# How do you evaluate  e^( ( pi)/4 i) - e^( ( 4 pi)/3 i) using trigonometric functions?

Nov 14, 2016

$\sin \frac{\pi}{4} + i \cos \frac{\pi}{4}$
- $\sin \frac{4 \pi}{3} + i \cos \frac{4 \pi}{3}$

#### Explanation:

In complex terms, $R \left(\sin \theta + i \cos \theta\right)$= $R {e}^{i \theta}$
where $R$ is the modulus and $\theta$ is the argument.

So
${e}^{\frac{\pi}{4} i} - {e}^{\frac{4 \pi}{3} i}$ = $1 \left(\sin \frac{\pi}{4} + i \cos \frac{\pi}{4}\right)$
- $1 \left(\sin \frac{4 \pi}{3} + i \cos \frac{4 \pi}{3}\right)$

= $\sin \frac{\pi}{4} + i \cos \frac{\pi}{4}$
- $\sin \frac{4 \pi}{3} + i \cos \frac{4 \pi}{3}$