# How do you perform the operation in trigonometric form (cos((5pi)/3)+isin((5pi)/3))/(cospi+isinpi)?

Dec 4, 2016

$\frac{\cos \left(\frac{5 \pi}{3}\right) + i \sin \left(\frac{5 \pi}{3}\right)}{\cos \pi + i \sin \pi} = \cos \left(\frac{2 \pi}{3}\right) + i \sin \left(\frac{2 \pi}{3}\right)$

= $- \frac{1}{2} + i \frac{\sqrt{3}}{2}$

#### Explanation:

A complex number in polar form such as $\left(r \cos \theta + i r \sin \theta\right)$ can be written in exponential form as

$r {e}^{i \theta}$

As such $\cos \left(\frac{5 \pi}{3}\right) + i \sin \left(\frac{5 \pi}{3}\right) = {e}^{\frac{5 \pi}{3} i}$

and $\cos \pi + i \sin \pi = {e}^{\pi i}$, and hence

$\frac{\cos \left(\frac{5 \pi}{3}\right) + i \sin \left(\frac{5 \pi}{3}\right)}{\cos \pi + i \sin \pi} = {e}^{\frac{5 \pi}{3} i} / {e}^{\pi i}$

= ${e}^{\frac{5 \pi}{3} i - \pi i} = {e}^{\frac{2 \pi}{3} i}$

= $\cos \left(\frac{2 \pi}{3}\right) + i \sin \left(\frac{2 \pi}{3}\right)$

= $- \frac{1}{2} + i \frac{\sqrt{3}}{2}$