SECTION 9.2 Polar Equations and Graphs 615 Solution To transform the equation to rectangular coordinates, multiply each side by r. r r x y x x x y x x y x y 2 cos 2 2 0 2 1 1 1 1 2 2 2 2 2 2 2 2 2 θ ( ) ( ) = − + = − + + = + + + = + + = This is the standard equation of a circle with center at 1, 0 ( ) − in rectangular coordinates and radius 1. Figure 29(a) shows the graph drawn by hand. Figure 29(b) shows the graph using Desmos. Multiply both sides by r. θ = + = r x y x r ; cos 2 2 2 Complete the square in x. Factor. Figure 29 θ =− r 2cos , or ( ) + + = x y 1 1 2 2 x O 2 4 5 u 5 0 u 5 p 1 3 1 2 3 4 5 y u 5 p– 2 u 5 3p –– 2 u 5 7p –– 4 u 5 p– 4 u 5 3p –– 4 u 5 5p –– 4 (a) (b) Based on Examples 6 and 7 and the Exploration to the left, we are led to the following results. (The proofs are left as exercises. See Problems 85–88.) THEOREM Suppose a is a positive real number. Then Equation Description • r a2 sinθ = Circle: radius a; center at a 0, ( ) in rectangular coordinates • r a2 sinθ = − Circle: radius a; center at a 0, ( ) − in rectangular coordinates • r a2 cosθ = Circle: radius a; center at a, 0 ( ) in rectangular coordinates • r a2 cosθ = − Circle: radius a; center at a, 0 ( ) − in rectangular coordinates Each circle passes through the pole. Now Work PROBLEM 23 The method of converting a polar equation to an identifiable rectangular equation to obtain the graph is not always helpful, nor is it always necessary. Usually, a table is created that lists several points on the graph. By checking for symmetry, it may be possible to reduce the number of points needed to draw the graph. 3 Test Polar Equations for Symmetry In polar coordinates, the points r, θ ( ) and r r , , θ π θ ( ) ( ) − = − − are symmetric with respect to the polar axis (and to the x -axis). See Figure 30(a). The points r, θ ( ) and r r , , π θ θ ( ) ( ) − = − − are symmetric with respect to the line 2 θ π = (the y -axis). See Figure 30(b). The points r, θ ( ) and r r , , θ θ π ( ) ( ) − = + are symmetric with respect to the pole (the origin). See Figure 30(c) on the next page. Exploration Be sure the mode of your graphing utility is set to polar coordinates and radian measure. Using a square screen, graph r r sin , 2sin , 1 2 θ θ = = and r 3sin . 3 θ = Do you see the pattern? Clear the screen and graph r r sin , 2sin , 1 2 θ θ =− =− and r 3sin . 3 θ =− Do you see the pattern? Clear the screen and graph θ θ = = r r cos , 2cos , 1 2 and θ = r 3cos . 3 Do you see the pattern? Clear the screen and graph θ θ =− =− r r cos , 2cos , 1 2 and r 3cos . 3 θ =− Do you see the pattern?
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