SECTION 9.1 Polar Coordinates 605 CAUTION Because of the restrictions on the inverse tangent function, it is important to plot the point obtained in order to make sure your answer lies in the quadrant found in Step 1. j How to Convert from Rectangular Coordinates to Polar Coordinates where the Point Lies in a Quadrant Find the polar coordinates of a point whose rectangular coordinates are ( ) − 2, 2. Step-by-Step Solution Step 1 Plot the point x, y ( ) and note the quadrant the point lies in or the coordinate axis the point lies on. Plot the point ( ) − 2, 2 in a rectangular coordinate system. See Figure 18. The point lies in quadrant IV. Figure 18 21 1 21 22 2 (x, y) 5 (2, 22) u x y r EXAMPLE 6 COMMENT Many calculators have the capability of converting from rectangular coordinates to polar coordinates. Consult your user’s manual for the proper keystrokes. j ( ) = + = + − = = r x y 2 2 8 2 2 2 2 2 2 Step 2 Find the distance r from the origin to the point. Step 3 Determine θ. Find θ by recalling that θ = y x tan , so θ π θ π = − < < − y x tan , 2 2 . 1 Because ( ) − 2, 2 lies in quadrant IV, this means that π θ − < < 2 0. As a result, θ π ( ) ( ) = = − = − = − − − − y x tan tan 2 2 tan 1 4 1 1 1 Polar coordinates for the point ( ) − 2, 2 are π ( ) − 2 2, 4 . Other possible representations include π ( ) 2 2, 7 4 and π ( ) −2 2, 3 4 . Figure 19 (x, y) 5 (21, 2 3) u x r y Converting from Rectangular Coordinates to Polar Coordinates Find polar coordinates of a point whose rectangular coordinates are ( ) − − 1, 3 . EXAMPLE 7 Solution Step 1 See Figure 19. The point lies in quadrant III. Step 2 The distance r from the origin to the point ( ) − − 1, 3 is ( ) ( ) = − + − = = r 1 3 4 2 2 2 Step 3 To find θ, use α π π α π = = − − = = − < < − − − y x tan tan 3 1 tan 3 3 , 2 2 1 1 1 Since the point ( ) − − 1, 3 lies in quadrant III and the inverse tangent function gives an angle in quadrant I, add π to the result to obtain an angle in quadrant III. Then θ π α π π π π = + = + = + = − tan 3 3 4 3 1 Polar coordinates for this point are π ( ) 2, 4 3 . Other possible representations include π ( ) −2, 3 and π ( ) − 2, 2 3 .
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