849 8.7 Polar Equations and Graphs Graph each polar equation. In Exercises 47–62, also identify the type of polar graph. See Examples 4–6. 47. r = 2 + 2 cos u 48. r = 4 - 4 cos u 49. r = 311 + sinu2 50. r = 511 - cos u2 51. r = 4 cos 2u 52. r = 3 cos 5u 53. r = 5sin3u 54. r = 2cos 5u 55. r = 3 + cos u 56. r = 2 - cos u 57. r = 8 + 6 cos u 58. r = 6 - 3 cos u 59. r2 = 4 cos 2u 60. r2 = 16cos 2u 61. r2 = 9sin2u 62. r2 = 4 sin2u 63. r = 2 sinu tanu (This is a cissoid.) 64. r = cos 2u cos u (This is a cissoid with a loop.) Graph each spiral of Archimedes. See Example 7. 65. r = u (Use both positive and nonpositive values.) 66. r = -4u (Use a graphing calculator in a window of 3-30, 304 by 3-30, 304, in radian mode, and u in 3-12p, 12p4.) For each equation, write an equivalent equation in rectangular coordinates, and graph. See Example 8. 67. r = 2 sinu 68. r = 2 cos u 69. r = 2 1 + sinu 70. r = 3 1 - sinu 71. r = -2 cos u - 2 sinu 72. r = 3 4 cos u - sinu 73. r = 2 sec u 74. r = -5 csc u 75. r = 2 cos u + sinu 76. r = 2 2 cos u + sinu Solve each problem. 77. Find the polar equation of the line that passes through the points 11, 0°2 and 12, 90°2. 78. Explain how to plot a point 1r, u2 in polar coordinates, if r 60 and u is in degrees. Concept Check The polar graphs in this section exhibit symmetry. Visualize an xy-plane superimposed on the polar coordinate system, with the pole at the origin and the polar axis on the positive x-axis. Then a polar graph may be symmetric with respect to the x-axis (the polar axis), the y-axis (the line u = p 2), or the origin (the pole). 79. Complete the missing ordered pairs in the graphs below. (a) (b) (c) y x (r, u) –u ( , ) u ( , ) ( , ) y x (r, u) –u u p – u ( , ) ( , ) y x (r, u) p + u u
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