Unformatted text preview: lar) coordinate system.
To do so, we identify the pole and polar axis in the polar system to the origin and positive x-axis,
respectively, in the rectangular system. We get the following result.
Theorem 11.7. Conversion Between Rectangular and Polar Coordinates: Suppose P is
represented in rectangular coordinates as (x, y ) and in polar coordinates as (r, θ). Then
• x = r cos(θ) and y = r sin(θ)
(provided x = 0)
In the case r > 0, Theorem 11.7 is an immediate consequence of Theorem 10.3 along with the
quotient identity tan(θ) = cos(θ) . If r < 0, then we know an alternate representation for (r, θ)
is (−r, θ + π ). Since cos(θ + π ) = − cos(θ) and sin(θ + π ) = − sin(θ), applying the theorem to
(−r, θ + π ) gives x = (−r) cos(θ + π ) = (−r)(− cos(θ)) = r cos(θ) and y = (−r) sin(θ + π ) =
(−r)(− sin(θ)) = r sin(θ). Moreover, x2 + y 2 = (−r)2 = r2 , and x = tan(θ + π ) = tan(θ), so the
theorem is true in this case, too. The remaining case is r = 0, in wh...
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