First Principles AI

Accedmychevron_right11thchevron_rightmathchevron_rightComplex Numberschevron_rightExercise 1.5

(ii)

Let $z_1=r_1(\cos\theta_1+i\sin\theta_1)$ and $z_2=r_2(\cos\theta_2+i\sin\theta_2)$ .

\frac{z_1}{z_2}=\frac{r_1}{r_2}[\cos(\theta_1-\theta_2)+i\sin(\theta_1-\theta_2)]

So:

\boxed{\operatorname{Arg}\left(\frac{z_1}{z_2}\right)=\theta_1-\theta_2=\operatorname{Arg}(z_1)-\operatorname{Arg}(z_2)}