# How can you use trigonometric functions to simplify  2 e^( ( 3 pi)/8 i )  into a non-exponential complex number?

##### 1 Answer
Mar 10, 2018

$2 {e}^{\frac{3 \pi}{8} i} = \sqrt{2 - \sqrt{2}} + i \sqrt{2 + \sqrt{2}}$

#### Explanation:

Using trigonometric functions $r {e}^{i \theta}$ can be written as

$r \left(\cos \theta + i \sin \theta\right)$ or $r \cos \theta + i r \sin \theta$

Hence $2 {e}^{\frac{3 \pi}{8} i} = 2 \left(\cos \left(\frac{3 \pi}{8}\right) + i \sin \left(\frac{3 \pi}{8}\right)\right)$

= $2 \left(\frac{\sqrt{2 - \sqrt{2}}}{2} + \left(\frac{\sqrt{2 + \sqrt{2}}}{2}\right) i\right)$

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

See details here for $\cos {67.5}^{\circ}$ and $\sin {67.5}^{\circ}$