# How do you evaluate e^( ( 13 pi)/8 i) - e^( ( pi)/2 i) using trigonometric functions?

Apr 11, 2016

${e}^{\frac{13 \pi}{8} i} - {e}^{\frac{\pi}{2} i} = 0.383 - 1.924 i$

#### Explanation:

According to Euler's formula,

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

Using values for $x$ from the equation, $\frac{13 \pi}{8}$ and $\frac{\pi}{2}$

$x = \frac{13 \pi}{8}$
${e}^{\frac{13 \pi}{8} i} = \cos \left(\frac{13 \pi}{8}\right) + i \sin \left(\frac{13 \pi}{8}\right)$
$= \cos 292.5 + i \sin 292.5$
$= 0.383 - 0.924 i$

$x = \frac{\pi}{2}$
${e}^{\frac{\pi}{2} i} = \cos \left(\frac{\pi}{2}\right) + i \sin \left(\frac{\pi}{2}\right)$
$= \cos 90 + i \sin 90$
$= i$

Inserting these two values back into the original question,

${e}^{\frac{13 \pi}{8} i} - {e}^{\frac{\pi}{2} i} = 0.383 - 0.924 i - i$
$= 0.383 - 1.924 i$