# How can you use trigonometric functions to simplify  11 e^( ( 4 pi)/3 i )  into a non-exponential complex number?

##### 1 Answer
Mar 28, 2016

${e}^{\frac{4 \pi}{3} i} = - \frac{\sqrt{3}}{2} - \frac{1}{2} i$

#### Explanation:

As ${e}^{i \theta} = \cos \theta + i \sin \theta$, we have

${e}^{\frac{4 \pi}{3} i} = \cos \left(\frac{4 \pi}{3}\right) + i \sin \left(\frac{4 \pi}{3}\right)$

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

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

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