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

Mar 25, 2018

The answer is $= 1 + i \sqrt{3}$

#### Explanation:

Apply Euler's Identity

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

Here,

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

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

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

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

Therefore,

$z = 2 \left(\frac{1}{2} + i \frac{\sqrt{3}}{2}\right)$

$= 1 + i \sqrt{3}$