840 CHAPTER 8 Applications of Trigonometry In Figure 57 the pole has been placed at the origin of a rectangular coordinate system so that the polar axis coincides with the positive x-axis. Point P has rectangular coordinates 1x, y2. Point P can also be located by giving the directed angle u from the positive x-axis to ray OP and the directed distance r from the pole to point P. The ordered pair 1r, u2 gives the polar coordinates of point P. If r 70 then point P lies on the terminal side of u, and if r 60 then point P lies on the ray pointing in the opposite direction of the terminal side of u, a distance r from the pole. Figure 58 shows rectangular axes superimposed on a polar coordinate grid. Rectangular and Polar Coordinates If a point has rectangular coordinates 1x, y2 and polar coordinates 1r, u2, then these coordinates are related as follows. x =r cos U y =r sin U r2 =x2 +y2 tan U = y x , if x ≠0 y x 0 1 1 P(2, 308) 308 Figure 59 EXAMPLE 1 Plotting Points with Polar Coordinates Plot each point in the polar coordinate system. Then determine the rectangular coordinates of each point. (a) P12, 30°2 (b) Q a-4, 2p 3 b (c) R a5, - p 4b SOLUTION (a) In the point P12, 30°2, r = 2 and u = 30°, so P is located 2 units from the origin in the positive direction on a ray making a 30° angle with the polar axis, as shown in Figure 59. We find the rectangular coordinates as follows. x = r cos u y = r sin u Conversion equations x = 2 cos 30° y = 2 sin 30° Substitute. x = 2¢ 23 2 ≤ y = 2 a 1 2b Evaluate the functions. x = 23 y = 1 Multiply. The rectangular coordinates are A 23, 1B. y x P(x, y) P(r, U) y x r 0 u r > 0 Figure 57 (r, u) (–r, u) O u x y r > 0 Figure 58
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