# How do you convert r = sec(θ + π/4) into cartesian form?

Jun 29, 2016

$x - y = \sqrt{2}$

#### Explanation:

A polar coordinate $\left(r , \theta\right)$ in cartesian coordinates is $\left(x , y\right)$ where $x = r \cos \theta$ and $y = r \sin \theta$. It is apparent that ${r}^{2} = {x}^{2} + {y}^{2}$.

Now $r = \sec \left(\theta + \frac{\pi}{4}\right) = \frac{1}{\cos} \left(\theta + \frac{\pi}{4}\right) = \frac{1}{\cos \theta \cos \left(\frac{\pi}{4}\right) - \sin \theta \sin \left(\frac{\pi}{4}\right)} = \frac{\sqrt{2}}{\cos \theta - \sin \theta}$

(as $\sin \left(\frac{\pi}{4}\right) = \cos \left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}$

Hence, $r \cos \theta - r \sin \theta = \sqrt{2}$ or

$x - y = \sqrt{2}$