# How do you multiply e^(( 7 pi )/ 4 i) * e^( 3 pi/2 i )  in trigonometric form?

Mar 2, 2016

${e}^{7 \frac{\pi}{4} i} = \left(\cos \left(7 \frac{\pi}{4}\right) + i \sin \left(7 \frac{\pi}{4}\right)\right) = \frac{1}{\sqrt{2}} - i \left(\frac{1}{\sqrt{2}}\right)$

${e}^{3 \frac{\pi}{2} i} = \left(\cos \left(3 \frac{\pi}{2}\right) + i \sin \left(3 \frac{\pi}{2}\right)\right) = - i$.

#### Explanation:

Trigonometric form of ${e}^{7 \frac{\pi}{4} i}$ can be written as

$\left(\cos \left(7 \frac{\pi}{4}\right) + i \sin \left(7 \frac{\pi}{4}\right)\right)$

As $\cos \left(7 \frac{\pi}{4}\right) = \cos \left(- \frac{\pi}{4}\right) = \cos \left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}$ and $\sin \left(7 \frac{\pi}{4}\right) = \sin \left(- \frac{\pi}{4}\right) = \sin \left(\frac{\pi}{4}\right) = - \frac{1}{\sqrt{2}}$

${e}^{7 \frac{\pi}{4} i}$ can be written as $\frac{1}{\sqrt{2}} - i \left(\frac{1}{\sqrt{2}}\right)$

and that of ${e}^{3 \frac{\pi}{2} i}$ can be written as

$\left(\cos \left(3 \frac{\pi}{2}\right) + i \sin \left(3 \frac{\pi}{2}\right)\right)$ and as $\cos \left(3 \frac{\pi}{2}\right) = 0$ and $\sin \left(\frac{\pi}{2}\right) = - 1$,

${e}^{3 \frac{\pi}{2} i}$ can be written as $- i$