How do you convert $r=1+2sin θ$ into a rectangular equations?

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Answer 1 · 2016-02-09T05:04:54

$x^2 + y^2 = (x^2 + y^2 - 2y)^2$

First, multiply both sides by $r$ .

$r^2 = r + 2 r sin theta$

Use the following identities

$r^2 = x^2 + y^2$
$y = r sin theta$

Substitute the rectangle forms in

$x^2 + y^2 = r + 2y$

Subtract $2y$ on both sides to make $r$ the subject

$r = x^2 + y^2 - 2y$

Square both sides to get the $r^2$ term

$r^2 = (x^2 + y^2 - 2y)^2$

Use the above identity again to remove the $r^2$

$x^2 + y^2 = (x^2 + y^2 - 2y)^2$

How do you convert r=1+2sin θr=1+2sin θ into a rectangular equations?