# How do you evaluate  e^( ( 7 pi)/4 i) - e^( ( 5 pi)/3 i) using trigonometric functions?

Jan 20, 2016

$0.207 + 0.159 i$

#### Explanation:

Euler's Theorem states that ${e}^{i \theta} = \cos \theta + i \sin \theta$.

Application of this theorem to the given question yields

$\cos \left(\frac{7 \pi}{4}\right) + i \sin \left(\frac{7 \pi}{4}\right) - \left(\cos \left(\frac{5 \pi}{3}\right) + i \sin \left(\frac{5 \pi}{3}\right)\right)$

$= \frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}} i - 0.5 + 0.866 i$

$= 0.207 + 0.159 i$