# How can you use trigonometric functions to simplify  23 e^( ( pi)/8 i )  into a non-exponential complex number?

Apr 8, 2016

$23 {e}^{\frac{\pi}{8} i} = - 20.08 - 11.2 i$

#### Explanation:

According to Euler's formula,

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

When $\frac{\pi}{8}$ is substituted as $x$, then

${e}^{\frac{\pi}{8} i} = \cos \left(\frac{\pi}{8}\right) + i \sin \left(\frac{\pi}{8}\right)$
$= \cos \left({22.5}^{o}\right) + i \sin \left({22.5}^{o}\right)$
$= - 0.873 - 0.487 i$,

which is the value of ${e}^{\frac{\pi}{8} i}$.

Multiply the whole thing by $23$ to find the entire thing:

$23 \left(- 0.873 - 0.487 i\right) = - 20.08 - 11.2 i$