Question

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

Answer 1

Use De Moivre’s theorem: $(\cos x+i \sin x)^n=\cos(nx )+i \sin(nx)$

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

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

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