Section 9.3 AN69 79. 3P 4 U 5 5P 4 U 5 3P 2 U 5 7P 4 U 5 P 2 U 5 P 4 U 5 y x 1 2 3 4 5 U 5 P U 5 0 81. 3P 4 U 5 5P 4 U 5 3P 2 U 5 7P 4 U 5 P 2 U 5 P 4 U 5 y x 1 2 3 4 5 U 5 P U 5 0 83. θ = = r a y a sin 85. θ θ ( ) = = + = + − = + − = r a r ar x y ay x y ay x y a a 2 sin 2 sin 2 2 0 2 2 2 2 2 2 2 2 Circle, radius a, center ( )a 0, in rectangular coordinates 87. θ θ ( ) = = + = − + = − + = r a r ar x y ax x ax y x a y a 2 cos 2 cos 2 2 0 2 2 2 2 2 2 2 2 Circle, radius a, center ( ) a, 0 in rectangular coordinates 89. (a) 5 knots (b) 6 knots (c) 10 knots (d) approximately ° 80 to ° 150 (e) ≈9 knots; approximately ° 90 to ° 100 91. ( ) + = − x y x y 2 2 2 2 2 95. { } ( ] < ≤ x x 3 8 , or 3, 8 96. ° 420 97. π = = Amplitude 2; period 2 5 98. Horizontal asymptote: = y 0; Vertical asymptote: = x 4 99. 3 100. ≈ 6 30 32.86 square units 101. { } 5 4 102. { } − 5 2 , 4 3 103. = − − y x4 5 104. ( ) = = − = − x x x x x x x cos cos cos cos 1 sin cos sin cos 3 2 2 2 Historical Problems (page 634) 1. (a) + + i i 1 4 , 1 (b) − +i 1, 2 9.3 Assess Your Understanding (page 634) 5. real; imaginary 6. magnitude; modulus; argument 7. θ θ ( ) + r r e i 1 2 1 2 8. F 9. three 10. T 11. c 12. a 13. 1 Imaginary axis Real axis 1 π π ( ) + π⋅ i e 2 cos 4 sin 4 ; 2 i 4 15. 1 2 Imaginary axis Real axis 1 21 π π ( ) + π ⋅ i e 2 cos 11 6 sin 11 6 ; 2 i 11 6 17. 3 23 Imaginary axis Real axis 3 23 π π ( ) + π ⋅ i e 3 cos 3 2 sin 3 2 ; 3 i 3 2 19. 2 Imaginary axis Real axis 2 4 22 24 4 π π ( ) + π ⋅ i e 4 2 cos 7 4 sin 7 4 ; 4 2 i 7 4 21. 1 22 Imaginary axis Real axis 1 22 24 3 ( ) + ⋅ i e 5 cos5.356 sin5.356 ; 5 i 5.356 23. 22 Imaginary axis Real axis 3 1 21 2 ( ) + ⋅ i e 13cos2.159 sin2.159; 13 i 2.159 25. − + i 1 3 27. − i 2 2 2 2 29. − i3 31. −7 33. − + i 0.035 0.197 35. + i 1.970 0.347 37. π π π π ( ) ( ) = + = + π π ⋅ ⋅ zw i e z w i e 8 cos 3 sin 3 or 8 ; 1 2 cos 9 sin 9 or 1 2 i i 3 9 39. π π π π ( ) ( ) = + = + π π ⋅ ⋅ zw i e z w i e 12 cos 2 9 sin 2 9 or 12 ; 3 4 cos 11 9 sin 11 9 or 3 4 i i 2 9 11 9 41. π π π π ( ) = + = + π π ⋅ ⋅ zw i e z w i e 4 cos 9 40 sin 9 40 or 4 ; cos 40 sin 40 or i i 9 40 40 43. π π π π ( ) ( ) = + = + π π ⋅ ⋅ zw i e z w i e 4 2 cos 12 sin 12 or 4 2 ; 2 cos 5 12 sin 5 12 or 2 i i 12 5 12 45. π ⋅ e 64 ; i 2 3 π π ( ) +i 64 cos 2 3 sin 2 3 ; − + i 32 32 3 47. π⋅ e 32 ; i 2 π π ( ) +i 32 cos 2 sin 2 ; i 32 49. π⋅ e 27 ; i 3 π π ( ) +i 27 cos 3 sin 3 ; + i 27 2 27 3 2 51. π ⋅ e 25 ; i 3 4 π π ( ) +i 25 cos 3 4 sin 3 4 ; − + i 25 2 2 25 2 2 53. π ⋅ e 4 2 ; i 3 4 π π ( ) +i 4 2 cos 3 4 sin 3 4 ; − + i 4 4 55. ⋅ e 27 ; i 2.590 ( ) +i 27 cos 2.590 sin 2.590 ; − + i 23 14.142 57. π⋅ e2 , i 6 12 π ⋅ e2 , i 6 3 4 π ⋅ e2 , i 6 17 12 π π ( ) +i 2 cos 12 sin 12 , 6 π π ( ) +i 2 cos 3 4 sin 3 4 , 6 π π ( ) +i 2 cos 17 12 sin 17 12 6 59. π ⋅ e8 , i 4 5 12 π ⋅ e8 , i 4 11 12 π ⋅ e8 , i 4 17 12 π ⋅ e8 , i 4 23 12 π π ( ) +i 8 cos 5 12 sin 5 12 , 4 π π ( ) +i 8 cos 11 12 sin 11 12 , 4 π π ( ) +i 8 cos 17 12 sin 17 12 , 4 π π ( ) +i 8 cos 23 12 sin 23 12 4
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