# How can you use trigonometric functions to simplify  32 e^( ( 3 pi)/4 i )  into a non-exponential complex number?

Mar 3, 2016

$- 16 \sqrt{2} + i 16 \sqrt{2}$

#### Explanation:

Trigonometric form of $32 {e}^{\frac{3 \pi}{4} i}$ can be written as

$32 \times \left(\cos \left(\frac{3 \pi}{4}\right) + i \sin \left(\frac{3 \pi}{4}\right)\right)$

But as $\cos \left(\frac{3 \pi}{4}\right) = - \frac{1}{\sqrt{2}}$ and $\sin \left(\frac{3 \pi}{4}\right) = \frac{1}{\sqrt{2}}$, this can be simplified to

$32 \times \left(- \frac{1}{\sqrt{2}} + i \times \frac{1}{\sqrt{2}}\right)$ or

= $- 16 \sqrt{2} + i 16 \sqrt{2}$