# How do you evaluate e^( ( pi)/12 i) - e^( ( 13 pi)/8 i) using trigonometric functions?

Mar 13, 2016

$= 0.58 + 0.38 i$

#### Explanation:

Euler's identity is a special case of Euler's formula from complex analysis, which states that for any real number x,
${e}^{i x} = \cos x + i \sin x$
using this formula we have

${e}^{i \frac{\pi}{12}} - {e}^{i 13 \frac{\pi}{12}}$
$= \cos \left(\frac{\pi}{12}\right) + i \sin \left(\frac{\pi}{12}\right) - \cos \left(13 \frac{\pi}{8}\right) - i \sin \left(13 \frac{\pi}{8}\right)$
$= \cos \left(\frac{\pi}{12}\right) + i \sin \left(\frac{\pi}{12}\right) - \cos \left(\pi + 5 \frac{\pi}{8}\right) - i \sin \left(\pi + 5 \frac{\pi}{8}\right)$
$= \cos \left(\frac{\pi}{12}\right) + i \sin \left(\frac{\pi}{12}\right) + \cos \left(5 \frac{\pi}{8}\right) + i \sin \left(5 \frac{\pi}{8}\right)$
$= 0.96 - 0.54 i - 0.38 + 0.92 i = 0.58 + 0.38 i$