# How do you convert r^2 = sin 2(theta) into cartesian form?

Apr 21, 2017

The equation is ${\left({x}^{2} + {y}^{2}\right)}^{2} = 2 x y$

#### Explanation:

To convert from polar coordinates $\left(r , \theta\right)$ to cartesian coordinates $\left(x , y\right)$, we apply the following equations

$\sin \theta = \frac{y}{r}$

$\cos \theta = \frac{x}{r}$

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

$S \in \left(2 \theta\right) = 2 \sin \theta \cos \theta$

The equation is

${r}^{2} = \sin \left(2 \theta\right)$

${r}^{2} = 2 \sin \theta \cos \theta$

${r}^{2} = 2 \cdot \frac{y}{r} \cdot \frac{x}{r}$

${r}^{4} = 2 x y$

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