First Principles AI

Solution

Given:

z_1=2(\cos 60^\circ+i\sin 60^\circ),\quad z_2=5(\cos 90^\circ+i\sin 90^\circ)

Use:

z_1z_2=r_1r_2\left(\cos(\theta_1+\theta_2)+i\sin(\theta_1+\theta_2)\right)

z_1z_2=(2\cdot 5)\left(\cos(60^\circ+90^\circ)+i\sin(60^\circ+90^\circ)\right) =10(\cos 150^\circ+i\sin 150^\circ)

\cos 150^\circ=-\frac{\sqrt{3}}{2},\quad \sin 150^\circ=\frac{1}{2}

\boxed{z_1z_2=10\left(-\frac{\sqrt{3}}{2}+\frac{1}{2}i\right)=-5\sqrt{3}+5i}