# What is the Cartesian form of (24,(5pi)/6))?

May 14, 2017

Answer: $\left(- 12 \sqrt{3} , 12\right)$

#### Explanation:

Consider the following formulas to convert from polar to Cartesian:
$x = r \cos \left(\theta\right)$
$y = r \sin \left(\theta\right)$

Since we are given the polar coordinate in $\left(r , \theta\right)$ form, we can simply substitute into the above formulas:
$x = 24 \cos \left(\frac{5 \pi}{6}\right)$
$y = 24 \sin \left(\frac{5 \pi}{6}\right)$

Now, we can simplify each individually:
$x = 24 \cos \left(\frac{5 \pi}{6}\right)$
$= 24 \left(- \frac{\sqrt{3}}{2}\right)$ using our special unit circle trig values
$= - 12 \sqrt{3}$

$y = 24 \sin \left(\frac{5 \pi}{6}\right)$
$= 24 \left(\frac{1}{2}\right)$ using our special unit circle trig values
$= 12$

Therefore, we have the point $\left(- 12 \sqrt{3} , 12\right)$ in Cartesian form.