First Principles AI

Accedmychevron_right11thchevron_rightmathchevron_rightComplex Numberschevron_rightExercise 1.5

(i)

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

\begin{aligned} z_1z_2 &=r_1r_2[\cos(\theta_1+\theta_2)+i\sin(\theta_1+\theta_2)] \end{aligned}

So:

\boxed{\operatorname{Arg}(z_1z_2)=\theta_1+\theta_2=\operatorname{Arg}(z_1)+\operatorname{Arg}(z_2)}