Rewriting a Polar Equation in Cartesian Form

Rewrite the polar equation $r = \sin (2 θ)$ in Cartesian form.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
$\begin{array}{l} r = \sin (2 θ) & Use the double angle identity for sine . \\ r = 2 \sin θ \cos θ \begin{matrix} \end{matrix} & Use \cos θ = \frac{x}{r} and \sin θ = \frac{y}{r} . \\ r = 2 (\frac{x}{r}) (\frac{y}{r}) & Simplify . \\ r = \frac{2 x y}{r^{2}} & Multiply both sides by r^{2} . \\ r^{3} = 2 x y \\ {(\sqrt{x^{2} + y^{2}})}^{3} = 2 x y & As x^{2} + y^{2} = r^{2}, r = \sqrt{x^{2} + y^{2}} . \end{array}$
This equation can also be written as

${(x^{2} + y^{2})}^{\frac{3}{2}} = 2 x y or x^{2} + y^{2} = {(2 x y)}^{\frac{2}{3}}$

How did it go?