A-43 Answers to Selected Exercises 41. 521cos 0° + i sin 0°2, 21cos 120° + i sin 120°2, 21cos 240° + i sin 240°26 43. 5cos 45° + i sin 45°, cos 135° + i sin 135°, cos 225° + i sin 225°, cos 315° + i sin 315°6 45. 5cos 22.5° + i sin 22.5°, cos 112.5° + i sin 112.5°, cos 202.5° + i sin 202.5°, cos 292.5° + i sin 292.5°6 47. 521cos 20° + i sin 20°2, 21cos 140° + i sin 140°2, 21cos 260° + i sin 260°26 49. 1, - 1 2 + 23 2 i, - 1 2 - 23 2 i 51. (a) yes (b) no (c) yes 53. 1, 0.30901699 + 0.95105652i, -0.809017 + 0.58778525i, -0.809017 - 0.5877853i, 0.30901699 - 0.9510565i 55. 5-1.8174 + 0.5503i, 1.8174 - 0.5503i6 57. 50.8771 + 0.9492i, -0.6317 + 1.1275i, -1.2675 - 0.2524i, -0.1516 - 1.2835i, 1.1738 - 0.5408i6 59. cos 2u + i sin 2u 60. 1cos2 u - sin2 u2 + i12 cos u sin u2 = cos 2u + i sin 2u 61. cos 2u = cos2 u - sin2 u 62. sin 2u = 2 sin u cos u Chapter 8 Quiz [8.4] 1. (a) 8-3, 129 (b) 8-14, 129 (c) 217 (d) 3 (e) 82.23° [8.3] 2. 30 lb [8.5] 3. -1 + 6i x y –4 –1 0 3 –1 + 6i 5 6 –4 + i 3 + 5i [8.5, 8.6] 4. -2 - 2i 5. (a) 41cos 270° + i sin 270°2 (b) 21cos 300° + i sin 300°2 (c) 210 1cos 198.4° + i sin 198.4°2 7. (a) 361cos 130° + i sin 130°2 (b) 223 + 2i (c) - 27 23 2 + 27 2 i [8.6] 8. 21cos 45° + i sin 45°2, 21cos 135° + i sin 135°2, 21cos 225° + i sin 225°2, 21cos 315° + i sin 315°2; 22 + i22, -22 + i22, -22 - i22, 22 - i22 8.7 Exercises 1. 3 2 3. { 22 5. II 7. IV 9. positive x-axis 11. negative y-axis Graphs for Exercises 13(a) –23(a) 908 1808 08 2708 Answers may vary in Exercises 13(b) – 23(b). 13. (b) 11, 405°2, 1-1, 225°2 (c) Q 22 2 , 22 2 R 15. (b) 1-2, 495°2, 12, 315°2 (c) A 22, - 2 B 17. (b) 15, 300°2, 1-5, 120°2 (c) Q5 2 , - 523 2 R 19. (b) 1-3, 150°2, 13, -30°2 (c) Q3 23 2 , - 3 2R 21. (b) A3, 11p 3 B , A -3, 2p 3 B (c) Q 3 2 , - 323 2 R 23. (b) A -2, 7p 3 B , A2, 4p 3 B (c) A -1, - 23 B Graphs for Exercises 25(a) – 35(a) y x 0 1 2 3 –3 –2 –1 2 3 –3–2–1 Answers may vary in Exercises 25(b) – 35(b). 25. (b) A 22, 315°B , A -22, 135°B 27. (b) 13, 90°2, 1-3, 270°2 29. (b) 12, 45°2, 1-2, 225°2 31. (b) A 23, 60°B , A -23, 240°B 33. (b) 13, 0°2, 1-3, 180°2 35. (b) 13, 240°2, 1-3, 60°2 37. r = 4 cos u - sin u x y 0 4 –4 –4 4 x – y = 4 r = 4 cos U – sin U 39. r = 4 or r = -4 x y 0 4 –4 –4 4 x2 + y2 = 16 r = 4 or r = –4 41. r = 5 2 cos u + sin u x y 2x + y = 5 r = 5 2 cos U + sin U 0 5 5 2 47. 1808 08 908 2708 4 r = 2 + 2 cos U cardioid 49. 1808 08 908 2708 6 r = 3(1 + sin U) cardioid 1808 08 908 2708 5 r = 5 sin 3U three-leaved rose 1808 08 908 2708 4 r = 4 cos 2U four-leaved rose 51. 53. 08 908 2708 1808 3 9 r = 8 + 6 cos U limaçon r = 3 + cos U 1808 08 908 2708 4 limaçon 55. 57. 43. C 45. A 6. (a) 2 + 2i23 (b) -3.2139+3.8302i (c) -7i, or 0 - 7i
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