Accedmychevron_right11thchevron_rightmathchevron_rightComplex Numberschevron_rightExercise 1.5

(iii) z1z2z_1\cdot z_2

z1z2=77(cos5π6+isin5π6)z_1z_2=77\left(\cos\frac{5\pi}{6}+i\sin\frac{5\pi}{6}\right) cos5π6=32,sin5π6=12\cos\frac{5\pi}{6}=-\frac{\sqrt{3}}{2},\quad \sin\frac{5\pi}{6}=\frac{1}{2} z1z2=7732+772i\boxed{z_1z_2=-\frac{77\sqrt{3}}{2}+\frac{77}{2}i}