SECTION 9.1 Polar Coordinates 601 NOTE Another way to plot a polar point θ ( ) r, , where < r 0, is to go out a distance r along the terminal side of θ π + . In other words, θ θ π ( ) ( ) = + r r , , , for < r 0. j Figure 4 O P 5 (r, u), r , 0 u )r * Figure 5 23, ( ) 2p –– 3 2p –– 3 O Plotting Points Using Polar Coordinates Plot the points with the following polar coordinates: (a) π ( ) 3, 5 3 (b) π ( ) − 2, 4 (c) ( ) 3, 0 (d) π ( ) −2, 4 Solution EXAMPLE 1 Figure 6 shows the points. Figure 6 3, 5p ––– 3 5p ––– 3 O (a) ( 2, O p–– 4 2 p–– 4 2 (b) ( O (3, 0) (c) O 22, p–– 4 p–– 4 (d) ( ) ) ) Notice that π ( ) −2, 4 identifies the same point as π π ( ) + 2, 4 . Now Work PROBLEMS 13, 25, AND 31 Recall that an angle measured counterclockwise is positive and an angle measured clockwise is negative. This convention has some interesting consequences related to polar coordinates. Figure 7 O (a) P 5 2, p–– 4 p–– 4 ( ) O P 5 2, 9p ––– 4 (b) 9p––– 4 ( ) P 5 2, 7p––– 4 O 2 2 7p ––– 4 (c) ( ) p–– 4 O P 5 22, 5p––– 4 (d) 5p––– 4 ( ) 2 2 2 2 Finding Several Polar Coordinates of a Single Point Consider again the point P with polar coordinates π ( ) 2, 4 , as shown in Figure 7(a). Because π 4 , π9 4 , and π − 7 4 all have the same terminal side, this point P also can be located by using the polar coordinates π ( ) 2, 9 4 or the polar coordinates π ( ) − 2, 7 4 , as shown in Figures 7(b) and (c). The point π ( ) 2, 4 can also be represented by the polar coordinates π ( ) −2, 5 4 . See Figure 7(d). EXAMPLE 2
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