First Principles AI

Solution

Given:

z_1=6(\cos 150^\circ+i\sin 150^\circ),\quad z_2=3(\cos 30^\circ+i\sin 30^\circ)

Use:

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

\frac{z_1}{z_2}=\frac{6}{3}\left(\cos(150^\circ-30^\circ)+i\sin(150^\circ-30^\circ)\right) =2(\cos 120^\circ+i\sin 120^\circ)

\cos 120^\circ=-\frac{1}{2},\quad \sin 120^\circ=\frac{\sqrt{3}}{2}

\boxed{\frac{z_1}{z_2}=2\left(-\frac{1}{2}+\frac{\sqrt{3}}{2}i\right)=-1+\sqrt{3}i}