How would I convert to polar coordinates?
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For a point represented as (x, y) in Cartesian coordinates and as $\displaystyle (r, \theta)$ in polar coordinates,
$\displaystyle x= r cos(\theta)$
$\displaystyle y= r sin(\theta)$
(I would think that would be one of the first things you would learn about polar coordinates.)
So $\displaystyle x^2+ y^2= (x^2+ y^2- x)^2$ immediately becomes $\displaystyle r^2 cos^2(\theta)+ r^2 sin^2(\theta)= (r^2 cos^2(\theta)+ r^2 sin^2(\theta)- r sin(\theta))^2$.
Of course $\displaystyle x^2+ y^2= r^2 cos^2(\theta)+ r^2 sin^2(\theta)= r^2(cos^2(\theta)+ sin^2(\theta))= r^2$ so this is $\displaystyle r^2= (r^2- r sin(\theta))^2$.
We can factor $\displaystyle r^2$ out of the right side and, as long as r is not 0, cancel on both sides. That leaves $\displaystyle (r- sin(\theta))^2= 1$.