606 CHAPTER 9 Polar Coordinates; Vectors Figure 20 r 5 x 2 1 y 2 u 5 tan 21 (a) x y (x, y) r 5 x 2 1 y 2 u 5 p 1 tan 21 (b) x y (x, y) r 5 x 2 1 y 2 u 5 p 1 tan 21 (c) x y (x, y) r 5 x 2 1 y 2 u 5 tan 21 (d) (x, y) x y u r u r u r u r y x y x y x y x To use equations (2) effectively, follow these steps: THEOREM Converting from Rectangular Coordinates to Polar Coordinates If P is a point with rectangular coordinates ( ) x y , , the polar coordinates θ ( ) r, of P are given by θ θ π = + = ≠ = = = r x y y x x r y x • tan if 0 • 2 if 0 2 2 2 (2) Steps for Converting from Rectangular to Polar Coordinates Step 1 Plot the point ( ) x y , , as shown in Examples 5, 6, and 7. Step 2 If = x 0 or = y 0, use the graph to find r . If ≠ x 0 and ≠ y 0, then = + r x y . 2 2 Step 3 Find θ. If = x 0 or = y 0, use the graph to find θ. If ≠ x 0 and ≠ y 0, note the quadrant in which the point lies. θ θ π = = + − − y x y x Quadrant I or IV: tan Quadrant II or III: tan 1 1 Figure 20 shows how to find polar coordinates of a point that lies in a quadrant when its rectangular coordinates ( ) x y , are given. Now Work PROBLEM 63 4 Transform Equations between Polar and Rectangular Forms Equations (1) and (2) can be used to transform equations from polar form to rectangular form, and vice versa. It is not always obvious how to convert from polar form to rectangular form.Two common strategies for transforming an equation from polar form to rectangular form are to • Multiply both sides of the equation by r • Square both sides of the equation Transforming an Equation from Polar to Rectangular Form Transform the equation θ = r 6 cos from polar coordinates to rectangular coordinates, and identify the graph. EXAMPLE 8
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