608 CHAPTER 9 Polar Coordinates; Vectors 5. If ( ) = P x y , is a point on the terminal side of the angle θ that is also on the circle + = x y r , 2 2 2 then θ = tan . (p. 407) 6. ( ) − = − tan 1 1 . (pp. 477–479) 3. To complete the square of + x x6 , 2 add . (p. A29) 4. Draw the angle π5 6 in standard position. (pp. 383–386) 10. Multiple Choice The point π ( ) 5, 6 can also be represented by which polar coordinates? (a) π ( ) − 5, 6 (b) π ( ) −5, 13 6 (c) π ( ) − 5, 5 6 (d) π ( ) −5, 7 6 11. True or False In the polar coordinates θ ( ) r, , r can be negative. 12. True or False The polar coordinates of a point are unique. 7. The origin in rectangular coordinates coincides with the in polar coordinates; the positive x-axis in rectangular coordinates coincides with the in polar coordinates. 8. If P is a point with polar coordinates θ ( ) r, , the rectangular coordinates ( ) x y , of P are given by = x and = y . 9. Multiple Choice In a rectangular coordinate system, where does the point with polar coordinates π ( ) − 1, 2 lie? (a) in quadrant IV (b) on the y-axis (c) in quadrant II (d) on the x-axis Concepts and Vocabulary Skill Building In Problems 13–20, match each point in polar coordinates with either A, B, C, or D on the graph. 13. π ( ) − 2, 11 6 14. π ( ) − − 2, 6 15. π ( ) −2, 6 16. π ( ) 2, 7 6 17. π ( ) 2, 5 6 18. π ( ) −2, 5 6 19. π ( ) −2, 7 6 20. π ( ) 2, 11 6 In Problems 21–34, plot each point given in polar coordinates. 21. π ( ) 3, 2 22. π ( ) 4, 3 2 23. ( ) −2, 0 24. π ( ) −3, 25. π ( ) 6, 6 26. π ( ) 5, 5 3 27. π ( ) −2, 3 4 28. π ( ) −3, 2 3 29. π ( ) − 4, 2 3 30. π ( ) − 2, 5 4 31. π ( ) − − 1, 3 32. π ( ) − − 3, 3 4 33. π ( ) − − 2, 34. π ( ) − − 3, 2 In Problems 35–42, plot each point given in polar coordinates, and find other polar coordinates θ ( ) r, of the point for which: (a) π θ > − ≤ < r 0, 2 0 (b) θ π < ≤ < r 0, 0 2 (c) π θ π > ≤ < r 0, 2 4 35. π ( ) 5, 2 3 36. π ( ) 4, 3 4 37. π ( ) −2, 3 38. π ( ) −3, 4 39. π ( ) 1, 2 40. π ( ) 2, 41. π ( ) − − 3, 4 42. π ( ) − − 2, 2 3 In Problems 43–58, polar coordinates of a point are given. Find the rectangular coordinates of each point. 43. π ( ) 3, 2 44. π ( ) 4, 3 2 45. ( ) −2, 0 46. π ( ) −3, 47. π ( ) 6, 5 6 48. π ( ) 5, 5 3 49. π ( ) −2, 3 4 50. π ( ) −2, 2 3 51. π ( ) − − 5, 6 52. π ( ) − − 6, 4 53. π ( ) − − 2, 54. π ( ) − − 3, 2 55. π ( ) 7.5, 11 18 56. π ( ) −3.1, 91 90 57. ( ) 6.3, 3.8 58. ( ) 8.1, 5.2 In Problems 59–70, the rectangular coordinates of a point are given. Find polar coordinates for each point. 59. ( ) 3, 0 60. ( ) 0, 2 61. ( ) −1, 0 62. ( ) − 0, 2 63. ( ) − 1, 1 64. ( ) −3, 3 65. ( ) 5, 5 3 66. − − ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ 3 2 , 1 2 67. ( ) − 1.3, 2.1 68. ( ) − − 0.8, 2.1 69. ( ) 8.3, 4.2 70. ( ) −2.3, 0.2 In Problems 71–78, the letters x and y represent rectangular coordinates. Write each equation using polar coordinates θ ( ) r, . 71. + = x y 2 2 3 2 2 72. + = x y x 2 2 73. = x y4 2 74. = y x2 2 75. = xy 2 1 76. = x y 4 1 2 77. = x 4 78. = − y 3 p 6 C D B A 2
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