Convert $$$y = x^{2}$$$ to polar coordinates

Convert $$$y = x^{2}$$$ to polar coordinates.

In polar coordinates, $$$x = r \cos{\left(\theta \right)}$$$ and $$$y = r \sin{\left(\theta \right)}$$$.

Thus, the input can be rewritten as $$$r \sin{\left(\theta \right)} = r^{2} \cos^{2}{\left(\theta \right)}$$$.

Simplify: the input now takes the form $$$r \left(- r \cos^{2}{\left(\theta \right)} + \sin{\left(\theta \right)}\right) = 0$$$.

Thus, $$$r = \frac{\sin{\left(\theta \right)}}{\cos^{2}{\left(\theta \right)}}$$$.

$$$y = x^{2}$$$A in polar coordinates is $$$r = \frac{\sin{\left(\theta \right)}}{\cos^{2}{\left(\theta \right)}}$$$A.