# What is the Cartesian form of ( 7 , (23pi)/3 ) ?

Feb 5, 2016

$\left(\frac{7}{2} , - \frac{7 \sqrt{3}}{2}\right)$

#### Explanation:

To find cartesian form of $\left(r , \theta\right)$ we use the formula

$x = r \cos \left(\theta\right)$ and $y = r \sin \left(\theta\right)$

We are given the $\left(7 , \frac{23 \pi}{3}\right)$

$r = 7$ and $\theta = \frac{23 \pi}{3}$

Let us simplify this $\frac{23 \pi}{3}$ into something which is easier to handle.
$\frac{23 \pi}{3} + \frac{\pi}{3} = \frac{24 \pi}{3}$
$\frac{23 \pi}{3} + \frac{\pi}{3} = 8 \pi$
$\frac{23 \pi}{3} = 8 \pi - \frac{\pi}{3}$

Also note $\cos \left(2 n \pi - \theta\right) = \cos \left(\theta\right)$
and $\sin \left(2 n \pi - \theta\right) = - \sin \left(\theta\right)$

$x = r \cos \left(\theta\right)$
$x = 7 \cos \left(8 \pi - \frac{\pi}{3}\right)$
$x = 7 \cos \left(\frac{\pi}{3}\right)$
$x = 7 \left(\frac{1}{2}\right)$
$x = \frac{7}{2}$

$y = r \sin \left(\theta\right)$
$y = 7 \sin \left(8 \pi - \frac{\pi}{3}\right)$
$y = 7 \left(- \sin \left(\frac{\pi}{3}\right)\right)$
$y = - 7 \left(\frac{\sqrt{3}}{2}\right)$
$y = - \frac{7 \sqrt{3}}{2}$

$\left(\frac{7}{2} , - \frac{7 \sqrt{3}}{2}\right)$