602 CHAPTER 9 Polar Coordinates; Vectors SUMMARY A point with polar coordinates θ ( ) r, , θ in radians, can also be represented by either of the following: θ π θ π π ( ) ( ) + − + + r k r k k , 2 or , 2 an integer The polar coordinates of the pole are θ ( ) 0, , where θ can be any angle. Finding Other Polar Coordinates of a Given Point Plot the point P with polar coordinates π ( ) 3, 6 , and find other polar coordinates θ ( ) r, of this same point for which: (a) π θ π > ≤ < r 0, 2 4 (b) θ π < ≤ < r 0, 0 2 (c) π θ > − ≤ < r 0, 2 0 EXAMPLE 3 The point π ( ) 3, 6 is plotted in Figure 8. (a) Add 1 revolution π ( ) 2 radians to the angle π 6 to get π π π ( ) ( ) = + = P 3, 6 2 3, 13 6 See Figure 9. (b) Add 1 2 revolution π( ) radians to the angle π 6 , and replace 3 by −3 to get π π π ( ) ( ) = − + = − P 3, 6 3, 7 6 . See Figure 10. (c) Subtract π2 from the angle π 6 to get π π π ( ) ( ) = − = − P 3, 6 2 3, 11 6 . See Figure 11. Figure 8 P 5 3, p–– 6 p–– 6 O ( ) Figure 10 P 5 23, 7p ––– 6 7p ––– 6 O ( ) Figure 11 P 5 3, 2 2 11p ––– 6 11p ––– 6 O ( ) Figure 9 P 5 3, 13p ––– 6 13p ––– 6 O ( ) Solution These examples show a major difference between rectangular coordinates and polar coordinates. A point has exactly one pair of rectangular coordinates; however, a point has infinitely many pairs of polar coordinates. Now Work PROBLEM 35 2 Convert from Polar Coordinates to Rectangular Coordinates Sometimes it is necessary to convert coordinates or equations in rectangular form to polar form, and vice versa.To do this, recall that the origin in rectangular coordinates is the pole in polar coordinates and that the positive x -axis in rectangular coordinates is the polar axis in polar coordinates. THEOREM Conversion from Polar Coordinates to Rectangular Coordinates If P is a point with polar coordinates θ ( ) r, , the rectangular coordinates ( ) x y , of P are given by x r y r cos sin θ θ = = (1)
RkJQdWJsaXNoZXIy NjM5ODQ=