# How can you use trigonometric functions to simplify  19 e^( ( 3 pi)/4 i )  into a non-exponential complex number?

Oct 29, 2017

$- \frac{19 \sqrt[2]{2}}{2} + i \frac{19 \sqrt[2]{2}}{2}$

#### Explanation:

We need to consider how $r {e}^{i \theta} = r \cos \theta + i r \sin \theta$

In this circumstance $\theta$ = $\frac{3 \pi}{4}$

And $r = 19$

Hence $19 {e}^{\frac{3 i \pi}{4}}$ = $19 \left(\cos \left(\frac{3 \pi}{4}\right) + i \sin \left(\frac{3 \pi}{4}\right)\right)$

Hence via evaluating $\cos \left(\frac{3 \pi}{4}\right)$ and $\sin \left(\frac{3 \pi}{4}\right)$ we get;

$19 \left(- \frac{\sqrt[2]{2}}{2} + i \frac{\sqrt[2]{2}}{2}\right)$

Hence yeilding our answer or $- \frac{19 \sqrt[2]{2}}{2} + i \frac{19 \sqrt[2]{2}}{2}$