Section 5.2
ANSWERS
C H A P T E R
5
Trigonometric Functions
5.1 Assess Your Understanding (page 366)
3. Standard position 4. ru;
11.
1 2
s ¨
r u 5. ;
2
t t
6. F 7. T
8. T
12.
9. T
10. F
13.
14.
135
120
30
60
15.
16.
540
17.
450
19.
20.
–
6
18.
4
3
3
4
21.
16
3
22.
21
4
– 2
3
23. 40.17° 24. 61.71° 25. 1.03° 26. 73.68° 27. 9.15° 28. 98.38° 29. 40°1912 30. 61°1424 31. 18°1518 32. 29°2440
p
2p
4p
11p
p
p
3p
3p
5p
p
33. 19°5924 34. 44°036 35.
36.
37.
38.
39. 40. 41. p 42.
43. 44. 45. 6
3
3
6
3
6
2
4
4
2
46. –p
47. 60° 48. 150° 49. 225° 50. –120° 51. 90° 52. 720° 53. 15° 54. 75° 55. –90° 56. –180° 57. –30° 58. –135°
59. 0.30 60. 1.27 61. 0.70 62. –0.89 63. 2.18 64. 6.11 65. 179.91° 66. 42.97° 67. 114.59° 68. 171.89° 69. 362.11°
4
p
L 1.047 in. 78. 2p≠6.283 m
70. 81.03° 71. 5 m 72. 12 ft 73. 6 ft 74. 24 cm 75. 0.6 radian 76. ≠1.333 radians 77.
3
3
4
p
L 1.047 in¤
79. 25 m¤ 80. 36 ft¤ 81. 2 13≠3.464 ft 82. 4 13≠6.928 cm 83. 0.24 radian 84. ≠0.444 radian 85.
9
3
86. 3∏≠9.425 m¤ 87. s=2.094 ft; A=2.094 ft¤ 88. s=2.094 m; A=4.189 m¤ 89. s=14.661 yd; A=87.965 yd¤
3p
40p
90. s=7.854 cm; A=35.343 cm¤ 91. 3p≠9.4248 in.; 5p≠15.7080 in. 92.
≠13.9626 in. 93. 2∏≠6.28 m¤ 94.
≠4.71 cm¤
9
2
675p
72
1
1
1
1
95.
≠1060.29 ft¤ 96.
≠4.58° 97. v =
radian/sec; v= cm/sec 98. v= radian/sec; v= m/sec
2
5p
60
12
8
4
99. Approximately 452.5 rpm 100. Approximately 282.7 in./sec; approximately 16.1 mi/hr 101. Approximately 359 mi
102. Approximately 554 mi
103. Approximately 898 mi/hr 104. Approximately 794 mi/hr 105. Approximately 2292 mi/hr
3
106. Approximately 66,633 mi/hr 107. rpm 108. Approximately 2.69 ft/sec; approximately 0.09 radian/sec
4
109. Approximately 2.86 mi/hr 110. Approximately 37.13 mi/hr; approximately 1034.26 rpm 111. Approximately 31.47 rpm
112. Approximately 5.8 min
113. Approximately 1037 mi/hr
114. Approximately 1.15 mi
115. radius ≠3979 miles; circumference ≠25,000 miles 116. v1=r1◊1, v2=r2◊2, and v1=v2, so r1◊1=r2◊2 1
r1
v2
=
.
r2
v1
5.2 Assess Your Understanding (page 384)
1
13
13
2 13
13
1
9. T 10. F 11. sin t= ; cos t=
; tan t=
; csc t=2; sec t=
; cot t= 13 12. sin t=; cos t= ;
2
2
3
3
2
2
2 13
13
121
2
121
5 121
5
tan t=- 13; csc t=; sec t=2; cot t=13. sin t=
; cos t=- ; tan t=; csc t=
; sec t=- ;
3
3
5
5
2
21
2
2 121
1
16
12
216
5 16
12
cot t=14. sin t=
; cos t=- ; tan t=- 2 16; csc t=
; sec t=–5; cot t=15. sin t=
; cos t=;
21
5
5
12
12
2
2
12
12
tan t=–1; csc t= 12; sec t=- 12; cot t=–1 16. sin t=
; cos t=
; tan t=1; csc t= 12; sec t= 12; cot t=1
2
2
7.
3
2
8. 0.91
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Prentice Hall
AN93
AN94
ANSWERS
Section 5.2
1
2 12
12
3 12
2
15
2 15
17. sin t=- ; cos t=
; tan t=; csc t=–3; sec t=
; cot t=- 2 12 18. sin t=- ; cos t=; tan t=
;
3
3
4
4
3
3
5
3
3 15
15
1
csc t=- ; sec t=; cot t=
19. –1 20. –1 21. 0 22. 0 23. –1 24. 1 25. 0 26. 0 27. –1 28. 0 29. (12 + 1)
2
5
2
2
1
1
13
2 13
1
30. (1 - 12) 31. 2 32. –1 33.
34.
35. 16 36.
37. 4 38. 8 39. 0 40. 12 + 3 41. 0 42. 13 +
2
2
2
3
2
4 13
2p
13
2p
1
2p
2p
2 13
=
= - ; tan
= - 13; csc
=
43. 2 12 +
44. 2 13 + 1 45. –1 46. –1 47. 1 48. –2 49. sin
; cos
;
3
3
2
3
2
3
3
3
2p
2p
5p
1
5p
5p
5p
13
13
13
5p
2 13
5p
= - 2; cot
= ; cos
= = = 2; sec
= = - 13
sec
=50. sin
; tan
; csc
; cot
3
3
3
6
2
6
2
6
3
6
6
3
6
1
13
13
2 13
51. sin 210°=- ; cos 210°=; tan 210°= ; csc 210°=–2; sec 210°=; cot 210°=13
2
2
3
3
13
1
2 13
13
52. sin 240°=; cos 240°=- ; tan 240°=13; csc 240°=; sec 240°=–2; cot 240°=
2
2
3
3
3p
12
3p
12
3p
3p
3p
3p
11p
12
12
11p
= = - 1; csc
= 12; sec
= - 12; cot
= - 1 54. sin
=
= 53. sin
= ; cos
; tan
; cos
;
4
2
4
2
4
4
4
4
4
2
4
2
11p
11p
11p
11p
8p
13
8p
1
8p
8p
213
= 12; sec
= - 12; cot
=
= - ; tan
= - 13; csc
=
tan
=–1; csc
=–1 55. sin
; cos
;
4
4
4
4
3
2
3
2
3
3
3
8p
8p
13p
1
13p
13
13p
13
13p
2 13
13
13p
13p
= - 2; cot
= = ; cos
=
=
=
= 13
sec
56. sin
; tan
; csc
=2; sec
; cot
3
3
3
6
2
6
2
6
3
6
6
3
6
12
12
57. sin 405°= ; cos 405°= ; tan 405°=1; csc 405°=12; sec 405°=12; cot 405°=1
2
2
2 13
1
13
13
58. sin 390°= ; cos 390°= ; tan 390°= ; csc 390°=2; sec 390°=
; cot 390°=13
2
2
3
3
13
213
p
1
p
p
13
p
p
p
59. sin a - b = - ; cos a - b =
; tan a - b = ; csc a - b = - 2; sec a - b =
; cot a - b = - 13
6
2
6
2
6
3
6
6
3
6
p
13
p
1
p
p
2 13
p
p
13
60. sin a - b = ; cos a - b = ; tan a - b = - 13; csc a - b = ; sec a - b =2; cot a - b = 3
2
3
2
3
3
3
3
3
3
12
12
61. sin(–45°)=; cos(–45°)=
; tan(–45°)=–1; csc(–45°)=- 12; sec(–45°)= 12; cot(–45°)=–1
2
2
13
1
2 13
13
62. sin(–60°)=; cos(–60°)= ; tan(–60°)=- 13; csc(–60°)=; sec(–60°)=2; cot(–60°)=2
2
3
3
5p
5p
5p
5p
5p
5p
63. sin
=1; cos
=0; tan
is undefined; csc
=1; sec
is undefined; cot
=0
2
2
2
2
2
2
64. sin(5∏)=0; cos(5∏)=–1; tan(5∏)=0; csc(5∏) is undefined; sec(5∏)=–1; cot(5∏) is undefined 65. sin 720°=0; cos 720°=1;
tan 720°=0; csc 720° is undefined; sec 720°=1; cot 720° is undefined 66. sin 630°=–1; cos 630°=0; tan 630° is undefined;
csc 630°=–1; sec 630° is undefined; cot 630°=0 67. 0.47 68. 0.97 69. 0.38 70. 0.36 71. 1.33 72. 1.22 73. 0.31 74. 0.92
4
3
4
5
75. 3.73 76. 5.67 77. 1.04 78. 1.07 79. 0.84 80. 1.56 81. 0.02 82. 0.02 83. sin ¨= ; cos ¨=- ; tan ¨=- ; csc ¨= ;
5
5
3
4
5
3
12
5
12
13
13
5
sec ¨=- ; cot ¨=84. sin ¨=- ; cos ¨= ; tan ¨=- ; csc ¨=- ; sec ¨= ; cot ¨=3
4
13
13
5
12
5
12
3 113
2 113
3
113
113
2
2 15
15
85. sin ¨=; cos ¨=
; tan ¨=- ; csc ¨=; sec ¨=
; cot ¨=86. sin ¨=; cos ¨=;
13
13
2
3
2
3
5
5
15
1
12
12
tan ¨=2; csc ¨=; sec ¨=- 15; cot ¨=
87. sin ¨=; cos ¨=; tan ¨=1; csc ¨=- 12; sec ¨=- 12; cot ¨=1
2
2
2
2
12
12
2 113
3 113
2
88. sin ¨=; cos ¨=
; tan ¨=–1; csc ¨=- 12; sec ¨= 12; cot ¨=–1 89. sin ¨=; cos ¨=; tan ¨= ;
2
2
13
13
3
113
113
3
12
12
csc ¨=; sec ¨=; cot ¨=
90. sin ¨=
; cos ¨=
; tan ¨=1; csc ¨= 12; sec ¨= 12; cot ¨=1
2
3
2
2
2
3
3
5
4
4
3
5
4
5
4
91. sin u=- ; cos u= ; tan u=- ; csc u=- ; sec u = ; cot u = 92. sin ¨=- ; cos ¨=- ; tan ¨= ; csc ¨=- ;
5
5
4
3
4
3
5
5
3
4
5
3
2 13
3
13
1
1
13
sec ¨=- ; cot ¨=
93. 0 94.
95. –0.1 96. –0.3 97. 3 98. –2 99. 5 100.
101.
102.
103.
104.
3
4
3
2
2
2
2
2
3
1
13
1
13
1
105.
106.
107.
108. 109. 13
110. 1 111. 112.
4
4
2
2
2
2
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Prentice Hall
Section 5.3
ANSWERS
113.
u
sin u
sin u
u
0.5
0.4794
0.4
0.3894
0.2
0.1987
0.1
0.0998
0.01
0.0100
0.001
0.0010
0.0001
0.0001
0.00001
0.00001
0.9589
0.9735
0.9933
0.9983
1.0000
1.0000
1.0000
1.0000
AN95
sin u
approaches 1 as u approaches 0.
u
114.
115.
118.
121.
123.
u
cos u-1
cos u - 1
u
0.5
–0.1224
0.4
–0.0789
0.2
–0.0199
0.1
–0.0050
0.01
–0.00005
0.001
0.0000
0.0001
0.0000
0.00001
0.0000
–0.2448
–0.1973
–0.0997
–0.0500
–0.0050
–0.0005
–0.00005
–0.000005
cos u - 1
approaches 0 as u approaches 0.
u
R≠310.56 ft; H≠77.64 ft 116. R≠1988.32 m; H≠286.99 m 117. R≠19,542 m; H≠2278 m
R≠1223.36 ft; H≠364.49 ft 119. (a) 1.20 s (b) 1.12 s (c) 1.20 s 120. 4.90 cm; 4.71 cm
(a) 1.9 hr; 0.57 hr (b) 1.69 hr; 0.75 hr (c) 1.63 hr; 0.86 hr (d) 1.67 hr; tan 90° is undefined 122. 251.42 cm‹; 117.88 cm‹; 75.4 cm‹
sin u - 0
(a) 16.56 ft (b)
(c) 67.5
124. Since L and M are parallel, mL=mM=
=tan ¨.
20
cos u - 0
45
90
0
125. (a) values estimated to the nearest tenth: sin 1≠0.8; cos 1≠0.5; tan 1≠1.6; csc 1≠1.3; sec 1≠2.0; cot 1≠0.6; actual values to the nearest
tenth: sin 1≠0.8; cos 1≠0.5; tan 1≠1.6; csc 1≠1.2; sec 1≠1.9; cot 1≠0.6 (b) values estimated to the nearest tenth: sin 5.1≠–0.9;
cos 5.1≠0.4; tan 5.1≠–2.3; csc 5.1≠–1.1; sec 5.1≠2.5; cot 5.1≠–0.4; actual values to the nearest tenth: sin 5.1≠–0.9; cos 5.1≠0.4;
tan 5.1≠–2.4; csc 5.1≠–1.1; sec 5.1≠2.6; cot 5.1≠–0.4 (c) values estimated to the nearest tenth: sin 2.4≠0.7; cos 2.4≠–0.7; tan 2.4≠–1.0;
csc 2.4≠1.4; sec 2.4 ≠–1.4; cot 2.4≠–1.0; actual values to the nearest tenth: sin 2.4≠0.7; cos 2.4≠–0.7; tan 2.4≠–0.9; csc 2.4≠1.5;
sec 2.4≠–1.4; cot 2.4≠–1.1
126. (a) values estimated to the nearest tenth: sin 2≠0.9; cos 2≠–0.4; tan 2≠–2.3; csc 2≠1.1; sec 2≠–2.5; cot 2≠–0.4; actual values to
the nearest tenth: sin 2≠0.9; cos 2≠–0.4; tan 2≠–2.2; csc 2≠1.1; sec 2≠–2.4; cot 2≠–0.5 (b) values estimated to the nearest tenth:
sin 4≠–0.8; cos 4≠–0.7; tan 4≠1.1; csc 4≠–1.3; sec 4≠–1.4; cot 4≠0.9; actual values to the nearest tenth: sin 4≠–0.8; cos 4≠–0.7;
tan 4≠1.2; csc 4≠–1.3; sec 4≠–1.5; cot 4≠0.9 (c) values estimated to the nearest tenth: sin 5.9≠–0.4; cos 5.9≠0.9; tan 5.9≠–0.4;
csc 5.9≠2.5; sec 5.9≠1.1; cot 5.9≠–2.5; actual values to the nearest tenth: sin 5.9≠–0.4; cos 5.9≠0.9; tan 5.9≠–0.4; csc 5.9≠2.7;
sec 5.9≠1.1; cot 5.9≠–2.5
127. (a) values estimated to the nearest tenth: sin 1.5≠1.0; cos 1.5≠0.1; tan 1.5≠10.0; csc 1.5≠1.0; sec 1.5≠10.0; cot 1.5≠0.1; actual values
to the nearest tenth: sin 1.5≠1.0; cos 1.5≠0.1; tan 1.5≠14.1; csc 1.5≠1.0; sec 1.5≠14.1; cot 1.5≠0.1 (b) values estimated to the nearest tenth:
sin 4.3≠–0.9; cos 4.3≠–0.4; tan 4.3≠2.3; csc 4.3≠–1.1; sec 4.3≠–2.5; cot 4.3≠0.4; actual values to the nearest tenth: sin 4.3≠–0.9;
cos 4.3≠–0.4; tan 4.3≠2.3; csc 4.3≠–1.1; sec 4.3≠–2.5; cot 4.3≠0.4 (c) values estimated to the nearest tenth: sin 5.3≠–0.8; cos 5.3≠0.6;
tan 5.3≠–1.3; csc 5.3≠–1.3; sec 5.3≠1.7; cot 5.3 = –0.8; actual values to the nearest tenth: sin 5.3≠–0.8; cos 5.3≠0.6; tan 5.3≠–1.5;
csc 5.3≠–1.2; sec 5.3≠1.8; cot 5.3≠–0.7
128. (a) values estimated to the nearest tenth: sin 2.7≠0.4; cos 2.7≠–0.9; tan 2.7≠–0.4; csc 2.7≠2.5; sec 2.7≠–1.1; cot 2.7≠–2.3; actual values
to the nearest tenth: sin 2.7≠0.4; cos 2.7≠–0.9; tan 2.7≠–0.5; csc 2.7≠2.3; sec 2.7≠–1.1; cot 2.7≠–2.1 (b) values estimated to the nearest
tenth: sin 3.9≠–0.7; cos 3.9≠–0.7; tan 3.9≠1.0; csc 3.9≠–1.4; sec 3.9≠–1.4; cot 3.9≠1.0; actual values to the nearest tenth: sin 3.9≠–0.7;
cos 3.9≠–0.7; tan 3.9≠0.9; csc 3.9≠–1.5; sec 3.9≠–1.4; cot 3.9≠1.1 (c) values estimated to the nearest tenth: sin 6.1≠–0.2; cos 6.1≠1.0;
tan 6.1≠–0.2; csc 6.1≠–5.0; sec 6.1≠1.0; cot 6.1≠–5.0; actual values to the nearest tenth: sin 6.1≠–0.2, cos 6.1≠1.0; tan 6.1≠–0.2;
csc 6.1≠–5.5; sec 6.1≠1.0; cot 6.1≠–5.4
5.3 Assess Your Understanding (page 399)
5. 2∏; ∏ 6. all real numbers except odd multiples of
12
12
20.
2
2
3
35. tan ¨=- ; cot ¨=4
18. 2
19.
21. 0
22. 1
p
2
23. 12 24. 1
4
5
5
; sec ¨= ; csc ¨=3
4
3
7. [–1, 1]
25.
13
3
8. T
26.
9. F 10. F 11.
2 13
3
12
2
12.
1
2
27. II 28. IV 29. IV
13. 1
14.
1
2
15. 1
16. –1
30. I 31. IV 32. III 33. II
5
4
3
5
36. tan ¨=- ; cot ¨=- ; sec ¨=- ; csc ¨=
3
4
3
4
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Prentice Hall
17. 13
34. II
AN96
ANSWERS
Section 5.3
1
15
1
15
13
213
37. tan ¨ 2; cot ¨ ; sec ¨ 15; csc ¨ 38. tan ¨= ; cot ¨=2; sec ¨=; csc ¨=- 15 39. tan ¨ ; cot ¨= 13; sec ¨=
;
2
2
2
2
3
3
13
2 13
12
3 12
csc ¨=2 40. tan ¨= 13; cot ¨=
; sec ¨=2; csc ¨=
41. tan ¨=; cot ¨=- 2 12; sec ¨=
; csc ¨=–3
3
3
4
4
12
3 12
5
12
13
13
5
42. tan ¨=–2 12; cot ¨=; sec ¨=–3; csc ¨=
43. cos u = - ; tan u = - ; csc u =
; sec u = - ; cot u = 4
4
13
5
12
5
12
4
4
5
5
3
3
3
5
5
4
44. sin u=- ; tan u=- ; csc u=- ; sec u= ; cot u=45. sin u = - ; tan u = ; csc u = - ; sec u = - ; cot u =
5
3
4
3
4
5
4
3
4
3
5
12
13
12
13
13
12
5
13
12
46. cos u=- ; tan u= ; csc u=- ; sec u=- ; cot u=
47. cos u = - ; tan u = - ; csc u =
; sec u = - ; cot u = 13
12
5
12
5
13
12
5
12
5
3
3
5
5
4
212
3 12
12
48. sin u=- ; tan u=- ; csc u=- ; sec u= ; cot u=49. sin u =
; tan u = - 212; csc u =
; sec u=–3; cot u = 5
4
3
4
3
3
4
4
2 15
15
3
15
3
3 15
15
2 15
315
50. cos u=; tan u=
; csc u=- ; sec u=; cot u=
51. cos u = ; tan u = ; csc u = ; sec u = ;
3
5
2
5
2
3
5
2
5
15
115
4 115
115
13
1
cot u = 52. sin u=; tan u=115; csc u=; sec u=–4; cot u=
53. sin u = ; cos u = ; tan u = - 13;
2
4
15
15
2
2
213
13
1
2 12
12
3 12
3
4
csc u = ; cot u = 54. sin u= ; cos u=; tan u=; sec u=; cot u=- 2 12 55. sin u = - ; cos u = - ;
3
3
3
3
4
4
5
5
4
3
110
5
5
3
4
5
5
3110
csc u = - ; sec u = - ; cot u =
56. sin u=- ; cos u=- ; tan u= ; csc u=- ; sec u=57. sin u =
; cos u = ;
3
4
3
5
5
4
3
4
10
10
110
13
1
2 13
13
13
13
csc u = 110; sec u = ; cot u=–3 58. sin u=; cos u=- ; tan u=13; csc u=; cot u=
59. 60.
3
2
2
3
3
2
2
13
12
12
13
2 13
61. 62. 63. 2 64. –2 65. –1 66. 0 67. –1 68. 0 69.
70. 71. 0 72. 1 73. - 12 74. –1 75.
3
2
2
2
3
213
76. 77. 1 78. 1 79. 1 80. 1 81. 0 82. 0 83. 1 84. 1 85. –1 86. 1 87. 0 88. 0 89. 0.9 90. 0.6 91. 9 92. –6
3
p
p
93. 0 94. –1 95. All real numbers 96. All real numbers 97. Odd multiples of
98. Multiples of p 99. Odd multiples of
2
2
100. Multiples of p 101. –1 y 1 102. –1 y 1 103. All real numbers 104. All real numbers 105. œyœ 1
106. œyœ 1
107. Odd; yes; origin 108. Even; yes; y-axis 109. Odd; yes; origin 110. Odd; yes; origin 111. Even; yes; y-axis
1
1
3
112. Odd; yes; origin 113. (a) (b) 1 114. (a)
(b)
115. (a) –2 (b) 6 116. (a) 3 (b) –9 117. (a) –4 (b) –12
3
4
4
118. (a) –2 (b) 6 119. About 15.81 min 120. About 2.75 hr
y
= a. Then
121. Let a be a real number and P=(x, y) be the point on the unit circle that corresponds to t. Consider the equation tan t =
x
1
a
y=ax. But x2+y2=1 so x2+a2x2=1. Thus, x = ;
and y = ;
; that is, for any real number a, there is a
21 + a2
21 + a2
point P=(x, y) on the unit circle for which tan t=a. In other words, the range of the tangent function is the set of all real numbers.
x
122. Let a be a real number and P=(x, y) be the point on the unit circle that corresponds to t. Consider the equation cot t= = a. Then
y
1
a
2
2
2 2
2
x=ay. But x +y =1 so a y +y =1. Thus, y=—
and x=—
; that is, for any real number a, there is a
21 + a2
21 + a2
point P=(x, y) on the unit circle for which cot t=a. In other words, the range of the cotangent function is the set of all real numbers.
123. Suppose that there is a number p, 0<p<2p, for which sin(u+p)=sin u for all u. If u=0, then sin(0+p)=sin p=sin 0=0,
p
p
p
3p
p
b = - 1 = sin a b =1. This is impossible.
so p=p. If u = , then sin a + p b =sin a b . But p=p. Thus, sin a
2
2
2
2
2
Therefore, the smallest positive number p for which sin(u+p)=sin u for all u is 2p.
124. Suppose that there is a number p, 0<p<2p, for which cos(u+p)=cos u for all u. If u=0, then cos(0+p)=cos p=cos 0=1.
But on the interval (0, 2p), cos p never equals 1. Therefore, the smallest positive number p for which cos(u+p)=cos u for all u is 2p.
1
1
125. sec u =
; since cos u has period 2p, so does sec u. 126. csc u=
; since sin u has period 2p, so does csc u.
cos u
sin u
127. If P=(a, b) is the point on the unit circle corresponding to u, then Q=(–a, –b) is the point on the unit circle corresponding to
-b
b
=
= tan u. Suppose that there exists a number p, 0<p<∏, for which tan(¨+p)=tan ¨ for all ¨.
u+p. Thus, tan(u + p) =
-a
a
Then, if ¨=0, then tan p=tan 0=0. But this means that p is a multiple of ∏. Since no multiple of ∏ exists in the interval (0, ∏),
this is a contradiction. Therefore, the period of f(¨)=tan ¨ is ∏.
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Prentice Hall
Section 5.4
ANSWERS
AN97
1
128. cot u=
; since tan u has period p, so does cot u. 129. Let P=(a, b) be the point on the unit circle corresponding to ¨.
tan u
a
1
1
1
1
1
1
=
=
Then csc ¨= =
; sec ¨= =
; cot u =
.
b
ba
tan u
a
cos u
b
sin u
cos u
b
sin u
a
130. If P=(a, b) is the point on the unit circle corresponding to ¨, then tan ¨= =
, and cot ¨= =
.
b
sin u
a
cos u
2
2
2
2
2
2
2
2
131. (sin ¨ cos Ï) +(sin u sin Ï) +cos ¨=sin ¨ cos Ï+sin ¨ sin Ï+cos ¨
=sin2 ¨(cos2 Ï+sin2 Ï) ± cos2 ¨=sin2 ¨+cos2 ¨=1
5.4 Assess Your Understanding (page 414)
p
p
∏
∏
x
+ 2∏k, k any integer 4. 3; ∏ 5. 3;
6. T 7. F 8. T 9. 0 10. 1 11. 12. 0 x p 13. 1 14. –1
2
2
3
2
p 3p
3p p
p 3p
, ; sin x=–1 for x = - ,
15. 0, p, 2p 16. ,
17. sin x=1 for x = 18. cos x=1 for –2p, 0, 2p; cos x=–1 for x=–p, p
2 2
2 2
2 2
3. 1;
19. B, C, F 20. A, D, E
21.
22.
y
23.
y
–
4
3
2
2
–
x
2
–
2
3
26.
x
2
3
x
–
27.
28.
y
1
2
–
x
x
–
2
2
2
1
30.
y
2
2
3
1
31.
y
2
x
x
2
3
y
6
4
1
1
32.
y
1
2
1
4
4
6
3
8
x
2
–
2
x
2
34.
y
6
4
4
2
2
4
6
2
x
2
–3
35.
y
4
2
y
4
2
2
4
–
–2
2
x
–2
–2
33.
3
–
3
–1
–1
2
y
1
–2
x
3
1
1
2
–1
y
2
1
2
–4
y
29.
y
–
1
–2
–3
25.
24.
x
3
y
1
x
4
2
x
2
2
4
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Prentice Hall
2
2
3
AN98
Section 5.4
ANSWERS
36.
y
6
37. Amplitude=2; Period=2∏
38. Amplitude=3; Period=2∏
39. Amplitude=4; Period=∏
40. Amplitude=1; Period=4∏
4
x
3 2 2p
42. Amplitude=3; Period=
3
4
44. Amplitude= ; Period=3∏
3
9
4
46. Amplitude= ; Period=
5
3
41. Amplitude=6; Period=2
2
2
2
1
4p
43. Amplitude= ; Period=
2
3
5
45. Amplitude = ; Period = 3
3
3
4
6
47. F 48. E 49. A 50. I 51. H 52. B 53. C 54. G 55. J 56. D 57. A 58. C 59. B 60. D
61.
62.
y
5
63.
y
64.
y
y
5
2
4
x
x
–
2
–
4
4
–
2
3
–
6
6
x
x
3
–1
1
2
–1
3
2
–2
–4
–5
–5
65.
1
66.
y
67.
y
y
4
3
3
68.
y
2
x
–1
1
2
2
1
x
–
x
1
2
69.
1
2
1
y
3
2
4
3
3
2
5
x
–4 –2
–4
–7
70.
3
2 4 6
1
y
2
3
x
x
4
–3
3
6
–2
3
32
–4
3
–2
1
71. y=—3 sin(2x) 72. y=—2 sin a x b 73. y=—3 sin(∏x)
2
p
1
74. y=—4 sin(2∏x) 75. y = 5 cos a x b 76. y=4 sin a x b
4
4
1
p
3
77. y = - 3 cos a x b 78. y=–2 sin a x b 79. y = sin(2px)
2
2
4
5
3
80. y=- cos(∏x) 81. y = - sin a x b 82. y=–∏ cos x
2
2
83. y = - cos a
4p
1
3
p
x b + 1 84. y=- sin a x b - 1 85. y = 3 sin a x b 86. y=–2 cos(∏x) 87. y=–4 cos(3x) 88. y=4 sin(2x)
3
2
2
2
1
1
1
89. Period =
; Amplitude=220
90. Period =
; Amplitude=120
91. (a) Amplitude=220; Period =
30
15
60
(b), (e)
I
V
I
1
15
t
1
30
–220
220
120
220
2
15
t
V
22
t
–22
1
15
–120
I
1
60
1
30
–220
(c) I=22 sin(120∏t)
(d) Amplitude=22; Period =
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Prentice Hall
1
60
Section 5.5
ANSWERS
92. (a) Amplitude=120; Period =
(b), (e)
1
60
93. (a) P =
120
t
1
60
I
1
30
–120
94. (a) Physical potential: ◊=
(b)
100
0
R
sin2(2pft)
V20
2p
p
2p
; Emotional potential: ◊= ; Intellectual potential: ◊=
23
14
33
(c) No
(d) Physical potential peaks at 15 days after 20th birthday. Emotional potential is 50%
at 17 days, with a maximum at 10 days and a minimum at 24 days. Intellectual potential
starts fairly high, drops to a minimum at 13 days, and rises to a maximum at 29 days.
36
0
95.
96.
y
1
y
1
x
–2
R
V20
and
2R
1
period
and is of the form y=A cos(◊t)+B,
2f
V20
V20
1
2p
and B =
then A=. Since = ,
2R
2R
2f
v
then ◊=4∏f. Therefore,
V20
V20
V20
=
P=cos(4∏ft)+
[1-cos(4∏ft)].
2R
2R
2R
1
(d) Amplitude=6; Period =
60
V
=
(b) Since the graph of P has amplitude
(c) I=6 sin(120∏t)
y
[V0 sin(2pft)]2
0
–
x
2
–2
–1
0
–
2
–1
5.5 Assess Your Understanding (page 423)
3. origin; odd multiples of
∏
2
4. y-axis; odd multiples of
∏
2
5. y=cos x 6. T
7. 0
11. sec x=1 for x=–2∏, 0, 2∏; sec x=–1 for x=–∏, ∏ 12. csc x=1 for x=13. -
3p p p 3p
,- , ,
2
2 2 2
21.
14. –2∏, –∏, 0, ∏, 2∏
22.
y
15. -
3p p p 3p
,- , ,
2
2 2 2
2
26.
y
2
x
3
–
27.
x
2
2
x
3
2
2 3
3
2
28.
y
y
10
3
2
x
2
–2
y
2
y
2
x
–
3
–2
25.
24.
2
x
–
3p p
p 3p
, ; csc x=–1 for x=- ,
2 2
2 2
y
2
2
10. No y-intercept
16. –2∏, –∏, 0, ∏, 2∏ 17. D 18. C 19. B 20. A
23.
y
8. No y-intercept 9. 1
2
3
x
x
2
–
–10
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Prentice Hall
AN99
AN100
Section 5.5
ANSWERS
29.
30.
y
31.
y
2
1
32.
y
y
2
2
1
1
x
–
x
2
–
2
x
3
1
2
–
3
–1
–
2
2
x
2
33.
34.
y
35.
y
36.
y
y
10
4
3
–
x
4
–
2
2
x
x
2
–4 –2
3
38.
y
39.
y
2
x
2
x y
2
2
2
–2
x
–
2
5
x
2
– 3
4
(4 , 1)
(4 , 3)
2
–
4
–2
4
x
y
6
–3
4
)
( 2 , 1)
–4
x y
2
2
3
4
5
6
7
44.
x x 2 x 3
2
6
5
4
3
2
1
4
x
5
2
5 , 2
2
, 2
2
1
2
3
4
5
x 9
4
x 5
4
(c)
t
1
2
t
2
3
2
– 4
3
4
4
2 , 1
5
7
4
4
3
2 , 1
(
( 4 , 2)
) (
(
7
4
) (
45. (a) L(u) =
(b)
3, 2
4
3
4
3
4 , 2
x 13
4
11, 2
4
11
4
x
)
7
4 , 2
)
3
4
+
= 3 sec u + 4 csc u
cos u
sin u
(c) 0.83
(d) 9.86 ft
25
x
0
0
(b) d=10 tan (pt) is undefined at e t =
d
25
–25
x
4
y
3
2
x
2
–6
5
4
3
2
1
3 2
2
4
(
2
x0
x – 2
(, 2)
3 , 1
4
x
–2 – 3 – 2
2
–5
42.
(54 , 3)
4
x
3
4
6
x 3
2
x 2
y
6
6
46. (a)
–
40.
y
5
–5
43.
–
4
–4
5
41.
2
–10
37.
2
2
k
∑k is an odd integer f .
2
d=10 tan (pt)
0
0
0.1
3.2492
0.2
7.2654
0.3
13.764
0.4
30.777
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Prentice Hall
Section 5.6
ANSWERS
AN101
d(0.2) - d(0.1)
d(0.3) - d(0.2)
d(0.4) - d(0.3)
= 32.492;
= 40.162;
= 64.986;
= 170.13
0.1
0.1
0.1
0.1
(e) The first differences represent the average rate of change of the beam of light against the wall measured in feet per second.
For example, between 0 and 0.1 second the beam of light is moving at 32.492 ft/sec, on average. The speed increases as d increases from 0 to 0.5.
(d)
47.
d(0.1) - d(0)
y
y
5
5
x
–
2
x
–
3
2
2
–5
y = tan x
3
2
–5
y cot(x + )
2
5.6 Assess Your Understanding (page 434)
1. phase shift 2. F
3. Amplitude=4
4. Amplitude=3
2p
Period=
3 p
Phase shift=
3
Period=p
p
Phase shift=
2
y
y
Period
4
3
3
3
–4
3
6
y
y
Phase 3
shift
2
–
4
4
x
–3
y
y Phase
shift
2
Phase
shift
3
Phase shift=-
2
–1
Period
x
1
2
–1 – –4
–2
Period
14. Amplitude=3
Period=p
p
Phase shift =
4
y
Period
Period
3
4
1
–3
–2
x
2
3
x
2
p
2
2
2– 13. Amplitude=3
Period=p
p
Phase shift =
4
x
2
2
p
Period
12. Amplitude=2
Period=1
2
Phase shift=
p
Period
10. Amplitude=2
Period=1
Phase y
shift
2
1– Period
y
2 1
2
–
–1
Phase
shift
–2
11. Amplitude=3
Period=2
2
Phase shift=
p
3
x
x
Period
Phase shift=-
2
2
2
4
2
Period
–
y
–
–3
x
9. Amplitude=4
Period=2
Phase
shift
2
2
Period
8. Amplitude=2
Period=p
p
Phase shift=
4
p
4
3
–2
Period
7. Amplitude=3
Period=p
–3
2
6
p
2
Phase shift=Phase y
shift
–
2
3
–3
Phase
shift
3
4
Period=p
x
–
3
2
Phase shift=-
6. Amplitude=3
Phase y
shift
Phase
shift
x
2
5. Amplitude=2
2p
Period=
3
p
Phase shift=6
x
4
–3
Phase
shift
Phase
shift
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Prentice Hall
x
3
4
5
4
AN102
Section 5.6
ANSWERS
15. y=2 sin c 2 a x -
1
b d or y=2 sin(2x-1)
2
16. y=3 sin c 4(x-2) d or y=3 sin(4x-8)
1
2
2
2
17. y=3 sin c a x + b d or y=3 sin a x + b 18. y=2 sin[2(x+2)] or y=2 sin(2x+4)
3
3
3
9
1
1
1
1
19. Period= ; Amplitude=120; Phase shift=
20. Period= ; Amplitude=220; Phase shift=
15
90
30
360
I
I
220
120
1
15
t
1
90
1
15
–120
21. (a)
–220
p
p
2p
b + 40.1
(b) y=15.9 sin c (x - 4) d + 40.1 or y=15.9 sin a x 6
6
3
(c) 60
(d) y=15.62 sin(0.517x-2.096)+40.377
(e)
60
60
0
20
13
0
20
22. (a)
13
0
20
13
p
p
2p
b +57.3
(b) y=22.7 sin c (x - 4) d + 57.3 or y=22.7 sin a x 6
6
3
(c) 80
(d) y=22.61 sin(0.503x-2.038)+57.17
80
(e)
0
20
t
1
60
2
15
80
13
0
20
13
0
13
20
23. (a)
p
p
2p
b + 50.45
(b) y=24.95 sin c (x - 4) d + 50.45 or y=24.95 sin a x 6
6
3
(c) 80
(d) y=25.693 sin(0.476x-1.814)+49.854
80
(e)
0
20
13
0
20
24. (a)
13
0
20
13
p
p
2p
b +54.4
(b) y=22.6 sin c (x - 4) d +54.4 or y=22.6 sin a x 6
6
3
(c) 80
(d) y=22.46 sin(0.506x-2.060)+54.35
(e) 80
80
0
20
80
13
0
20
13
0
13
20
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Prentice Hall
Review Exercises
ANSWERS
25. (a) 4:08 PM
26. (a) 8:41 P.M.
4p
(b) y=4.4 sin c (x - 0.5083) d + 3.8 or
25
4p
y=4.4 sin c x - 0.2555 d + 3.8
25
(c) y
(d) 8.2 ft
(b) y=5.5 sin c
4p
(x - 5.0583) d + 7.7 or
25
4p
y=5.5 sin a x - 2.5426 b + 7.7
25
(c) y
(d) 13.2 ft
14
12
10
8
6
4
2
9
7
5
3
1
x
2
4
6
27. (a) y=1.0835 sin c
y=1.0835 sin a
(b) 11.86 hr (c)
8
12
x
1
2p
(x - 80.75) d + 11.6665 or
365
2p
x - 1.3900 b + 11.6665
365
3
5
7
28. (a) y=2.2915 sin c
2p
(x - 80.75) d + 11.3585 or
365
y=2.2915 sin a
(b) 11.76 hr (c)
y
13
9 11 13
2p
x - 1.3900 b + 11.3585
365
y
14
13
12
12
11
11
10
10
0
29. (a) y=5.3915 sin c
y=5.3915 sin a
(b) 11.79 hr (c)
9
x
100
200
300
8
400
2p
(x - 80.75) d + 10.8415 or
365
2p
x - 1.3900 b + 10.8415
365
30. (a) y=0.992 sin c
y=0.992 sin a
x
0
100
200
300
400
2p
(x - 80.75) d + 11.775 or
365
2p
x - 1.3900 b + 11.775
365
(b) 11.95 hr (c) y
y
17
14
15
13
13
11
12
9
11
7
5
x
0
100
200
300
400
10
x
0
100
200
300
400
Review Exercises (page 440)
1.
13.
27.
32.
34.
36.
38.
p
1
1
12
3 12
4 13
5. 135° 6. 120° 7. 450° 8. –270° 9.
10.
11.
12. 2+313
+
12
2
2
2
2
3
3 13
15. 3 16. 1 + 4 12 17. 0 18. 1 19. 0 20. –2 21. 1 22. 1 23. 1 24. 1 25. 1 26. 1
- 3 12 - 2 13 14.
2
3
4
5
5
3
–1 28. –1 29. 1 30. –1 31. cos u = ; tan u = ; csc u = ; sec u = ; cot u =
5
3
4
3
4
4
4
5
5
3
12
5
13
13
5
sin u= ; tan u= ; csc u= ; sec u= ; cot u= 33. sin u = - ; cos u = - ; csc u = - ; sec u = - ; cot u =
5
3
4
3
4
13
13
12
5
12
5
12
5
13
13
3
4
3
5
4
sin u=- ; cos u=- ; tan u= ; csc u=- ; sec u=35. sin u = ; cos u = - ; tan u = - ; csc u = ; cot u = 13
13
12
5
12
5
5
4
3
3
3
4
3
5
4
5
12
13
13
5
sin u=- ; cos u= ; tan u=- ; sec u= ; cot u=37. cos u = - ; tan u = - ; csc u =
; sec u = - ; cot u = 5
5
4
4
3
13
5
12
5
12
4
4
5
5
3
12
5
13
13
12
sin u=- ; tan u= ; csc u=- ; sec u=- ; cot u= 39. cos u =
; tan u = - ; csc u = - ; sec u =
; cot u = 5
3
4
3
4
13
12
5
12
5
3p
4
2.
7p
6
3.
p
10
4.
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Prentice Hall
AN103
AN104
Review Exercises
ANSWERS
5
5
13
13
12
110
3 110
; tan u=- ; csc u=- ; sec u= ; cot u=41. sin u = ; cos u = ; csc u = - 110;
13
12
5
12
5
10
10
2 113
113
110
3 113
113
3
sec u = ; cot u = 3 42. sin u=
; cos u=; csc u=
; sec u=; cot u=3
13
13
2
3
2
2 12
1
3 12
12
43. sin u = ; cos u = ; tan u = - 2 12; csc ¨=; cot u = 3
3
4
4
1
115
115
4 115
44. sin u=- ; cos u=; tan u=
; sec u=; cot u=115
4
4
15
15
15
2 15
1
15
45. sin u =
; cos u = ; tan u = - ; csc u = 15; sec u = 5
5
2
2
2 15
15
15
1
46. sin u=; cos u= ; csc u=; sec u=15; cot u=5
5
2
2
47.
48.
49. y
50. y
y
y
40. sin u=-
2
3
x
x
␲
–
␲
⫺–
␲
␲
–
2
2
␲
⫺3
52.
y
4
3␲
⫺–––
2
␲
–
␲
⫺–
x
␲
⫺4
55.
2␲
56.
␲
–
2
⫺␲
6
␲
–
x
␲
–
3
␲
2
⫺10
57.
58.
y
63. Amplitude=4
2p
Period=
3
Phase shift=0
5
␲
–
4
x
x
␲
–
⫺␲
60. Amplitude=1; period=∏
x
3
⫺4
x
3␲
⫺2
Period=4∏
p
Phase Shift=
2
y
3
2
2␲
–––
3
2
62. Amplitude=2; period=
3
66. Amplitude=1
Period=∏
y
␲
–
⫺5
65. Amplitude=2
Phase shift=0
4
2
61. Amplitude=8; period=4
Period=6∏
6␲
Phase Shift=–∏
y
2
⫺2␲
x
⫺␲
␲
⫺3
x
␲
–
⫺␲
⫺5
64. Amplitude=2
y
y
5
⫺5
59. Amplitude=4; period=2∏
␲
2
2
⫺5
x
␲
–
⫺8
⫺ –4
x
␲
–
6
y
5
5
y
10
␲
⫺–
⫺3
y
54.
y
8
x
2␲
⫺3
53.
y
2
2
␲
⫺2
3
⫺␲
x
x
4␲
2␲
2
⫺2
51.
3
2
2
x
2
2␲
2
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Prentice Hall
ANSWERS
1
67. Amplitude=
2
4p
Period=
3
2p
Phase shift=
3
y
1–
2
2␲
–––
3
␲
––
3
3
68. Amplitude=
2
p
Period=
3
Phase shift=-
␲
2
69. Amplitude=
3
70. Amplitude=7
Period=2
Period=6
6
Phase shift=
p
y
p
2
y
5␲
–––
3
Phase shift=-
␲
––
6
x
␲
––
3
6
––
␲
x
3
12
––
␲
2
⫺ –3
3
⫺ –2
1
⫺ –2
4
p
7
x
x
2␲
6
⫺7
x
x
p
p
72. y=4 sin
73. y=–6 cos a x b 74. y=–7 sin a x b 75. 0.38 76. 1.02
4
4
4
4
Sine, Cosine, Cosecant and Secant: Negative; Tangent and Cotangent: Positive 78. Quadrant IV
212
312
1
12
sin ¨=
; cos ¨=- ; tan ¨=- 212; csc ¨=
; sec ¨=–3; cot ¨=3
3
4
4
4
3
4
p
sin t= , cos t=- , tan t=81. Domain: e x ` x odd multiple of f ; range: {yœœyœ 1}; period = 2(∏)
5
5
3
2
p
p
16p
(a) 32.34 (b) 631048 83. ≠1.05 ft; L 1.05 ft¤ 84. 8∏≠25.13 in.;
L 16.76 in.
3
3
3
p
Approximately 114.59 revolutions/hr 86. Approximately 5.24 ft/s 87. 0.1 revolution/s= radian/s
5
Approximately 945.38 rpm; yes; approximately 1080.43 rpm
1
1
1
(a) 120 (b)
(c)
90. (a)
(b) 220 (c) (d)
E
I
60
15
180
120
71. y=5 cos
77.
79.
80.
82.
85.
88.
89.
220
t
1
––
60
1
––
30
t
1
––
15
⫺120
2
––
15
⫺220
91. (a)
p
2p
(b) y=19.5 sin a x b + 70.5
6
3
(c)
(d) y=19.518 sin(0.541x-2.283)+71.014
y
100
(e)
95
y
100
90
90
80
80
70
70
60
50
92. (a)
y
80
70
60
50
40
30
20
x
0 1 2 3 4 5 6 7 8 9 10 11 12
60
50
x
x
0 1 2 3 4 5 6 7 8 9 10 11 12
80
70
60
50
40
30
20
0
0 1 2 3 4 5 6 7 8 9 10 11 12
p
2p
(b) y=25 sin a x b + 50
6
3
(c) y
x
0 1 2 3 4 5 6 7 8 9 10 11 12
AN105
y
2–
3
3–
2
4␲
–––
3
Review Exercises
13
45
(d) y=25.815 sin(0.521x-2.175)+50.46
(e) 80
0
20
13
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Prentice Hall
AN106
93. (a) y=1.85 sin a
(b)
Review Exercises
ANSWERS
2p
357
p b + 11.517
x 365
146
(c) 11.83 hr
y
14
94. (a) y=2.775 sin a
(b)
y
14
13
12
11
10
9
8
13
12
11
10
9
x
0
100
200
300
400
2p
357
p b + 11.192
x 365
146
(c) 11.66 hr
x
0
100
200
300
400
Chapter Test (page 443)
1.
13
9
2. -
20
9
13
180
3.
15.≠– 1.524 16.≠2.747 17.
2 16
7
5
5 16
tan u==12
2 16
7
csc u=
5
8. 0
1
2
10. -
sin u
cos u
tan u
sec u
csc u
cot u
u in QI
+
+
+
+
+
+
+
-
-
-
+
-
u in QIII
-
-
+
-
-
+
u in QIV
-
+
-
+
-
-
sec u=
2 15
2
=5
15
26.
y
y tan(x + ) + 2
4
( 23 , 2 )
13
5
cot u=-
5
12
24. -
1
2
27. y=–3 sin a 3x +
y
x
2
–2
7 153
53
51146
23. 146
22.
13
12
sec u=-
13.≠0.292 14.≠0.309
2
4
( 76 , 0 )
3(1 - 12)
3
18. 5
csc u=
3
2
cot u=-
12.
21. sin u=
3
3 15
=5
15
csc u=-
11. 2
12
13
5
cos u=13
20. sin u=-
7
7 16
=12
2 16
2 16
cot u=5
( 6 , 0 )
13
3
u in QII
sec u=-
2
9. -
15
3
15
tan u=2
19. cos u=-
25.
1
2
4. – 22.5° 5. 810° 6. 135° 7.
, 0)
( 13
6
2
( 53 , –2 )
x
–2
–
28. 121.19 ft2 29. 143.5 rpm 30. (a) 2633 ft (b) 818 ft
Cumulative Review (page 446)
1
1. e - 1, f
2
2. y-5=–3(x+2) or y=–3x-1
2
4. A line. Slope ; intercepts (6, 0) and (0, –4)
3
3. x¤+(y+2)¤=16
5. A circle. Center (1, –2); Radius 3
y
6.
y
2
4
y
7
x
3
5
x
6
(2, 3)
(1, 2)
6
(4, 3)
(3, 2)
4
6
1
1
x
5
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Prentice Hall
3p
b
4
Cumulative Review
ANSWERS
7. (a)
(b)
y
6
(c)
y
4
(1, 1)
(1, 1)
3
(d)
(1, 1)
3
(0, 0)
1
3
(1, 1)
x
(
1
–1, e
4
(0, 0)
(e)
)
2
y
3
2,1
1
(
)
2 , 1
(
1
(x + 2)
3
9. –2
x
x
(0, 0)
5
(1, 0)
1 , 1
e
( 4 , 1)
x
10.
3
11. 3-
y
4
2
(0, 0)
( 4 , 1)
1
)
x
2
1
(f)
y
2
(0, 1)
( )
(e, 1)
8. f- 1(x) =
(1, e)
x
3
y
3
2
y
5
2
3 13
2
p
12. y=2(3x) 13. y=3 cos a x b
6
x
4
14. (a) f(x)=–3x-3; m=–3; (–1, 0), (0, –3)
y
4
(2, 3)
x
5 (1, 0)
5
(0, 3)
6
(b) f(x)=(x-1)¤-6; (0, –5), (- 16 + 1, 0); ( 16 + 1, 0)
y
(2, 3)
3
x
5
5
(1.45, 0)
(1, 6)
(3.45, 0)
7
(1, 6)
(c) We have that y=3 when x=–2 and y=–6 when x=1. Both points satisfy y=aex. Therefore, for (–2, 3)
we have 3=ae–¤ which implies that a=3e¤. But for (1, –6) we have –6=ae⁄, which implies that a=–6e–⁄.
Therefore, there is no exponential function y=aex that contains (–2, 3) and (1, –6).
1
15. (a) f(x)= (x + 2)(x - 3)(x - 5)
6
y
10
(0.08, 5.01)
(2, 0)
4
10
(b) R(x)=-
(x + 2)(x - 3)(x - 5)
3(x - 2)
y
10
(3, 0)
(5, 0)
x
6
(4.08, 1.01)
(0, 5)
(2, 0)
4
(3.82, 1.03)
(5, 0)
x
(3, 0) 6
10
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AN107