# How do you convert r=1+0.5cos2(theta) into cartesian form?

Sep 19, 2016

${x}^{2} / \sqrt{{x}^{2} + {y}^{2}} - {x}^{2} / \left({x}^{2} + {y}^{2}\right) = \frac{1}{2}$

#### Explanation:

The pass equations are

$\left(\begin{matrix}x = r \cos \theta \\ y = r \sin \theta\end{matrix}\right)$

and also keeping in mind that $\cos 2 \theta = 1 - 2 {\sin}^{2} \theta$

$r = 1 + \frac{1 - 2 {\sin}^{2} \theta}{2} = 1 + \frac{1}{2} - {\sin}^{2} \theta = \frac{1}{2} + {\cos}^{2} \theta$

then

$\frac{x}{\cos} \theta = \frac{1}{2} + {\cos}^{2} \theta$

but $\cos \theta = \frac{x}{\sqrt{{x}^{2} + {y}^{2}}}$

so

${x}^{2} / \sqrt{{x}^{2} + {y}^{2}} - {x}^{2} / \left({x}^{2} + {y}^{2}\right) = \frac{1}{2}$