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

Oct 17, 2016

${e}^{\frac{5 i \pi}{4}} - {e}^{\frac{7 i \pi}{4}} = - \sqrt{2}$

#### Explanation:

Use the definition ${e}^{i \theta} = \cos \theta + i \sin \theta$
${e}^{\frac{5 i \pi}{4}} = \cos \left(\frac{5 \pi}{4}\right) + i \sin \left(\frac{5 \pi}{4}\right)$
${e}^{\frac{7 i \pi}{4}} = \cos \left(\frac{7 \pi}{4}\right) + i \sin \left(\frac{7 \pi}{4}\right)$
$\cos \left(\frac{5 \pi}{4}\right) = - \frac{\sqrt{2}}{2}$
$\cos \left(\frac{7 \pi}{4}\right) = \frac{\sqrt{2}}{2}$
$\sin \left(\frac{7 \pi}{4}\right) = - \frac{\sqrt{2}}{2}$
$\sin \left(\frac{5 \pi}{4}\right) = - \frac{\sqrt{2}}{2}$
The result is
${e}^{\frac{5 i \pi}{4}} - {e}^{\frac{7 i \pi}{4}} = \cos \left(\frac{5 \pi}{4}\right) + i \sin \left(\frac{5 \pi}{4}\right) - \left(\cos \left(\frac{7 \pi}{4}\right) + i \sin \left(\frac{7 \pi}{4}\right)\right)$
${e}^{\frac{5 i \pi}{4}} - {e}^{\frac{7 i \pi}{4}} = - \frac{\sqrt{2}}{2} - i \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2}$
${e}^{\frac{5 i \pi}{4}} - {e}^{\frac{7 i \pi}{4}} = - \sqrt{2}$