SECTION 9.1 Polar Coordinates 607 Solution Multiplying both sides by r makes it easier to use equations (1) and (2). θ θ = = + = r r r x y x 6 cos 6 cos 6 2 2 2 Multiply both sides by r . r x y x r ; cos 2 2 2 θ = + = This is the equation of a circle. Complete the square to obtain the standard form. ( ) ( ) ( ) + = − + = − + + = − + = x y x x x y x x y x y 6 6 0 6 9 9 3 9 2 2 2 2 2 2 2 2 General form Complete the square in x . Factor. This is the standard form of the equation of a circle with center ( ) 3, 0 and radius 3. Now Work PROBLEM 79 Graphing calculators can graph polar equations. Graphing both the polar equation and the rectangular equation is a good way to check your work. See Figures 21(a) and (b). We discuss graphing polar equations in the next section. Now Work PROBLEM 73 r1 5 6 cosu (b) Y1 5 92 (x 2 3) 2 Y2 52 92 (x 2 3) 2 Transforming an Equation from Rectangular to Polar Form Transform the equation = xy 4 9 from rectangular coordinates to polar coordinates. EXAMPLE 9 Solution Use θ = x r cos and θ = y r sin . θ θ θ θ ( )( ) = = = xy r r r 4 9 4 cos sin 9 4 cos sin 9 2 x r y r cos , sin θ θ = = This is the polar form of the equation. It can be simplified as follows: θ θ θ ( ) ( ) = = r r 2 2 sin cos 9 2 sin 2 9 2 2 Factor out r2 . 2 Double-angle Formula 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure 2. The distance between points ( ) = P x y , 1 1 1 and ( ) = P x y , 2 2 2 is ( ) = d P P , 1 2 . (pp. 13–16) 1. Plot the point whose rectangular coordinates are ( ) − 3, 1. What quadrant does the point lie in? (p. 2) ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 9.1 Assess Your Understanding (a) (b) Figure 21
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