Accedmychevron_right11thchevron_rightmathchevron_rightComplex Numberschevron_rightExercise 1.5

(iii) z1z2z_1\cdot z_2

z1z2=45(cos19π12+isin19π12)z_1z_2=45\left(\cos\frac{19\pi}{12}+i\sin\frac{19\pi}{12}\right) cos19π12=624,sin19π12=6+24\cos\frac{19\pi}{12}=\frac{\sqrt{6}-\sqrt{2}}{4},\quad \sin\frac{19\pi}{12}=-\frac{\sqrt{6}+\sqrt{2}}{4} z1z2=45(62)445(6+2)4i\boxed{z_1z_2=\frac{45(\sqrt{6}-\sqrt{2})}{4}-\frac{45(\sqrt{6}+\sqrt{2})}{4}i}