841 8.7 Polar Equations and Graphs (b) In the point Q A -4, 2p 3 B , r is negative, so Q is 4 units in the opposite direction from the pole on an extension of the 2p 3 ray. See Figure 60. The rectangular coordinates are x = -4 cos 2p 3 = -4 a- 1 2b = 2 and y = -4 sin 2p 3 = -4 ¢ 23 2 ≤ = -223. (c) Point R A5, - p 4 B is shown in Figure 61. Because u is negative, the angle is measured in the clockwise direction. x = 5 cos a- p 4b = 522 2 and y = 5 sin a- p 4b = - 522 2 S Now Try Exercises 13(a), (c), 15(a), (c), and 21(a), (c). y x 1 –1 0 Q(–4, ) 3 2P 2p 3 Figure 60 y x 1 –1 0 R(5, – ) 4 P p 4 – Figure 61 While a given point in the plane can have only one pair of rectangular coordinates, this same point can have an infinite number of pairs of polar coordinates. For example, 12, 30°2 locates the same point as 12, 390°2, 12, -330°2, and 1-2, 210°2. LOOKING AHEAD TO CALCULUS Techniques studied in calculus associated with derivatives and integrals provide methods of finding slopes of tangent lines to polar curves, areas bounded by such curves, and lengths of their arcs. EXAMPLE 2 Giving Alternative Forms for Coordinates of Points Determine the following. (a) Three other pairs of polar coordinates for the point P13, 140°2 (b) Two pairs of polar coordinates for the point with rectangular coordinates 1-1, 12 SOLUTION (a) Three pairs that could be used for the point are 13, -220°2, 1-3, 320°2, and 1-3, -40°2. See Figure 62. P 0 –408 –2208 1408 3208 y x Figure 62 y x 0 (–1, 1) Figure 63 (b) As shown in Figure 63, the point 1-1, 12 lies in the second quadrant. Because tan u = 1 -1 = 1, one possible value for u is 135°. Also, r = 2x2 + y2 = 21-122 + 12 = 22. Two pairs of polar coordinates are A 22, 135°B and A -22, 315°B . S Now Try Exercises 13(b), 15(b), 21(b), and 25.
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