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

Feb 9, 2016

${x}^{2} + {y}^{2} = {\left({x}^{2} + {y}^{2} - 2 y\right)}^{2}$

Explanation:

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

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

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

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

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

${r}^{2} = {\left({x}^{2} + {y}^{2} - 2 y\right)}^{2}$

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

${x}^{2} + {y}^{2} = {\left({x}^{2} + {y}^{2} - 2 y\right)}^{2}$