Convert $$$r = 2 \sin{\left(\theta \right)}$$$ to rectangular coordinates

Convert $$$r = 2 \sin{\left(\theta \right)}$$$ to rectangular coordinates.

From $$$x = r \cos{\left(\theta \right)}$$$ and $$$y = r \sin{\left(\theta \right)}$$$, we have that $$$\cos{\left(\theta \right)} = \frac{x}{r}$$$, $$$\sin{\left(\theta \right)} = \frac{y}{r}$$$, $$$\tan{\left(\theta \right)} = \frac{y}{x}$$$, and $$$\cot{\left(\theta \right)} = \frac{x}{y}$$$.

The input becomes $$$r = \frac{2 y}{r}$$$.

Simplify: the input now takes the form $$$r^{2} - 2 y = 0$$$.

In rectangular coordinates, $$$r = \sqrt{x^{2} + y^{2}}$$$ and $$$\theta = \operatorname{atan}{\left(\frac{y}{x} \right)}$$$.

Thus, the input can be rewritten as $$$x^{2} + y^{2} - 2 y = 0$$$.

$$$r = 2 \sin{\left(\theta \right)}$$$A in rectangular coordinates is $$$x^{2} + y^{2} - 2 y = 0$$$A.