AN70 Answers: Chapter 9 61. π ⋅ e2 , i 3 8 π ⋅ e2 , i 7 8 π ⋅ e2 , i 11 8 π ⋅ e2 , i 15 8 π π ( ) +i 2 cos 3 8 sin 3 8 , π π ( ) +i 2 cos 7 8 sin 7 8 , π π ( ) +i 2 cos 11 8 sin 11 8 , π π ( ) +i 2 cos 15 8 sin 15 8 63. π⋅ e , i 10 π⋅ e , i 2 π ⋅ e , i 9 10 π ⋅ e , i 13 10 π ⋅ e , i 17 10 π π +i cos 10 sin 10 , π π +i cos 2 sin 2 , π π +i cos 9 10 sin 9 10 , π π +i cos 13 10 sin 13 10 , π π +i cos 17 10 sin 17 10 65. − − i i 1, , 1, 1 21 Imaginary axis Real axis i 2i 67. Look at formula ( )7 . = z r k n for all k. 69. Look at formula ( )7 . The zk are spaced apart by an angle of π n 2 . 71. Since the sine and cosine functions each has period π θ θ π θ θ π ( ) ( ) = + = + k k 2 , cos cos 2 and sin sin 2 , k an integer. Then, θ θ θ π θ π ( ) ( ) ( ) ( ] [ = + = + + + = θ θ π ( ) + re r i r k i k re cos sin cos 2 sin 2 , i i k2 k an integer. 73. Assume the theorem is true for ≥ n 1. For = n 0: θ θ [ ] [ ] ( ) ( ) [ ] = = ⋅ + ⋅ = ⋅ + = ⋅ + = θ ( ) ⋅ z r e r i i cos 0 sin 0 1 1 cos0 sin0 1 1 1 0 1 1 True i 0 0 0 0 For negative integers: θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ [ ] ( ) [ ] [ ] [ ] [ ] ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = = = + ≥ = + = + ⋅ − − = − + = − = − = − + − = θ θ ( ) ( ) − − − − − − − − z z r e r n i n n r n i n r n i n n i n n i n n i n r n n n i n r r n i n r n i n r e cos sin with 1 1 cos sin 1 cos sin cos sin cos sin cos sin (cos sin cos sin cos sin cos sin n n n i n n n n n n n n n i n 1 1 1 2 2 Thus, De Moivre’s Theorem is true for all integers. 75. π ( ) + = + x iy i k k ln7 2 , aninteger 77. ≈40.50 78. π 4 3 79. y x y 2 3 3 2 2 80. Minimum: ( ) = − f 6 5 16 5 81. ≈ ° ≈ ° ≈ ° A B C 26.4 , 36.3 , 117.3 82. x y z loga 3 2 5 83. { } 621 84. ( )( ) = − f g x x x 75 20 6 3 85. = − + y x 3 2 8 86. − = − = = x x x x 16 sec 16 16(sec 1 16 tan 4 tan 2 2 2 9.4 Assess Your Understanding (page 647) 1. vector 2. 0 3. unit 4. position 5. horizontal; vertical 6. resultant 7. T 8. F 9. a 10. b 11. v 1 w v w 13. 3v 15. v 2 w v 2w 17. 3v u 22w 3v 1 u 2 2w 19. T 21. F 23. F 25. T 27. = + v i j 3 4 29. = + v i j 2 4 31. = − v i j 8 33. = − + v i j 35. 5 37. 2 39. 13 41. 1 43. j− 45. 89 47. 34 13 − 49. i 51. i j 3 5 4 5 − 53. i j 2 2 2 2 − 55. 14 57. v i j 8 5 5 4 5 5 , = + or v i j 8 5 5 4 5 5 = − − 59. 2 21, 2 21 { } − + − − 61. = + v i j 5 2 5 3 2 63. v i j 7 7 3 = − + 65. v i j 25 3 2 25 2 = − 67. 45° 69. 150° 71. 333.4° 73. 258.7° 75. = + F i j 20 3 20 77. F i j 20 3 30 2 20 30 2 ( ) ( ) = + + − 79. (a) v j v i j 550 ; 50 2 50 2 a w = = + (b) g ( ) = + + v i j 502 550 502 (c) g = ° v 624.7 mph; N6.5 E 81. v i j 250 2 30 250 2 30 3 ; 518.8 km h; N38.6 E ( ) ( ) = − + + ° 83. Approximately 4031 lb 85. 8.6° left of direct heading across the river; 1.52 min 87. (a) N7.05 E° (b) 12 min 89. Tension in right cable: 1000 lb; tension in left cable: 845.2 lb 91. Tension in right part: 1088.4 lb; tension in left part: 1089.1 lb 93. 0.36 μ = 95. 13.68 lb 97. The truck must pull with a force of 4635.2 lb. 99. (a) 1, 4 ( ) − (b) y 5 x 5 u9 v u v (3, 21) (24, 5) (21, 4) 101. P F2 F1 F4 F3 103. About 8.2° north of east. 107. 29 { } 108. x x x 3 2 6 ( )( ) − + − 109. 3 110. Amplitude 3 2 ; period 3 π = = Phase shift 2 π = − y 2.5 x P 3 3 2 P 3 , 2 111. 15 112. x y 10 2 49 2 2 ( ) ( ) − + + = 113. x–intercepts: 3, 2, − − 3; y–intercept: 18 − 114. 5 11, 5 11 { } − + 115. x x x x x 3 9 or 3 9 27 2 3 2 ( ) ( ) + + + + + 116. θ θ θ θ θ θ ( ) ( )( ) ( ) = − = − = − = = f g 25 5 sin 25 25 sin 25 1 sin 25 cos 5 cos 2 2 2 2 Historical Problem (page 657) a b c d ac bd i j i j ( ) ( ) + ⋅ + = + Real part a bi c di a bi c di ac adi bci bdi ac bd real part real part 2 ( ) [ ] ( ) ( )( ) [ ] ⎡ ⎣ + + ⎤⎦ = − + = + − − = +
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