2

0, �

4,

2,

3�

4, �,

5�

4,

3�

2,

7�

4

0

−2

2�

2

0, �

4,

2,

3�

4, �,

5�

4,

3�

2,

7�

4

��2��2

�3 � 12

�3 � 12

2 sin x cos �

2� 0�2 sin � sin

2

2 cos�� � �� cos � � �2 cos�� � ��

2 cos � sin �2 sin 3x cos 2x

2 cos 4x cos 2x2 sin 2� cos �

2 cos 4� sin �12�cos 2� � cos 2�� �

12�1 � cos 2��

12 �sin 2� � sin 2��

12�sin 2x � sin 2y�1

2�cos 2y � cos 2x�

12 �cos 2� � cos 6��5

2�cos 8� � cos 2��

32�cos � � cos 5��1

2 �sin 10� � sin 2��

3�sin 60 � sin 30�5�cos 60 � cos 90�

2�sin 7�

6� sin

2�3�sin �

2� sin 0�

0

−3

2�

3

0

−2

2�

2

0, �

2,

3�

2�

3, �,

5�

3

−2

0 2�

1

0

−2

2�

2

0, �

3,

5�

3�

��sin�x � 1

2 ����tan 4x��cos 2x�

u

2�

3�5

5

u

2�

�70

14

�sin 3x�u

2�

3�14

14

u

2� �3

u

2� �3 � �10

u

2� �

�10

10

u

2� �

1

2�10 � 3�10

5

u

2�

3�10

10

u

2�

1

2 �10 � 3�10

5

333202CB05_AN.qxd 1/1/70 09:40 AM Page 11

Page 12: Chapter 5 x Section 5.1 (page 379) - Higher Ed eBooks ...college.cengage.com/mathematics/larson/precalculus_limits/1e2/... · 4 5 cos 2 2 3 5 sin 2 2 cot x x 3 3 3 sec x 2 3 3 sec

Cop

yrig

ht ©

Hou

ghto

n M

iffl

in C

ompa

ny. A

ll ri

ghts

res

erve

d.

Precalculus with Limits, Answers to Section 5.5 12

(Continued)

117. 118. 119.

120. (a)

(b)

The area is maximum when

121. (a) (b) 0.4482

(c) 760 miles per hour; 3420 miles per hour

(d)

122.

123. False. For

124. False. when is in the second

quadrant.

125. (a) (b)

Maximum:

126. (a) (b)

Minima:

Maxima:

127. (a) (b)

(c) (d)

(e) No. There is often more than one way to rewrite atrigonometric expression.

128. (a)

(b) (c) Answers will vary.

129. (a)

(b) (c) Midpoint:

130. (a)

(b) (c) Midpoint:

131. (a)

(b) (c) Midpoint:

132. (a)

(b) (c) Midpoint:

133. (a) Complement: supplement:

(b) No complement; supplement: 18

12535;

��13, � 5

12�Distance �16�233

y

x

( )13

, 23

( )32

−1−2 1

−1

−2

1

−1, −

�23, 32�Distance �

23�13

y

x−1 1 2

1

2

3

( )43

, 52

( )0, 12

�1, 72�Distance � �269

y

x−4−6 2 4 6 8 10

−2

2

4

6

8

10

12

(−4, −3)

(6, 10)

�2, 3�Distance � 2�10

y

x−1−2−3 1 2 3 4 5

−1

−2

1

2

3

5

6

(−1, 4)

(5, 2)

g�x� � sin 2x

−2

−2� 2�

2

1 �12 sin2 2x1 � 2 sin2 x cos2 x

2 cos4 x � 2 cos2 x � 114�3 � cos 4x�

� 7�

6,

3

2� , � 11�

6,

3

2�

��

2, �3� , � 3�

2, 1�

2,

7�

6,

3�

2,

11�

6

−3

0 2�

2

��, 3�

020 �

4

u2

sin u2

��1 � cos u2

� 2 sin u cos u.

� �2��sin u� cos u

� �2 sin��u� cos��u�

sin 2u � �sin��2u�

u < 0,

x � 2r�1 � cos ��

� � 2 sin�1� 1M�

� � ��2.

A � 50 sin �

A � 100 sin �

2 cos

2

23.852x2 � 12x�1 � x2

333202CB05_AN.qxd 1/1/70 09:41 AM Page 12

Page 13: Chapter 5 x Section 5.1 (page 379) - Higher Ed eBooks ...college.cengage.com/mathematics/larson/precalculus_limits/1e2/... · 4 5 cos 2 2 3 5 sin 2 2 cot x x 3 3 3 sec x 2 3 3 sec

Precalculus with Limits, Answers to Section 5.5 13

(Continued)

134. (a) No complement; supplement:

(b) Complement: supplement:

135. (a) Complement: supplement:

(b) Complement: supplement:

136. (a) Complement: supplement: 2.19

(b) No complement; supplement: 0.38

137. September: $235,000; October: $272,600

138. gallons

139. �127 feet

�15.7

0.62;

11�

20�

20;

17�

184�

9;

10212;

71

Cop

yrig

ht ©

Hou

ghto

n M

iffl

in C

ompa

ny. A

ll ri

ghts

res

erve

d.

333202CB05_AN.qxd 1/1/70 09:41 AM Page 13

Page 14: Chapter 5 x Section 5.1 (page 379) - Higher Ed eBooks ...college.cengage.com/mathematics/larson/precalculus_limits/1e2/... · 4 5 cos 2 2 3 5 sin 2 2 cot x x 3 3 3 sec x 2 3 3 sec

Review Exercises (page 420)1. 2. 3.4. 5. 6.

7. 8.

10.

11. 12. 13. 1 14.

15. 16. 17. 18. 1

19. 20.

21. 22.

23–32. Answers will vary.

33. 34.

35. 36.

37. 38.

39. 40.

41. 42. 0

43. 44.

45.

46. No solution 47. 48.

49.

50.

51.

52.

53.

54.

55. 56. 57.

58. 59.

60. 61.

62. 63.

64. 65–70. Answers will vary.

71. 72.

73. 74.

75. 76.

77. 78.

79. 80.

81. 82.1 � cos 2x

23 � 4 cos 2x � cos 4x

4�1 � cos 2x�

1 � cos 6x2

1 � cos 4x1 � cos 4x

−1

−2� 2�

4

−2

−2� 2�

2

tan 2u � �43tan 2u � �

247

cos 2u �35cos 2u � �

725

sin 2u � �45sin 2u �

2425

0, ��

6,

11�

6

3�

2�

4,

7�

4

�960 � 507�71121

152

�5�7 � 36�12�7 � 1552

152

�5�7 � 36�960 � 507�71121

�352�5 � 4�7 �tan��47�

tan 35cos 165sin 15

tan 19�

12� �2 � �3

cos 19�

12�

�24

��3 � 1�

sin 19�

12� �

�24

��3 � 1�

tan 25�

12� 2 � �3

cos 25�

12�

�24

��3 � 1�

sin 25�

12�

�24

��3 � 1�

tan 345 � �2 � �3

cos 345 ��24

�1 � �3 �

sin 345 ��24

�1 � �3 �

tan 285 � �2 � �3

cos 285 ��24

��3 � 1�

sin 285 � ��24

��3 � 1�

3�

4,

7�

4, arctan��5� � �, arctan��5� � 2�

� � arctan 3

arctan��4� � �, arctan��4� � 2�, arctan 3,

2,

3�

20, �

0, �

8,

3�

8,

5�

8,

7�

8,

9�

8,

11�

8,

13�

8,

15�

8

0, �

3,

2�

3, �,

4�

3,

5�

3�

8,

3�

8,

9�

8,

11�

8

0, �

2, �

6,

2,

5�

60,

2�

3,

4�

3

6� n�,

5�

6� n�

3� n�,

2�

3� n�

3� 2n�,

5�

3� 2n�

6� n�

3� 2n�,

5�

3� 2n�

3� 2n�,

2�

3� 2n�

1 � sin x�2 tan2 �

2 tan2 x � 2 sec x tan x � 1sec x � 2 sin x

cot2 xtan u sec ucot �

cos2 xsec � csc �sin2 x

cot � ��520

sec � � 9

csc � �9�520

tan � � 4�5

cos � �19

cot � �32

cot x �43

csc � ��13

2sec x �

54

cos � �3�13

13csc x �

53

sin � �2�13

13tan x �

34

�sec x�cot xcot xcos xcsc xsec x

Precalculus with Limits, Answers to Review Exercises 14

Cop

yrig

ht ©

Hou

ghto

n M

iffl

in C

ompa

ny. A

ll ri

ghts

res

erve

d.

9.

cot x � �1sec x � �2csc x � ��2tan x � �1

cos x ��22

333202CB05_AN.qxd 1/1/70 09:41 AM Page 14

Page 15: Chapter 5 x Section 5.1 (page 379) - Higher Ed eBooks ...college.cengage.com/mathematics/larson/precalculus_limits/1e2/... · 4 5 cos 2 2 3 5 sin 2 2 cot x x 3 3 3 sec x 2 3 3 sec

Precalculus with Limits, Answers to Review Exercises 15

(Continued)

83.

84.

85.

86. 87.

88.

89. 90.

91. 92. 93.

94. 95.

96. 97.

98. 99.

100. 101.

102. (a)

(b)

Volume is maximum when

103.

104.

105. feet 106. cycles per second

107. False. If then The sign ofdepends on the quadrant in which lies.

108. False. Using the sum and difference formula,

109. True.

110. True by the product-to-sum formula

111. Reciprocal identities:

Quotient identities:

Pythagorean identities:

112. No. For an equation to be an identity, the equation must be true for all real numbers has an infinitenumber of solutions but is not an identity.

113. for all

114. (a) (b)

115.

116.

117.

118. �3.1395, �2.0000, �0.4378, 2.0000

�1.8431, 2.1758, 3.9903, 8.8935, 9.8820

y1 � 1 � y2

y1 � y2 � 1

54.73561649.91479, 59.86118

x�1 ≤ sin x ≤ 1

sin � �12x.

1 � cot2 � � csc2 �1 � tan2 � � sec2 �,

sin2 � � cos2 � � 1,

tan � �sin �

cos �, cot � �

cos �

sin �

cot � �1

tan �sec � �

1cos �

,csc � �1

sin �,

tan � �1

cot �,cos � �

1sec �

,sin � �1

csc �,

� �2 sin 2x

� �2�2 sin x cos x�

� �4 sin x cos x

4 sin��x� cos��x� � 4��sin x� cos x

sin x cos y � cos x sin y.sin�x � y� �

��2cos���2�cos���2� > 0.���2� < � < �,

4��12�10

y �12�10 sin�8t � arctan 13�

−2

0 2�

2

� � ��2.

V �12 sin � cubic meters

V � sin �

2 cos

2 cubic meters

� � 15 or �

122 cos x sin

4

�2 sin x sin �

62 cos

5�

2 cos

2

2 cos 3� sin �2�sin 5� � sin ��

12�cos 2� � cos 8��3cos��30� � cos 60

12

sin �

3tan 3x��cos 5x�

tan u2

�7

�35�

�355

tan u2

�3�5

5

cos u2

�� 512

��15

6cos

u2

��7014

sin u2

�� 712

��21

6sin

u2

�3�14

14

tan u2

��89 � 8

5

cos u2

� ���89 � 8

2�89���178�8�89 � 89�

178

sin u2

���89 � 8

2�89��178�8�89 � 89�

178

tan u2

�13

tan��17�

12 � � �2 � �3

cos u

2�

3�1010

cos��17�

12 � � �12�2 � �3

sin u2

��1010

sin��17�

12 � �12�2 � �3

tan 19�

12� �2 � �3

cos 19�

12�

12�2 � �3

sin 19�

12� �

12�2 � �3

tan 15 � 2 � �3

cos 15 �12�2 � �3

sin 15 �12�2 � �3

tan��75� � �2 � �3

cos��75� �12�2 � �3

sin��75� � �12�2 � �3

Cop

yrig

ht ©

Hou

ghto

n M

iffl

in C

ompa

ny. A

ll ri

ghts

res

erve

d.

333202CB05_AN.qxd 1/1/70 09:41 AM Page 15

Page 16: Chapter 5 x Section 5.1 (page 379) - Higher Ed eBooks ...college.cengage.com/mathematics/larson/precalculus_limits/1e2/... · 4 5 cos 2 2 3 5 sin 2 2 cot x x 3 3 3 sec x 2 3 3 sec

Chapter Test (page 423)

1. 2. 1 3. 1 4.

5.

6. 7–12. Answers will vary.

13.

14.

15.

16.

17.

18.

19.

20.

21. 22.

23.

24. Day 123 to day 223

25.

1.83 minutes

1.52 minutes

1.20 minutes

0.89 minute

0.58 minute

t � 0.26 minute

sin 2u �45 , tan 2u � �

43, cos 2u � �

35

�2 � �6

4�2.938, �2.663, 1.170

6,

5�

6,

3�

2

6,

5�

6,

7�

6,

11�

6

6,

2,

5�

6,

3�

2

0, 3�

4, �,

7�

4

�2 cos 7�

2 sin

2

2�sin 6� � sin 2��

tan 2�

116�

10 � 15 cos 2x � 6 cos 4x � cos 6x1 � cos 2x �

y1 � y2

−3

−2� 2�

3

3�

2< � < 2�� � 0,

2< � ≤ �,

cot � �2

3

sec � � ��13

2

csc � � ��13

3

cos � � �2�13

13

csc � sec �sin � � �3�13

13

Precalculus with Limits, Answers to Chapter Test 16

Cop

yrig

ht ©

Hou

ghto

n M

iffl

in C

ompa

ny. A

ll ri

ghts

res

erve

d.

333202CB05_AN.qxd 1/1/70 09:41 AM Page 16

Page 17: Chapter 5 x Section 5.1 (page 379) - Higher Ed eBooks ...college.cengage.com/mathematics/larson/precalculus_limits/1e2/... · 4 5 cos 2 2 3 5 sin 2 2 cot x x 3 3 3 sec x 2 3 3 sec

Precalculus with Limits, Answers to Problem Solving 17

Problem Solving (page 427)1. (a) (b)

2–3. Answers will vary.

4. (a)

(b)

is periodic with period

(c)

(d) Maximum:

Minimum:

5. 6.

7.

8. (a)

(b) (c) Maximum:

Minimum:

9. (a) (b) SpringEquinox and Fall Equinox

(c) Seward; The amplitudes:6.4 and 1.9

(d) 365.2 days

10. (a) High tides: 6:12 A.M., 6:36 P.M.

Low tides: 12:00 A.M., 12:24 P.M.

(b) The water depth never falls below 7 feet.

(c)

11. (a) (b)

(c)

(d)

12. (a) (b)

13. (a)

(b)

14. (a)

(b)

15. (a) (b) 233.3 times per second

00 1

15

cos 4� � cos4 � � 6 sin2 � cos2 � � sin4 �

cos 3� � cos � � 4 sin2 � cos �

�tan u � tan v � tan w � tan u tan v tan w

1 � tan u tan v � tan u tan w � tan v tan w

tan�u � v � w�

� cos u cos v sin w� cos u sin v cos w

� sin u sin v sin w� sin u cos v cos w

sin�u � v � w�

� � 76.5n �12�cot

2� �3�

5�

4≤ x ≤ 2�0 ≤ x ≤

4,

3�

2< x < 2�

2< x < �,

2�

3≤ x ≤

4�

3�

6≤ x ≤

5�

6

00 24

70

t � 91, t � 274;

00 365

20

� � 90

� � 0

00 90

550

F �0.6 W cos �

sin 12

tan �

2�

sin �1 � cos �

cos �

2��1 � cos �

2

sin �

2��1 � cos �

2

y �164 v0

2 sin2 �u � v � w

�0.00357, �1.19525��0.00024, 1.19524�

�0.00382, 0��0.00285, 0�,�0, 0�, �0.00096, 0�, �0.00191, 0�

1262.p

p5: 1

1310p2: 1

524

p6: 1

1572p3: 1

786p1: 1

262

p6(t)

−1.4

−0.006 0.006

1.4

p5(t)

−1.4

−0.006 0.006

1.4

p3(t)

−1.4

−0.006 0.006

1.4

p2(t)

−1.4

−0.006 0.006

1.4

−1.4

−0.006 0.006

1.4

p1(t)

p6�t� �16 sin�3144 �t�

p5�t� �15 sin�2620 �t�

p3�t� �13 sin�1572 �t�

p2�t� �12 sin�1048 �t�

p1�t� � sin�524 �t�

cot � � ±cos �

�1 � cos2 �csc � �

1sin �

sec � �1

cos �sec � � ±

1�1 � sin2 �

csc � � ±1

�1 � cos2 �cot � � ±

�1 � sin2 �sin �

tan � � ±�1 � cos2 �

cos �tan � � ±

sin ��1 � sin2 �

sin � � ±�1 � cos2 �cos � � ±�1 � sin2 �

Cop

yrig

ht ©

Hou

ghto

n M

iffl

in C

ompa

ny. A

ll ri

ghts

res

erve

d.

333202CB05_AN.qxd 1/1/70 09:41 AM Page 17


Recommended