Answer to Question #117766 in Complex Analysis for Asubonteng Isaac Adjei

Question

Use De Moivre’s theorem to simplify the following (a) (cosπ/5+isinπ/5)10, (b) (cosπ/9+isinπ/9)−3, (c) {cos(−π/6) + isin(−π/6)}−4

Answer 1

De Moivre's formula is $(cos(x)+i\,sin(x))^n=(cos(nx)+i\,sin(nx)),\quad n\in{\mathbb{Z}}$

By applying it, we obtain:

(a) $(cos(\frac{\pi}{5})+i\,sin(\frac{\pi}{5}))^{10}=cos(2\pi)+i\,sin(2\pi)=1$

(b) $(cos(\frac{\pi}{9})+i\,sin(\frac{\pi}{9}))^{-3}=cos(-\frac{\pi}{3})+i\,sin(-\frac{\pi}{3})=\frac{1}{2}-i\,\frac{\sqrt{3}}{2}$

(c) $(cos(-\frac{\pi}{6})+i\,sin(-\frac{\pi}{6}))^{-4}=cos(\frac{2\pi}{3})+i\,sin(\frac{2\pi}{3})=-\frac{1}{2}+i\,\frac{\sqrt{3}}{2}$