SECTION 9.1 Polar Coordinates 603 Proof Suppose that P has the polar coordinates θ ( ) r, . We seek the rectangular coordinates ( ) x y , of P. Refer to Figure 12. • If = r 0, then, regardless of θ, the point P is the pole, for which the rectangular coordinates are ( ) 0, 0 . Formula (1) is valid for = r 0. • If > r 0, the point P is on the terminal side of θ, and ( ) = = + r d O P x y , . 2 2 Because θ θ = = x r y r cos sin this means x r y r cos sin θ θ = = • If < r 0 and θ is in radians, then the point θ ( ) = P r, can be represented as π θ ( ) − + r, , where − > r 0. Because x r y r cos cos sin sin π θ θ π θ θ ( ) ( ) + = − = − + = − = − this means x r y r cos sin θ θ = = ■ Figure 12 P O x x y y r u Converting from Polar Coordinates to Rectangular Coordinates Find the rectangular coordinates of the points with polar coordinates: (a) π ( ) 6, 6 (b) π ( ) − − 4, 4 EXAMPLE 4 Figure 13 x 3 3 6, 6 3 p– 6 p– 6 (a) y ( ) 4 x 22 2 24, 2 2 2 p– 4 p– 4 (b) 2 y ( ) Solution Use equations (1): θ = x r cos and θ = y r sin . (a) Figure 13(a) shows π ( ) 6, 6 plotted. Notice that π ( ) 6, 6 lies in quadrant I of the rectangular coordinate system. So both the x-coordinate and the y-coordinate will be positive. Substituting = r 6 and θ π = 6 gives x r y r cos 6cos 6 6 3 2 3 3 sin 6 sin 6 6 1 2 3 θ π θ π = = = ⋅ = = = = ⋅ = The rectangular coordinates of the point π ( ) 6, 6 are ( ) 3 3, 3 , which lies in quadrant I, as expected. (b) Figure 13(b) shows π ( ) − − 4, 4 plotted. Notice that π ( ) − − 4, 4 lies in quadrant II of the rectangular coordinate system. Substituting = − r 4 and 4 θ π = − gives x r y r cos 4cos 4 4 2 2 2 2 sin 4 sin 4 4 2 2 2 2 θ π θ π ( ) ( ) = = − − = − ⋅ = − = = − − = − − ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ = The rectangular coordinates of the point π ( ) − − 4, 4 are 2 2,2 2, ( ) − which lies in quadrant II, as expected. Most calculators have the capability of converting from polar coordinates to rectangular coordinates. Consult your user’s manual for the proper keystrokes. In most cases this procedure is tedious, so you will probably find using equations (1) faster.
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