How can you use trigonometric functions to simplify  2 e^( ( pi)/4 i )  into a non-exponential complex number?

Dec 30, 2015

$\sqrt{2} \left(1 + i\right)$

Explanation:

${e}^{i \Theta} = \cos \left(\Theta\right) + i \sin \left(\Theta\right)$

Therefore:
$2 {e}^{\frac{\pi}{4} i} = 2 \left(\cos \left(\frac{\pi}{4}\right) + i \sin \left(\frac{\pi}{4}\right)\right)$

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