SECTION 9.2 Polar Equations and Graphs 611 Now Work PROBLEM 15 1 Identify and Graph Polar Equations by Converting to Rectangular Equations One method that can be used to graph a polar equation is to convert the equation to rectangular coordinates. In the following discussion, x y , ( ) represents the rectangular coordinates of a point P, and r, θ ( ) represents polar coordinates of the point P. Identifying and Graphing a Polar Equation (Circle) Identify and graph the equation: r 3 = Solution EXAMPLE 1 Figure 23 = + = r x y 3 or 9 2 2 x u 5 0 u 5 p p– 2 3p –– 2 7p –– 4 p– 4 u 5 u 5 u 5 u 5 u 5 u 5 3p –– 4 5p –– 4 1 2 3 4 5 O y Convert the polar equation to a rectangular equation. r r x y 3 9 9 2 2 2 = = + = Square both sides. r x y 2 2 2 = + The graph of r 3 = is a circle, with center at the pole and radius 3, and consists of the points 3, , θ ( ) where θ can be any angle. See Figure 23. Identifying and Graphing a Polar Equation (Line) Identify and graph the equation: 3 θ π = Solution EXAMPLE 2 Convert the polar equation to a rectangular equation. y x y x 3 tan tan 3 3 3 θ π θ π = = = = Find the tangent of both sides. y x tan ; tan 3 3 θ π = = The graph of 3 θ π = is a line passing through the pole making an angle of 3 π with the polar axis. See Figure 24. Figure 24 θ π = = y x 3 or 3 x 1 2 3 4 5 O 3 5 y p– 3 u 5 0 u 5 p u 5 p– 2 u 5 3p –– 2 u 5 7p –– 4 u 5 p– 4 u 5 3p –– 4 u 5 5p –– 4 Now Work PROBLEM 17
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