# How do I convert polar coordinates (10, -pi/4) to rectangular coordinates?

Aug 14, 2014

The answer is $\left(\sqrt{50} , - \sqrt{50}\right)$.

This problem is solved using trigonometry. I like to use the mnemonic SYR, CXR, TYX (pronounced sir kicks'er ticks). $\sin \theta = \frac{y}{r} , \cos \theta = \frac{x}{r} , \tan \theta = \frac{y}{x}$.

So rearranging with algebra, we get $x = r \cos \theta , y = r \sin \theta$.

Next, substitute the polar coordinates. Make sure that your calculator is in RAD mode. However, $\frac{\pi}{4}$ is a special angle that you should memorize the ratios.

$x = 10 \cos \left(- \frac{\pi}{4}\right) , y = 10 \sin \left(- \frac{\pi}{4}\right)$

$- \frac{\pi}{4}$ is in the fourth quadrant, this corresponds with $\left(\sqrt{50} , - \sqrt{50}\right)$.