# How can you use trigonometric functions to simplify  18 e^( ( 5 pi)/3 i )  into a non-exponential complex number?

Dec 13, 2016

The answer is $= 9 \left(1 - i \sqrt{3}\right)$

#### Explanation:

We use

${e}^{i \theta} = \cos \theta + i \sin \theta$

$\cos \left(\frac{5}{3} \pi\right) = \frac{1}{2}$

$\sin \left(\frac{5}{3} \pi\right) = - \frac{\sqrt{3}}{2}$

$18 {e}^{\frac{5}{3} i \pi} = 18 \left(\cos \left(\frac{5}{3} \pi\right) + i \sin \left(\frac{5}{3} \pi\right)\right)$

$= 18 \left(\frac{1}{2} - i \frac{\sqrt{3}}{2}\right)$

$= 9 \left(1 - i \sqrt{3}\right)$