# Let z1 = √2(cos 7pi/4 + i sin 7pi/4) and z2 = 2(cos 5pi/3 + i sin 5pi/3), how do you find z1/z2?

Apr 13, 2016

${z}_{1} / {z}_{2} = \frac{1}{\sqrt{2}} \left[\cos \left(\frac{\pi}{12}\right) + i \sin \left(\frac{\pi}{12}\right)\right]$

#### Explanation:

A complex number of form $r \cos \theta + i r \sin \theta$ can also be written as $r \cdot {e}^{i \theta}$.

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

and ${z}_{2} = 2 \left(\cos \left(5 \frac{\pi}{3}\right) + i \sin \left(5 \frac{\pi}{2}\right)\right) = 2 {e}^{i 5 \frac{\pi}{3}}$

Hence ${z}_{1} / {z}_{2} = \frac{\sqrt{2}}{2} x {e}^{i \left(7 \frac{\pi}{4} - 5 \frac{\pi}{3}\right)} = \frac{1}{\sqrt{2}} \cdot {e}^{i \left(\frac{21 \pi - 20 \pi}{12}\right)} = \frac{1}{\sqrt{2}} \cdot {e}^{i \frac{\pi}{12}}$

= $\frac{1}{\sqrt{2}} \left[\cos \left(\frac{\pi}{12}\right) + i \sin \left(\frac{\pi}{12}\right)\right]$