# How do you evaluate e^( ( pi)/12 i) - e^( ( 11 pi)/8 i) using trigonometric functions?

Apr 19, 2016

$1.139 + 1.183 i$

#### Explanation:

Euler's formula says that

${e}^{i x} = \cos x + i \sin x$.

Using values from the question of $x = \frac{\pi}{12}$ and $x = \frac{11 \pi}{8}$,

${e}^{\frac{\pi}{12} i} = \cos \left(\frac{\pi}{12}\right) + i \sin \left(\frac{\pi}{12}\right)$
$= \cos 15 + i \sin 15$
$= 0.966 + 0.259 i$

${e}^{\frac{11 \pi}{8} i} = \cos \left(\frac{11 \pi}{8}\right) + i \sin \left(\frac{11 \pi}{8}\right)$
$= \cos 247.5 + i \sin 247.5$
$= - 0.383 - 0.924 i$

Substituting these values into the question,

$\left(0.966 + 0.259 i\right) - \left(- 0.383 - 0.924 i\right)$
$= 0.966 + 0.383 + 0.259 i + 0.924 i$
$= 1.349 + 1.183 i$