# What is the Cartesian form of (12,(5pi )/3)?

Jan 20, 2018

$\left(6 , - 6 \sqrt{3}\right)$

#### Explanation:

Remember that polar coordinates are of the form $\left(r , \theta\right)$.

Also, remember that our $x$-values correspond with cosine and $y$-values with sine.

Then, remember that our sine and cosine values come from the unit circle, where $r = 1$. So, when changing our coordinates from polar to cartesian coordinates we are taking $\left(r , \theta\right) \to \left(r \cos \left(\theta\right) , r \sin \left(\theta\right)\right)$.

Notice that $\frac{5 \pi}{3} = 2 \pi - \frac{\pi}{3}$. So, we can say that $\theta = - \frac{\pi}{3}$, which is in the fourth quadrant. This means that cosine is positive, and sine is negative.

Then we can essentially say that $\left(12 , \frac{5 \pi}{3}\right) \to \left(12 \cos \left(\frac{\pi}{3}\right) , - 12 \sin \left(\frac{\pi}{3}\right)\right)$
$= \left(12 \left(\frac{1}{2}\right) , - 12 \left(\frac{\sqrt{3}}{2}\right)\right) = \left(6 , - 6 \sqrt{3}\right)$