If we have two complex numbers, in polar form, as the numerator and denominator of a fraction, and we are asked to write them as a single complex number, is there an easier way to deal with them rather than the usual procedure? (By usual procedure I mean first expanding them by writing the value of each term and then realizing the denominator, etc.)
Thank you.
On
$\def\cis{\operatorname{cis}}$ If we have the fraction $$ \frac{r_{1}\cis(\theta_{1})}{r_{2}\cis(\theta_{2})} $$
then in polar form this is the complex number $$ \frac{r_{1}}{r_{2}}\cis(\theta_{1}-\theta_{2}) $$
It follows from the fact that $|\cis(\alpha)|=1$ hence $|r\cis(\alpha)|=r$ and from the trigonometric identities
Do you mean you have $\frac{r_1 e^{i \theta_1}}{r_2 e^{i \theta_2}}$? In that case you can divide them as you might expect to yield $\left(\frac{r_1}{r_2}\right) e^{i ( \theta_1 - \theta_2)}$.