How do I find the quotient of two complex numbers in polar form?

Oct 18, 2014

If complex numbers ${z}_{1}$ and ${z}_{2}$ in polar form are

$\left\{\begin{matrix}{z}_{1} = {r}_{1} \left(\cos {\theta}_{1} + i \sin {\theta}_{1}\right) \\ {z}_{2} = {r}_{2} \left(\cos {\theta}_{2} + i \sin {\theta}_{2}\right)\end{matrix}\right.$,

then we can write in exponential form

$\left\{\begin{matrix}{z}_{1} = {r}_{1} {e}^{i {\theta}_{1}} \\ {z}_{2} = {r}_{2} {e}^{i {\theta}_{2}}\end{matrix}\right.$.

So, the quotient ${z}_{1} / {z}_{2}$ can be written as

${z}_{1} / {z}_{2} = \frac{{r}_{1} {e}^{i {\theta}_{1}}}{{r}_{2} {e}^{i {\theta}_{2}}} = {r}_{1} / {r}_{2} {e}^{i \left({\theta}_{1} - {\theta}_{2}\right)}$

$= {r}_{1} / {r}_{2} \left[\cos \left({\theta}_{1} - {\theta}_{2}\right) + i \sin \left({\theta}_{1} - {\theta}_{2}\right)\right]$

I hope that this was helpful.