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d.Precalculus with Limits, Answers to Section 6.1 1
Chapter 6Section 6.1 (page 436)
Vocabulary Check (page 436)
1. oblique 2. 3.
1.
2.
3.
4.
5.
6. Two solutions:
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20. No solution 21. No solution
22.
23. Two solutions:
24. No solution
25. (a) (b)
(c)
26. (a) (b)
(c)
27. (a) (b)
(c)
28. (a) (b)
(c)
29. 10.4 30. 474.9 31. 1675.2 32. 4.533. 3204.5 34. 159.3 35. 15.3 meters36. (a)
(b) (c) 9 meters
37. 38. 39. 77 meters40. (a) (b) 4385.71 feet
(c) 3061.80 feet
41. (a) (b) 22.6 miles(c) 21.4 miles(d) 7.3 miles
42. From Pine Knob: 42.4 kilometersFrom Colt Station: 15.5 kilometers
43. 3.2 miles
44. (a) (b)
(c)
(d)
45. True. If an angle of a triangle is obtuse greater than then the other two angles must be acute and therefore lessthan The triangle is oblique.90�.
90��,�
d �58.36 sin�84.64 � ��
sin �
� � sin�1�d sin �58.36 �� � 5.36�
17.5°18.8°
9000 ft y
zx
Not drawn to scale
40°
3000
ft
r
r
s
240�16.1�
16
sin 70��
h
sin 32�
20°
12°16
32°70°
h
b >315.6sin 88�
315.6 < b <315.6sin 88�
b ≤ 315.6, b �315.6sin 88�
b >10.8
sin 10�
10.8 < b <10.8
sin 10�b ≤ 10.8, b �
10.8sin 10�
b >10
sin 60�
10 < b <10
sin 60�b ≤ 10, b �
10
sin 60�
b >5
sin 36�
5 < b <5
sin 36�b ≤ 5, b �
5
sin 36�
B � 107.79�, C � 14.21�, c � 3.30
B � 72.21�, C � 49.79�, c � 10.27
B � 36.82�, C � 67.18�, c � 32.30
B � 48.74�, C � 21.26�, c � 48.23
A � 48�, b � 2.29, c � 4.73
C � 83�, a � 0.62, b � 0.51
A � 44�14�, B � 50�26�, b � 38.67
B � 18�13�, C � 51�32�, c � 40.06
B � 75.48�, C � 4.52�, b � 122.87
A � 25.57�, B � 9.43�, a � 10.53
A � 174�41�, C � 2�34�, a � 11.99
A � 10�11�, C � 154�19�, c � 11.03
C � 166�5�, a � 3.30, c � 8.05
B � 42�4�, a � 22.05, b � 14.88
B � 101.1�, a � 1.35, b � 3.23
B � 60.9�, b � 19.32, c � 6.36
B � 14.21�, C � 105.79�, b � 2.55
B � 45.79�, C � 74.21�, b � 7.45
B � 21.55�, C � 122.45�, c � 11.49
A � 35�, a � 36.50, b � 11.05
C � 120�, b � 4.75, c � 7.17
A � 35�, a � 11.88, b � 13.31
C � 105�, b � 28.28, c � 38.64
12
ac sin Bb
sin B
d 324.1 154.2 95.2 63.8 43.3 28.1
60�50�40�30�20�10��
333202CB06_AN.qxd 4/13/06 5:41 PM Page 1
(Continued)46. False. Two angles and one side determine a unique triangle,
while two sides and one opposite angle do not necessarilydetermine a unique triangle.
47. (a)(b) Domain:
Range:
(c)
(d) Domain:
Range:
(e
As increases from 0 to , increases and thendecreases, and decreases from 27 to 9.
48. (a)
(b)
(c) Domain: The area would increase and the domain wouldincrease in length.
49. 50. 51. 52. sec2 xsin2 xtan xcos x
0 ≤ � ≤ 1.6690
00 1.7
170
A � 20�15 sin 3�
2� 4 sin
�
2� 6 sin ��
c��
9 < c < 27
0 < � <
00 �
27
c �18 sin� � � � arcsin�0.5 sin ���
sin �
0 < � <
6
0 < � <
00 �
1
� � arcsin�0.5 sin ��
Precalculus with Limits, Answers to Section 6.1 2
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0.4 0.8 1.2 1.6
0.1960 0.3669 0.4848 0.5234
25.95 23.07 19.19 15.33c
�
�
2.0 2.4 2.8
0.4720 0.3445 0.1683
12.29 10.31 9.27c
�
�
333202CB06_AN.qxd 4/13/06 5:41 PM Page 2
Precalculus with Limits, Answers to Section 6.2 3
Section 6.2 (page 443)
Vocabulary Check (page 443)1. Cosines 2.3. Heron’s Area Formula
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.a b c d
17. 5 8 12.07 5.69
18. 25 35 52.20 31.22
19. 10 14 20 13.86
20. 40 60 63.30 80
21. 15 16.96 25 20
22. 35.18 25 50 35
23. 16.25 24. 54 25. 10.4
26. 1350.2 27. 52.11 28. 0.61
29.
30. 1357.8 miles,
31. 373.3 meters 32. 33.
34. 35. 43.3 miles
36. 131.1 feet, 118.6 feet
37. (a) (b)
38. (a) (b) 39. 63.7 feet
40. 103.9 feet 41. 24.2 miles 42. 3.8 miles
43.
44. (a)(b)(c) (d) 8.5 inches
45.
46. 3.95 feet 47. 46,837.5 square feet
48. 6577.8 square meters 49. $83,336.37
50. $62,340.71
51. False. For to be the average of the lengths of the threesides of the triangle, would be equal to
52. False. To solve an SSA triangle, the Law of Sines is needed.
53. False. The three side lengths do not form a triangle.
54. (a) and (b) Answers will vary.
55. (a) 570.60 (b) 5910 (c) 177
56. 405.2 feet 57–58. Answers will vary.
�a b c�3.ss
00 2�
10
x �12�3 cos � 9 cos2 � 187 �
x2 � 3x cos � � 46.75 � 0
PQ � 9.4, QS � 5, RS � 12.8
N 72.8� EN 59.7� E
S 81.5� WN 58.4� W
127.2�
72.3�41.2�, 52.9�
56�
Franklin
Centerville
Rosemount
75°
32° 648 miles
810 miles
S
EW
N
N 37.1� E, S 63.1� E
C
AB 3700 m
1700
m 3000 m
S
EW
N
111.2�68.8�
102.8�77.2�
75.6�104.5�
111.8�68.2�
120�60�
135.1�45�
��
A � 23.65�, B � 53.35�, c � 0.91
A � 33.80�, B � 103.20�, c � 0.54
A � 157�2�, B � 7�43�, c � 4.21
A � 27�10�, C � 27�10�, b � 56.94
A � 37�6�, C � 67�34�, b � 9.94
A � 141�45�, C � 27�40�, b � 11.87
B � 16.53�, C � 108.47�, a � 8.64
B � 13.45�, C � 31.55�, a � 12.16
A � 86.68�, B � 31.82�, C � 61.50�
A � 92.94�, B � 43.53�, C � 43.53�
A � 39.35�, B � 16.75�, C � 123.90�
A � 31.99�, B � 42.39�, C � 105.63�
A � 53.73�, B � 21.27�, c � 11.98
B � 23.79�, C � 126.21�, a � 18.59
A � 61.22�, B � 19.19�, C � 99.59�
A � 23.07�, B � 34.05�, C � 122.88�
b2 � a2 c2 � 2ac cos B
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(inches) 9 10 12 13 14
(degrees)
(inches) 20.88 20.28 18.99 18.28 17.48s
109.6�98.2�88.0�69.5�60.9��
d
(inches) 15 16
(degrees)
(inches) 16.55 15.37s
139.8�122.9��
d
333202CB06_AN.qxd 4/13/06 5:41 PM Page 3
(Continued)
59. 60.
61. 62.
63. 64.
65. 66.
67. 68.
69. 70.
is undefined.
71. 72.
73. 74. 2 cos x sin��
2��2 sin 7
12 sin
4
csc � � ±23
3csc � � �2
sec � � 2sec � �23
3
12 � 6 sec �tan � � �33
csc � � �2csc �
sec � � 2sec � � 1
sin � � �22
cos � � 1
4 � �x � 1�2
21
x � 2
1 � 9x2
3x1
1 � 4x2
5
6�
3
�
3
3
2�
2
Precalculus with Limits, Answers to Section 6.2 4
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333202CB06_AN.qxd 4/13/06 5:41 PM Page 4
Precalculus with Limits, Answers to Section 6.3 5
Section 6.3 (page 456)
Vocabulary Check (page 456)1. directed line segment 2. initial; terminal3. magnitude 4. vector5. standard position 6. unit vector7. multiplication; addition 8. resultant9. linear combination; horizontal; vertical
1.
and have the same magnitude and direction, so they areequal.
2.and have the same magnitude and direction, so they are
equal.
3.
4. 5.
6. 7.
8. 9.
10. 11.
12.
13.
14.
15. 16.
17. 18.
19. 20.
21. (a) (b)
(c)
22. (a) (b)
(c)
x642
10
8
4
2
2u2 3u v−
y
−3v
−2−4
−4−6−8
−6−8−10
��8, 6�
x
uu v−
21
6
5
4
2
y
−1−2−3−4−5
−v
u
v
u + v
8642
8
6
4
2
x
y
��2, 3��6, 3�
x642−2−4−6
2
−6
−10
2u
−3v2 3u v−
y
�1, �7�
x3
3
2
1
21−1−2−3
u
−vu v−
y
x
5
4
3
2
1
−154321−1
u
u v+v
y
�1, �2��3, 4�
x
v
u− 12
v u− 12
y
xu
2v
u + 2v
y
x
u
− v
u v−
y
x
u
v
y
u + v
v
5v
x
y
x
v
v−
y
v � �7, �24�; v � 25
v � ��9, �12�; v � 15
v � �12, 29�; v � 985
v � �8, 6�; v � 10v � �8, �8�; v � 82
v � �16, 7�; v � 305v � �7, 0�; v � 7
v � �0, 5�; v � 5v � �4, 6�; v � 213
v � ��3, 2�; v �13v � ��4, �2�; v � 25
v � �3, 2�; v � 13
vu u � v � 73, slopeu � slopev �
83
vu
slopeu � slopev �14 u � v � 17,
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333202CB06_AN.qxd 4/13/06 5:41 PM Page 5
(Continued)23. (a) (b)
(c)
24. (a) (b)
(c)
25. (a) (b)
(c)
26. (a) (b)
(c)
27. (a) (b)
(c)
x321−1
1
−1
−2
−3
−4
y
2u − 2v−3v
2u
4i � 3j
3−1
u
− vu v−
1
−1
−2
−3
x
y
x321−1
3
2
1
−1
uv
u v+
y
2i � j2i j
x
3
2
1
4321
y
2u − 3v
2u
−3v
−1−3−4−5
−5
−6
−7
−6
�i � 4 j
x
2
1
21
u
y
−1
−1
−2
−2−v
u − vx
4
3
2
−1
1
u
v
y
−1−2−3−4
u + v
�i � j�3i 3j
x
12
10
8
−2
2u
−3v
2 3u v−
642−2−4−6−8
y
�4i 11j
x321−1−2−3
u
−v
u v−5
4
−1
y
x3
3
2
1
−1
−2
−3
−1−2−3
v
u
u v+
y
�i 4j3i � 2j
x1
1
y
−3v = 2u − 3v
−1
−2
−2−3−4−5−6−7
−3
−4
−5
−6
−7
2u
��6, �3�
x1
1
u
y
−1
−2
−2
−3
−3
−v = u − v
x
3
2
1
321
v u v= +
u
y
−1
−1
��2, �1��2, 1�
x2−2−4−6−8−10−12
12
10
8
6
4
2
− 2
−3v
2 2 3u u v= −
y
��10, 6�
x1−1−2−3−4−5−6−7
u u v= −
v
7
6
5
4
3
2
1
y
x1−1−2−3−4−5−6−7
u u v= +
v
7
6
5
4
3
2
1
y
��5, 3���5, 3�
Precalculus with Limits, Answers to Section 6.3 6
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d.Precalculus with Limits, Answers to Section 6.3 7
(Continued)
28. (a) (b)
(c)
29. 30. 31.
32. 33.
34. 35. 36.
37. 38.
39. 40.
41. 42. 43.
44. 45. 46.
47. 48.
49. 50.
51. 52.
53. 54.
55. 56.
57. 58.
59. 60.
x45°
1
2
3
y
1 2 3
x
4
3
2
−1
1−1−2−3−4
150°
y
v � �52
4,
52
4 �v � ��73
4,
74�
x
45°
1
1
y
x
−1
1
2
1 2 3
y
v � �2
2, 2
2 �v � �3, 0�
v � 41; � � 141.3� v � 62; � � 315�
v � 8; � � 135� v � 3; � � 60�
x
u
y
−1−2
−2
−3
−4
2 3
u − 2w
−2w
x
w12
u23
12 (3u + w)
1
−1
−2
2
4
y
v � �0, �5�v � �72, �1
2�
x
w2
3
y
−1−2 1
−u + w
−ux
u
2w
3 4 5
−1
1
2
3
4
y
u + 2w
v � ��1, 3�v � �4, 3�
x2
2
1
1
w
w34
y
−1
−1
x1
1
2 3
−1
−2
u
32 u
y
v � �34, 32�v � �3, �3
2�6i � 3j3i 8j3i 8j
7i 4j��10, 0��182929
, 4529
29 �
��32, 32 ��522
, 52
2 �
�358
58 i
75858
j55
i �25
5 j
�ij22
i 22
j
31010
i �1010
j� 513
, �1213�
��22
, 22 ��0, �1��1, 0�
x2
8
4
2
y
−2
−2−4−6−8
2u − 3v
2u
−3v
�6i 6j
x
2
1
u
−v
y
−1
−1
−2−3 1
u − v
x3
3
2
1
21
u
v
y
u + v
−1
−1
�2i 3j2i 3j
333202CB06_AN.qxd 4/13/06 5:41 PM Page 7
(Continued)
61. 62.
63. 64.
65. 66.
67. 68. 69.
70. 71. 72.
73. 398.32 newtons 74. 2396.2 newtons
75. 228.5 pounds 76. 58.6 pounds
77. Vertical component: feet per second
Horizontal component: feet per second
78. Vertical component: feet per second
Horizontal component: feet persecond
79. 80.
81. 3154.4 pounds 82.53.2 pounds
83. 138.7 kilometers per hour
84. (a)
(b) (c)(d) 600.3 miles per hour (e)
85. 1928.4 foot-pounds 86. pounds; 1 pound
87. True. See Example 1.
88. True, by the definition of a unit vector.
89. (a) (b)(c) No. The magnitude is at most equal to the sum when
the angle between the vectors is
90. (a)
(b)
(c) Range:Maximum is 15 when Minimum is 5 when
(d) The magnitudes of and are not the same.
91–92. Answers will vary.
93. 94.
95. 96. 97.
98. 99.
100.
101.
102.
2 n,
5
4 2n,
7
4 2n
n,
6 2n,
11
6 2n
n, 5
4 2n,
7
4 2n
2 n, 2n125 tan3 �
6 sec �8 cos �8 tan �
�10, 50� or ��10, �50��1, 3� or ��1, �3�
F2F1
� � .� � 0.
�5, 15�
00 2
15
�
55 4 cos �
0�.
180�0�
2
337.5�
580�cos 118�, sin 118��60�cos 45�, sin 45��
x
28°
45°
y
S
EW
N
60 mph
580 mph
N 21.4� E;
100 lb
20°20°
TBC � 2169.5 poundsTBC � 1305.4 pounds
TAC � 3611.1 poundsTAC � 1758.8 pounds
1200 cos 6� � 1193.4
1200 sin 6� � 125.4
70 cos 35� � 57.34
70 sin 35� � 40.15
37.5�;71.3�;
8.7�;12.8�;
47.4�62.7�90�
90��33.04, 53.19��102 � 50, 102 ��2, 4 23 ��5, 5�
x
1
2
3
y
−1
−1 1 2 3
x2
1
2
3
y
1−1
v � �95
, 125 �v � �10
5,
310
5 �
x
10
8
6
4
2
642
90°
y
−2−2
−4−6x
150°
−5 − 4 −3 −2 −
1
11
2
3
4
5
y
−1
v � �0, 43 �v � ��36
2,
32
2 �
Precalculus with Limits, Answers to Section 6.3 8
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333202CB06_AN.qxd 4/13/06 5:41 PM Page 8
Precalculus with Limits, Answers to Section 6.4 9
Section 6.4 (page 467)
Vocabulary Check (page 467)
1. dot product 2. 3. orthogonal
4. 5.
1. 2. 9 3. 4. 5. 6
6. 13 7. 8. 9. 8; scalar
10. 6; scalar 11. vector
12. vector 13. vector
14. vector 15. scalar
16. scalar 17. 4; scalar
18. 13; scalar 19. 13 20. 21.
22. 20 23. 6 24. 21 25. 26.
27. 28. 29. 30.
31. 32. 33. 34.
35. 36.
37. 38.
39. 40.
41. 42.
43. 44. 45. 46. 24
47. Parallel 48. Neither 49. Neither 50. Neither
51. Orthogonal 52. Orthogonal
53. 54.
55. 56.
57. 0 58. 0 59.
60. 61.
62. 63. 32 64. 14
65. (a) $58,762.50This value gives the total revenue that can be earned byselling all of the units.
(b)66. (a) $8732.50; answers will vary. (b)67. (a)
(b)
(c) 29,885.8 pounds68. pounds; 5318.0 pounds69. 735 newton-meters 70. 12,000 foot-pounds71. 779.4 foot-pounds72. 1,048,514.62 kilogram-meters (10,282,651.78 newton-
meters)73. 21,650.64 foot-pounds 74. 1174.62 foot-pounds75. False. Work is represented by a scalar.76. True.
77. (a) (b) (c)
78. (a) and are parallel. (b) and are orthogonal.79–80. Answers will vary.81. 82. 83. 84.
85. 86. 87.
88. 89. 90.
91. 92. �253204
204325
�323325
�253325
7
6,
3
2,
11
6
0,
2,
5
4,
3
2,
7
40,
6, ,
11
6
�242�261214127
vuvu
2< � ≤ 0 ≤ � <
2� �
2
cos 90� � 0
937.7
Force � 30,000 sin d1.025v
1.05v
3i �52 j, �3i
52 j
23 i
12 j, �2
3 i �12 j�3, 8�, ��3, �8�
��5, 3�, �5, �3�
1417��4, �1�, 5
17�1, �4�45229�2, 15�, 6
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333202CB06_AN.qxd 4/13/06 5:41 PM Page 9
Section 6.5 (page 478)
Vocabulary Check (page 478)1. absolute value2. trigonometric form; modulus; argument
3. DeMoivre’s 4. th root
1. 2.
7 73. 4.
135. 6.
7. 8.
9. 10.
11. 12.
13. 14.
15. 16.
17. 18.
19. 20.
10�cos 5.96 i sin 5.96�65 �cos 2.62 i sin 2.62�
Realaxis3
1
Imaginaryaxis
−1
−2
2
3 − i
Realaxis
−7 + 4 i
−2
−2
2
4
− 4
− 4
− 6− 8
Imaginaryaxis
4�cos
2 i sin
2�5�cos 3
2 i sin
3
2 �
axis
Real
5
4
3
2
1
321
Imaginary
−1−1−2−3 axis
4i
axis
Realaxis42−2−4
−2
−4
−6
−8
−5i
Imaginary
5�cos 11
6 i sin
11
6 �4�cos 4
3 i sin
4
3 �
Realaxis
( )52
1
2
2
3
Imaginaryaxis
−1−1
−2
−3
−4
4 5
3 − i
Realaxis
Imaginary
−1−2
−2
−3
−3
− 4
− 4−2(1+ 3i)
axis
8�cos 5
3 i sin
5
3 �2�cos
6 i sin
6�
Real
8642
2
Imaginary
−2
−2
−4
−6
−8
axis
axis
4 − 4 3 i
Realaxis
Imaginaryaxis
−1
−1
1
2
1 2
3 + i
22�cos
4 i sin
4�32�cos 7
4 i sin
7
4 �
Real
1
1
2
3
2 3 axis
axisImaginary
2 + 2i
Realaxis
3 − 3i
1 2 3
−1
−2
−3
Imaginaryaxis
2�cos 2
3 i sin
2
3 �10 �cos 5.96 i sin 5.96�
2�cos i sin �3�cos
2 i sin
2�7385
Realaxis
2
4
6
Imaginaryaxis
−2−2
−4
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−8 + 3i
Realaxis
6 − 7i
2 4 6 8
−2
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− 6
− 8
Imaginaryaxis
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42 6
Imaginaryaxis
−2−2
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5 − 12i
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Imag naryaxis
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8
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Precalculus with Limits, Answers to Section 6.5 10
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333202CB06_AN.qxd 4/13/06 5:41 PM Page 10
Cop
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d.Precalculus with Limits, Answers to Section 6.5 11
(Continued)
21. 22.
23. 24.
25. 26.
27. 28.
29. 30.
31. 32.
33. 34.
35. 36.
37. 38.
7
39. 40.
�3.8165 � 4.6297i2.8408 0.9643i
1
Real
−3.8165 − 4.6297i
axis
axisImaginary
1−1−2
−2
−3
−4
−5
−3−4−5
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2.8408 + 0.9643i
−2
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1
2
Imaginaryaxis
1 2 3 4
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Realaxis
7
2
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Imaginaryaxis
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yaxis
Realaxis
10
8
6
4
2
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8i
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1.5529 5.7956i�152
8
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4
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642
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1.5529 + 5.7956i
axis
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−2
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15 28
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− i+
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2
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32
− 3+ 3 i2
Imaginar
11�cos 3.75 i sin 3.75�139�cos 3.97 i sin 3.97�
axis
Realaxis
−9 − 2 10 i
Imaginary
−2−4−6−8−10−2
−4
−6
−8
−10
axis
Realaxis
−8 − 5 3 i
Imaginary
−2−4−6−8−10−2
−4
−6
−8
−10
73�cos 0.36 i sin 0.36�29�cos 0.38 i sin 0.38�
y
Real
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axis
axis
8 + 3i
−2 2 4 6 8−2
−4
2
4
6
y
Real
5
4
3
2
1
−154321−1
Imaginar
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axis
5 + 2i
10�cos 1.25 i sin 1.25�10 �cos 3.46 i sin 3.46�
Real
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1
2
2
3
3
Imaginaryaxis
−1
−1 axis
1 + 3iRealaxis
−1
−2
−2
−3
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−3 − i
− 4
− 4
Imaginaryaxis
3�cos 5.94 i sin 5.94�23�cos
6 i sin
6�
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3
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4321−1
4
3
2
1
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3 3+ i
Imaginary
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Realaxis
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1
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333202CB06_AN.qxd 4/13/06 5:41 PM Page 11
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Precalculus with Limits, Answers to Section 6.5 12
(Continued)
41. 42.
43. 44.
45. 46.
The absolute value of The absolute value ofeach is 1. each is 1.
47. 48.
49. 50.
51. 52.
53. 54.
55. 56.
57. 58.
59. (a)
(b) (c) 4
60. (a)
(b)
(c)
61. (a)
(b)
(c)
62. (a)
(b)
(c)
63. (a)
(b)
(c)
64. (a)
(b)
(c)
65. (a)
(b)
(c)
66. (a)
(b)
(c)
67. 68.
69. 70.
71. 72. 73.
74. 75. 76.
77. 78. 79.
80. 256 81. 82. 1
83. 84.
85. 86. 87.
88.
89. (a)
(b)
31
1
3
−3
−3 −1
Realaxis
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5 �cos 240� i sin 240��5 �cos 60� i sin 60��
�322 322 i
32i44.45 252.11i81
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−1−1
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axis
42
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s
Realaxis
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−4 −2
Realaxis
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−4
−2
1
2
4
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1
3
Realaxis
Imaginaryaxis
−1 1
25
�45
i � 0.400 � 0.800i
25
�cos 5.18 i sin 5.18� � 0.403 � 0.798i
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2 i sin
2��
10
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�0.018 0.298i
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3��
��0.982 2.299i
5
2�cos 1.97 i sin 1.97� � �0.982 2.299i
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3 ���5�cos 0.93 i sin 0.93��
4 � 43 i
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3 � � 4 � 43 i
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3 i sin
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3 ���2i � 2i 2 � �2i 2 � 2 � 2i
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4 � � 2 � 2i
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2 ���3 � 1� �3 1�i � 0.732 2.732i
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12 i sin
5
12� � 0.732 2.732i
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4���2�cos
6 i sin
6��4 �cos 0 i sin 0� � 4
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4 ��67�cos 300� i sin 300��4�cos 302� i sin 302��
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12�cos 80� i sin 80��cos 30� i sin 30�
cos 25� i sin 25�0.27�cos 150� i sin 150��0.4�cos 40� i sin 40��10
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z = (1 + i) 22
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z2 = i
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4.7693 7.6324i�2.9044 0.7511i
3.0902 9.5106i4.6985 1.7101i
333202CB06_AN.qxd 4/13/06 5:41 PM Page 12
Precalculus with Limits, Answers to Section 6.5 13
(Continued)
(c)
90. (a)
(b)
(c)
91. (a)
(b)
(c)
92. (a)
(b)
(c)
93. (a) (b)
(c)
94. (a) (b)
(c)
95. (a)
(c)
96. (a) (b)
(c)
97. (a)
2�cos 3
2 i sin
3
2 �2�cos i sin �
2�cos
2 i sin
2�2�cos 0 i sin 0�
0.5176 � 1.9319i
�1.9319 0.5176i,
2 2i,
2�cos 19
12 i sin
19
12 �
2�cos 11
12 i sin
11
12 � 3
3
−3
−3
Realaxis
Imaginaryaxis
2�cos
4 i sin
4�
3.8302 � 3.2140i�4.6985 � 1.7101i,0.8682 4.9240i,
5�cos 16
9 i sin
16
9 �5�cos
10
9 i sin
10
9 �5�cos
4
9 i sin
4
9 �
1.9134 � 4.6194i�1.9134 4.6194i, �4.6194 � 1.9134i,4.6194 1.9134i,
5�cos 13
8 i sin
13
8 �5�cos
9
8 i sin
9
8 �5�cos
5
8 i sin
5
8 � 6
4
−6
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−4
−22 6
Realaxis
Imaginaryaxis
5�cos
8 i sin
8�
52
2�
52
2i
�52
2
52
2i
5�cos 7
4 i sin
7
4 �
−2−6
6
4
4
2
−2
−4
−6
2 6
Imaginaryaxis
Realaxis
5�cos 3
4 i sin
3
4 ��0.8135 � 1.8271i, 1.4863 � 1.3383i
3 i, �0.4158 1.9563i, �1.9890 0.2091i,
3
3−3
−3
Realaxis
Imaginaryaxis
2�cos 53
30 i sin
53
30 �2�cos
41
30 i sin
41
30 �2�cos
29
30 i sin
29
30 �2�cos
17
30 i sin
17
30 �2�cos
6 i sin
6�0.3473 � 1.9696i1.5321 1.2856i, �1.8794 0.6840i,
axis
Realaxis
3
1
−1
−3
31−1−3
Imaginary
2�cos 14
9 i sin
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9 �2�cos
8
9 i sin
8
9 �2�cos
2
9 i sin
2
9 �23 2i, �23 � 2i
2
6
6
2
−6
−6−2
Realaxis
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4 �cos 210� i sin 210��4�cos 30� i sin 30��
5
2
15
2i, �
5
2�
15
2i
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(b)
6
4 6
2
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−6
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Imaginaryaxis
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333202CB06_AN.qxd 4/13/06 5:41 PM Page 13
Cop
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Hou
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d.
Precalculus with Limits, Answers to Section 6.5 14
(Continued)
(b) (c)
98. (a) (b)
(c)
99. (a)
(b)
(c)
100. (a)
(b)
(c)
101. (a)
(b)
(c)
102. (a)
(b)
(c)
103. (a)
22�cos 7
4 i sin
7
4 �
22�cos 27
20 i sin
27
20 �22�cos
19
20 i sin
19
20 �
22�cos 11
20 i sin
11
20 �
22�cos 3
20 i sin
3
20�
1 i, �1 i, �1 � i, 1 � i
2
1
21
Real
Imaginary
−1
−1
−2
−2
axis
axis
2�cos 7
4 i sin
7
4 �
2�cos 5
4 i sin
5
4 �
2�cos 3
4 i sin
3
4 �
2�cos
4 i sin
4�
5
2
53
2i, �5,
5
2�
53
2i
Real
642
6
4
2
−4
−6
Imaginary
−2−6 axis
axis
5�cos 5
3 i sin
5
3 �5�cos i sin �
5�cos
3 i sin
3�
10, �5 53 i, �5 � 53 i
64
8
−2−4−6
−6
−8
−8
2 4 6 8
Realaxis
Imaginaryaxis
10�cos 4
3 i sin
4
3 �10�cos
2
3 i sin
2
3 �10�cos 0 i sin 0�
0.3090 � 0.9511i�0.8090 0.5878i, �0.8090 � 0.5878i,1, 0.3090 0.9511i,
cos 8
5 i sin
8
5
cos 6
5 i sin
6
5
cos 4
5 i sin
4
52
−2
−2
2
Imaginaryaxis
Realaxis
cos 2
5 i sin
2
5
cos 0 i sin 0
0.3827 � 0.9239i�0.3827 0.9239i, �0.9239 � 0.3827i,0.9239 0.3827i,
cos 13
8 i sin
13
8
cos 9
8 i sin
9
8
cos 5
8 i sin
5
8
2
2
−2
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Realaxis
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cos
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8
2, 2i, �2, �2iaxis
3
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−3
31−1−3
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Imaginary
333202CB06_AN.qxd 4/13/06 5:41 PM Page 14
Precalculus with Limits, Answers to Section 6.5 15
(Continued)(b)
(c)
104. (a)
(b)
(c)
105.
106.
107.
108.
109.
110.
2�cos 23
12 i sin
23
12 �2�cos
19
12 i sin
19
12 �
2�cos 5
4 i sin
5
4 �
2�cos 11
12 i sin
11
12 �
2�cos 7
12 i sin
7
12�Realaxis−1−3 1 3
−1
−3
1
3
Imaginaryaxis
2�cos
4 i sin
4�
2�cos 15
8 i sin
15
8 �
2�cos 11
8 i sin
11
8 �
2�cos 7
8 i sin
7
8 �3
1
−3
3−3 −1
Real
Imaginary
axis
axis2�cos 3
8 i sin
3
8 �
3�cos 4
3 i sin
4
3 �
3�cos 2
3 i sin
2
3 �
Real
4
2
421
Imaginary
−4
−2
−2−4 −1
axis
axis
3�cos 0 i sin 0�
3�cos 9
5 i sin
9
5 �
3�cos 7
5 i sin
7
5 �3�cos i sin �
3�cos 3
5 i sin
3
5 �4
−4
−2− 4 2 4
Imaginaryaxis
Realaxis
3�cos
5 i sin
5�
cos 5
3 i sin
5
3
cos i sin
2
2
−2
−2
Realaxis
Imaginaryaxiscos
3 i sin
3
cos 15
8 i sin
15
8
cos 11
8 i sin
11
8
cos 7
8 i sin
7
812
12
−
Imaginaryaxis
Realaxis
cos 3
8 i sin
3
8
�0.5176 � 1.9319i, 2 � 2 i
�2 2 i, �1.9319 � 0.5176i,
1.9319 0.5176i, 0.5176 1.9319i,
3
1
31
Real
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−3 axis
axis
2�cos 7
4 i sin
7
4 �
2�cos 17
12 i sin
17
12 �
2�cos 13
12 i sin
13
12 �
2�cos 3
4 i sin
3
4 �
2�cos 5
12 i sin
5
12�
2�cos
12 i sin
12�
�2.7936 0.4425i, �1.2841 � 2.5201i, 2 � 2i
2.5201 1.2841i, �0.4425 2.7936i,
1
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21−2 −1
Real
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axis
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d.
333202CB06_AN.qxd 4/13/06 5:41 PM Page 15
(Continued)
111.
112.
113. True, by the definition of the absolute value of a complexnumber.
114. False. They are equally spaced around the circle centeredat the origin with radius
115. True. ifand only if and/or
116. False. The complex number needs to be converted totrigonometric form before using DeMoivre’s Theorem.
117–118. Answers will vary.
119. (a) (b)
120–122. Answers will vary.
123. (a) (b)
124. (a) (b)
125.
126.
127.
128.
129.
130.
131. 16; 2 132.
133. 134.
135. 136. sin 7� � sin 3�3�sin 11� sin 5��
112; 1
60116; 45
18; 1
24
A � 8�30�, a � 1.01, b � 6.73
B � 47�45�, a � 7.53, b � 8.29
A � 84�, a � 2009.43, c � 2020.50
B � 60�, a � 65.01, c � 130.02
A � 24�, b � 75.24, c � 82.36
B � 68�, b � 19.80, c � 21.36
3�cos 315� i sin 315��3�cos 225� i sin 225��3�cos 135� i sin 135��
�813�cos 45� i sin 45��
2�cos 270� i sin 270��2�cos 150� i sin 150��
8i2�cos 30� i sin 30��
cos 2� i sin 2�r 2
�4 6 i�8� �22�cos 0.55 i sin 0.55��8
r2 � 0.r1 � 0z1z2 � r1r2�cos��1 �2� i sin��1 �2�� � 0
nr.
82�cos 29
16 i sin
29
16 �
82�cos 21
16 i sin
21
16 �
82�cos 13
16 i sin
13
16 �2
2
−2
−2
Realaxis
Imaginaryaxis82�cos
5
16 i sin
5
16�
62�cos 23
12 i sin
23
12 �
62�cos 5
4 i sin
5
4 �−2 2
2
−2
Imaginaryaxis
Realaxis
62�cos 7
12 i sin
7
12�
Precalculus with Limits, Answers to Section 6.5 16
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d.Precalculus with Limits, Answers to Review Exercises 17
Review Exercises (page 482)
1.
2.
3.
4.
5.
6.
7.
8.
9.
10. No solution
11.
12. Two solutions:
13. 7.9 14. 15.8 15. 33.5 16. 44.1
17. 31.1 meters 18. 4.8 19. 31.01 feet
20. 586.4 feet 21.
22.
23.
24.
25.
26.
27.
28.
29. feet, feet
30. meters, meters
31. 615.1 meters
32.
1135.5 miles33. 9.80 34. 36.98 35. 8.36 36. 242.63
37.
38.
39. 40. 41. 42.
43. 44.
45. (a) (b) (c)
(d)
46. (a) (b) (c)
(d)
47. (a) (b) (c)
(d)
48. (a) (b) (c)
(d)
49. (a) (b) (c)
(d)
50. (a) (b) (c)
(d)
51. (a) (b) (c) (d)
52. (a) (b) (c) (d)
53. 54.
55. 56.
57. 58. 59.
60. 61.
62. 63. v � 7; � � 60�17�cos 346� i sin 346� j�
102�cos 135� i sin 135�j�7i � 16j
6i 4j�6i � 8j�3i 4j
x
2
4
6
8
12
v
v
y
−2
2 4 6 8 10
x10 20 30
10
−10
20
3v
v
y
�5, 32��30, 9�
x
20
20
y
−60
−60
−40
−40−5v 4u
4u − 5v
30252010−5x
2
2 +u v
v
2u
y
−2
−4
−6
−8
−10
−12
��26, �35��22, �7�
2i � 28j�18ji � 7ji � 5j
18i 12j12i5i � 6j3i 6j
�27i � 17j
�21i � 9j�11i � 4j�3i � 4j
20i j
6i � 3j�3i � 4j7i 2j
�11, �44��3, �24���2, �6��4, �10�
��17, 18���15, 6���9, �2���1, 6�
�20, 23��12, 15��4, 6��4, 4�
��11, �3���3, �9��2, �9���4, 3�
��24
, �24 ���4, 43 �
�14, 4��7, �7��6, 52��7, �5�
u � v � 210, slopeu � slopev � �3
u � v � 61, slopeu � slopev �56
67°
5°850
d
1060
S
EW
N
�33.5�11.3
�12.6� 4.3
B � 35.20�, C � 82.80�, a � 17.37
A � 45.76�, B � 91.24�, c � 21.42
A � 9.90�, C � 20.10�, b � 29.09
A � 35�, C � 35�, b � 6.55
A � 101.47�, B � 31.73�, C � 46.80�
A � 29.92�, B � 86.18�, C � 63.90�
A � 53.13�, B � 36.87�, C � 90�
A � 29.69�, B � 52.41�, C � 97.90�
A � 139.08�, C � 15.92�, c � 2.60
A � 40.92�, C � 114.08�, c � 8.64
B � 39.48�, C � 65.52�, c � 48.24
A � 20.41�, C � 9.59�, a � 20.92
A � 80�, b � 334.95, c � 219.04
B � 108�, a � 11.76, c � 21.49
C � 40�, a � 162.42, b � 115.29
C � 66�, a � 2.53, b � 9.11
A � 150�, a � 48.24, b � 16.75
A � 26�, a � 24.89, c � 56.23
C � 37�, b � 38.90, c � 27.31
C � 74�, b � 13.19, c � 13.41
333202CB06_AN.qxd 4/13/06 5:41 PM Page 17
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d.
Precalculus with Limits, Answers to Review Exercises 18
(Continued)
64. 65.
66. 67.
68.
69. The resultant force is 133.92 pounds and from the 85-pound force.
70. 180 pounds each 71. 422.30 miles per hour;
72. 740.5 kilometers per hour; 73. 45
74. 75. 76. 77. 50; scalar
78. 5; scalar 79. vector 80. scalar
81. 82. 83. 84.
85. Orthogonal 86. Parall 87. Neither
88. Orthogonal 89.
90. 91.
92. 93. 48 94.
95. 72,000 foot-pounds 96. 281.9 foot-pounds
97. 98.
7 699. 100.
101.
102.
103. 104.
105. (a)
(b)
106. (a)
(b)
107. 108.
109. 110. 16
111. (a)
(b)
(c)
112. (a)
4�cos 13
8 i sin
13
8 �4�cos 9
8 i sin
9
8 �
4�cos 5
8 i sin
5
8 �4�cos
8 i sin
8�0.7765 � 2.898i, 2.898 � 0.7765i
�2.898 0.7765i, �32
2�
322
i,
322
32
2i, �0.7765 2.898i,
4
−4
−2
−2−4 4
Imaginaryaxis
Realaxis
3�cos 23
12 i sin
23
12 �
3�cos 19
12 i sin
19
12 �
3�cos 5
4 i sin
5
4 �
3�cos 11
12 i sin
11
12 �
3�cos 7
12 i sin
7
12�
3�cos
4 i sin
4�2035 � 828i
�16 � 163 i625
2
6253
2i
z1
z2�
32
4 �cos 13
12 i sin
13
12 �z1z2 � 122�cos
17
12 i sin
17
12 �z2 � 4�cos
6 i sin
6�z1 � 32�cos
5
4 i sin
5
4 �
z1
z2�
2
5 �cos
3 i sin
3�z1z2 � 40�cos
10
3 i sin
10
3 �z2 � 10�cos
3
2 i sin
3
2 �
z1 � 4�cos 11
6 i sin
11
6 �7�cos i sin �6�cos
5
6 i sin
5
6 �13�cos 1.176 i sin 1.176�
52�cos 7
4 i sin
7
4 �22934
y
Real
6
4
2
Imaginar
−2
−4
−6
−6−8−10−12
−10 − 4i
axis
axisy
Real
5
4
3
2
1
−154321−1
5 3+ i
Imaginar
axis
axis
Real
8642
8
6
4
2
−6i
Imaginary
−2−4
−4
−6
−6
−8
−8
axis
axis
10
8
6
4
2
642
7i
Real
Imaginary
−2−2−4−6 axis
axis
�1322529��5, 2�, 19
29�2, 5�
52��1, 1�, 92�1, 1��5, 0�, �0, 6�
�1317�4, 1�, 16
17��1, 4�
22.4�160.5�105�11
12
�6;�6, �8�;
�136�2�140
32.1�
130.4�
5.6�
v � 65; � � 352.9�
v � 32; � � 225� v � 65; � � 119.7�
v � 41; � � 38.7� v � 3; � � 150�
333202CB06_AN.qxd 4/13/06 5:41 PM Page 18
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d.Precalculus with Limits, Answers to Review Exercises 19
(Continued)
(b)
(c)
113. (a)
(b)
(c)
114. (a)
(b)
(c)
115.
116.
117.
2�cos 11
6 i sin
11
6 � � 3 � i
2�cos 7
6 i sin
7
6 � � �3 � i
2�cos
2 i sin
2� � 2i
3
1
3
−3
−1−3 1
Realaxis
Imaginaryaxis
2�cos 8
5 i sin
8
5 � � 0.6180 � 1.9021i
2�cos 6
5 i sin
6
5 � � �1.6180 � 1.1756i
2�cos 4
5 i sin
4
5 � � �1.6180 1.1756i
2�cos 2
5 i sin
2
5 � � 0.6180 1.9021i
2�cos 0 i sin 0� � 2
2
4
−4
−2
−2−4 2 4
Imaginaryaxis
Realaxis
3�cos 7
4 i sin
7
4 � �32
2�
32
2i
3�cos 5
4 i sin
5
4 � � �32
2�
32
2i
3�cos 3
4 i sin
3
4 � � �32
2
32
2i
3�cos
4 i sin
4� �32
2
32
2i
�1.236 � 3.804i, 3.236 � 2.351i
3.236 2.351i, �1.236 3.804i, �4,
532
5
1Real
Imaginary
axis
axis
−5
−1−2−3
4�cos 9
5 i sin
9
5 �
4�cos 7
5 i sin
7
5 �4�cos i sin �
4�cos 3
5 i sin
3
5 �
4�cos
5 i sin
5�
2, �1 3 i, �1 � 3 i
−3
−3
3
−1 1 3
Imaginaryaxis
Realaxis
2�cos 4
3 i sin
4
3 �
2�cos 2
3 i sin
2
3 �2�cos 0 i sin 0�
�3.696 � 1.531i, 1.531 � 3.696i
3.696 1.531i, �1.531 3.696i,
5321
5
3
1Real
Imaginary
axis
axis
−5
−3
−3
−2
−1
333202CB06_AN.qxd 4/13/06 5:41 PM Page 19
(Continued)
118.
119. True. is defined in the Law of Sines.120. False. There may be no solution, one solution, or two
solutions.
121. True. By definition, so
122. False. If then
123. False. The solutions to are and
124.
125.
126. Direction and magnitude 127. and
128. a; The angle between the vectors is acute.
129. If the direction is the same and the magnitude is times as great.
If the result is a vector in the opposite directionand the magnitude is times as great.
130. The diagonal of the parallelogram with and as its adja-cent sides
131. (a) (b)
132. (a) (b)
133.
134. (a) 3 roots
(b) On the circle and from the positive axisx-
300�120�, 210�,
� �cos 2� � i sin 2�
z1z2 � �4; z1
z2� cos�2� � � i sin�2� � �
4�cos 330� i sin 330��
4�cos 240� i sin 240��
4�cos 150� i sin 150��
�128 � 1283 i4�cos 60� i sin 60��
4�cos 300� i sin 300��
4�cos 180� i sin 180��
�644�cos 60� i sin 60��
vu
�k�k < 0,
kk > 0,
CA
c2 � a2 b2 � 2ab cos Cb2 � a2 c2 � 2ac cos B,a2 � b2 c2 � 2bc cos A,
asin A
�b
sin B�
csin C
x � �2 � 2i.x � 2 2ix2 � 8i � 0
a � b � 0.v � ai bj � 0,
v � v u .u �v
v ,
sin 90�
2
2
−2
−2
Realaxis
Imaginaryaxis
cos 3
2 i sin
3
2� �i
cos 4
3 i sin
4
3� �
1
2�
3
2i
cos 2
3 i sin
2
3� �
1
2
3
2i
cos
2 i sin
2� i
cos 0 i sin 0 � 1
3
1
3
−3
−3−1
Imaginaryaxis
Realaxis
Precalculus with Limits, Answers to Review Exercises 20
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Precalculus with Limits, Answers to Chapter Test 21
Chapter Test (page 486)
1.
2.
3. Two solutions:
4. No solution 5.
6.
7. 2052.5 square meters 8. 606.3 miles;
9. 10.
11. 12.
13. 14.
15. 250.15 pounds 16. 17. No
18. 19. pounds
20. 21.
22. 23.
24.
25.
3�cos 3
2 i sin
3
2 �
3�cos 5
6 i sin
5
6 �
421
4
2
1
y
Real
Imaginar
−4
−4
−2
−2 −1 axis
axis3�cos
6 i sin
6�
4 42�cos 19
12 i sin
19
12 �
4 42�cos 13
12 i sin
13
12 �
4 42�cos 7
12 i sin
7
12�
4 42�cos
12 i sin
12�
5832i�6561
2�
656132
i
�3 33 i52�cos 7
4 i sin
7
4 �
�1043726�5, 1�; 29
26��1, 5�
135�14.9�;
x
42
36
30
24
18
12
6
423630246
5 3u v−5u
y
−6 −3v
�45, �3
5��36, 22�
x
u v−
12
10
8
4
2
6
121082
u
−v
y
−2−2
x
8
4
2
42
u v+
u
v
y
−2
−2−4−6
�10, 4���4, 6�
�183417
, �3034
17 ��14, �23�
29.1�
A � 23.43�, B � 33.57�, c � 86.46
A � 39.96�, C � 40.04�, c � 15.02
B � 150.88�, C � 5.12�, c � 2.46
B � 29.12�, C � 126.88�, c � 22.03
A � 43�, b � 25.75, c � 14.45
C � 88�, b � 27.81, c � 29.98
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333202CB06_AN.qxd 4/13/06 5:41 PM Page 21
Cumulative Test for Chapters 4–6 (page 487)
1. (a) (b)
(c)
(d)
(e)
2. 3.
4. 5.
6. 7.
8. 9. 6.7 10.
11. 12. 1 13.
14–16. Answers will vary. 17.
18. 19. 20. 21.
22. 23.
24.
25.
26.
27.
28.
29. 36.4 square inches 30. 85.2 square inches
31. 32. 33.
34. 35.
36.
37.
38.
39. radians per minute; inches per minute
40. square yards
41. 5 feet 42. 43.
44. 543.9 kilometers per hour 45. 425 foot-pounds32.6�;
d � 4 cos
4t22.6�
Area � 63.67
� 8312.6� 395.8
3�cos 9
5 i sin
9
5 �
3�cos 7
5 i sin
7
5 �3�cos i sin �
3�cos 3
5 i sin
3
5 �
3�cos
5 i sin
5�
cos 4
3 i sin
4
3� �
12
�32
i
cos 2
3 i sin
2
3� �
12
32
i
cos 0 i sin 0 � 1
�123 12i
22�cos 3
4 i sin
3
4 ��1
13�1, 5�;
2113
�5, �1�
�5�22
, 22 �3i 5j
A � 26.38�, B � 62.72�, C � 90.90�
B � 60�, a � 5.77, c � 11.55
B � 52.48�, C � 97.52�, a � 5.04
B � 26.39�, C � 123.61�, c � 15.0
2 cos 6x cos 2x
52�sin
5
2� sin �5
5,
255
43
1663
3
2
6,
5
6,
7
6,
11
6
3,
2,
3
2,
5
3
2 tan �1 � 4x2
34
x
6
5
4
3
2
π
y
−1
−2
−3
a � �3, b � , c � 0
π2π−πx
y
−1
−2
−3
−4
3
4
x
3
y
−1
−2
−3
π32
π2
x1 2 3 4 5 6 7 8
1
2
3
4
6
y
−1
−2
35134.6�
cot��120�� �3
3tan��120�� � 3
sec��120�� � �2cos��120�� � �1
2
csc��120�� � �23
3sin��120�� � �
3
2
60�
�2
3
240�
x
−120°
y
Precalculus with Limits, Answers to Cumulative Test for Chapter 4–6 22
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333202CB06_AN.qxd 4/13/06 5:41 PM Page 22
Precalculus with Limits, Answers to Problem Solving 23
Problem Solving (page 493)
1. 2.01 feet 2. yards
3. (a)
(b) Station A: 27.45 miles; Station B: 53.03 miles
(c) 11.03 miles;
4. (a) (b) 50.5 feet
(c) 22 bags
5. (a) (i) (ii) (iii) 1
(iv) 1 (v) 1 (vi) 1
(b) (i) 1 (ii) (iii)
(iv) 1 (v) 1 (vi) 1
(c) (i) (ii) (iii)
(iv) 1 (v) 1 (vi) 1
(d) (i) (ii) (iii)
(iv) 1 (v) 1 (vi) 1
6. (a)
(b) (c) 126.5 miles perhour; The magni-tude gives thevelocity of the skydiver’s fall.
(d)
(e) 123.7 miles per hour
7.
8. and
9. (a) (b)
The amount of work done The amount of work doneby is equal to the amount is times as great asof work done by the amount of work done
by
10. (a)
(b) No. Find the square root of the sum of the squares ofthe vertical and horizontal components.
(c) (i) 150 miles per hour(ii) 150 miles per hour
F1.F2.
3F2F1
P
F2
Q
30°
60°
F1
P
F1
F2
Q
θ1
θ2
� 0 � c�u � v� d�u � w� � u � cv u � dwu � �cv dw�
u � w � 0u � v � 0
w �12�u v�; w �
12�v � u�
−20−60 20 40 60 80 100
80
60
100
120
140
Up
EW
Down
u
s
v
108.43�
−20−60 20 40 60 80 100
20
40
60
80
100
120
140
Up
EW
Down
u s
v
u � �0, �120�, v � �40, 0�
525225
852
1352
1332
52
46 ft
52 ft
65°
S 21.7� E
15°135°30°
60°75°
A B
Lost partyx y
75 mi
S 22.09� E; 1025.88
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d.
0.873 1.745 2.618
99.996 99.985 99.966 v cos �
v sin �
1.5�1.0�0.5��
3.490 4.362 5.234
99.939 99.905 99.863 v cos �
v sin �
3.0�2.5�2.0��
333202CB06_AN.qxd 4/13/06 5:41 PM Page 23
