# How can you use trigonometric functions to simplify  26 e^( ( 3 pi)/2 i )  into a non-exponential complex number?

Jan 3, 2016

$26 {e}^{\frac{3 \pi}{2} i} = - 26 i$

#### Explanation:

Use Euler's formula:
${e}^{i \phi} = \cos \phi + i \sin \phi$
where $\phi$ is a number, which can be interpreted as angle in radians.

In this question $\phi = \frac{3 \pi}{2}$:
$26 {e}^{\frac{3 \pi}{2} i} = 26 \cos \left(\frac{3 \pi}{2}\right) + 26 i \sin \left(\frac{3 \pi}{2}\right)$

Now substract period $2 \pi$ from the arguements.
$26 \cos \left(- \frac{\pi}{2}\right) + 26 i \sin \left(- \frac{\pi}{2}\right)$
$= 26 \cos \left(\frac{\pi}{2}\right) - 26 i \sin \left(\frac{\pi}{2}\right) = 26 \cdot 0 - 26 i \cdot 1 = - 26 i$