Suppose $\displaystyle z_1=r_1(\cos(\theta_1)+i\sin(\theta_1))$ and $\displaystyle z_2=r_2(\cos(\theta_2)+i\sin(\theta_2))$. If $\displaystyle z_1=z_2$, then how are $\displaystyle r_1 \ \mbox{and} \ r_2$ related? How are $\displaystyle \theta_1 \ \mbox{and} \ \theta_2$ related?

If $\displaystyle z_1=z_2$, then $\displaystyle a_1=a_2 \ \mbox{and} \ b_1=b_2$. Therefore, $\displaystyle r_1=r_2$ and $\displaystyle \theta_1=\theta_2$ since $\displaystyle a^2+b^2=r^2 \ \mbox{and} \ \theta=\tan\frac{b}{a}$.

Is this correct?