# How can you use trigonometric functions to simplify  4 e^( ( 5 pi)/4 i )  into a non-exponential complex number?

Jun 14, 2016

The value of this complex number is $- 2 \sqrt{2} - 2 \sqrt{2} i$

#### Explanation:

To express a complex number without exponents you use the following formula:

$| z | \cdot {e}^{\varphi i} = | z | \left(\cos \varphi + i \sin \varphi\right)$

In the example above you have:

$| z | = 4$

$\varphi = \frac{5 \pi}{4}$

So you have:

$4 {e}^{\frac{5 \pi}{4} \cdot i} = 4 \cdot \left(\cos \left(\frac{5 \pi}{4}\right) + i \cdot \sin \left(\frac{5 \pi}{4}\right)\right) =$

$4 \cdot \left(\cos \left(\pi + \frac{\pi}{4}\right) + i \cdot \sin \left(\pi + \frac{\pi}{4}\right)\right) =$
$4 \cdot \left(- \cos \left(\frac{\pi}{4}\right) - i \sin \left(\frac{\pi}{4}\right)\right) =$
$4 \cdot \left(- \frac{\sqrt{2}}{2} - i \cdot \frac{\sqrt{2}}{2}\right) = - 2 \sqrt{2} - 2 \sqrt{2} \cdot i$