# How can you use trigonometric functions to simplify  12 e^( ( pi)/3 i )  into a non-exponential complex number?

Jun 29, 2016

$12 {e}^{\frac{\pi}{3} i} = 6 + 6 \sqrt{3} i$

#### Explanation:

A complex number in polar form can be written in two ways -

(a) $r {e}^{i \theta}$ or (b) $r \cos \theta + i r \sin \theta$

Hence $12 {e}^{\frac{\pi}{3} i} = 12 \cos \left(\frac{\pi}{3}\right) + i 12 \sin \left(\frac{\pi}{3}\right)$

= $12 \times \frac{1}{2} + i \times 12 \times \frac{\sqrt{3}}{2}$

= $6 + 6 \sqrt{3} i$