Answer to Question #117389 in Complex Analysis for Amoah Henry

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

De Moivre’s Theorem:

$(\cos \theta +i \sin \theta )^n=\cos n\theta +i \sin n\theta, \ \$ for all integers $n$ .

a) $(\cosπ/5+i\sin⁡π /5)^{10}=\cos 2\pi +i \sin 2\pi=1+i \times 0=1$

b) $(\cos⁡π/9+i\sin⁡π/9)^{-3}= \cos ⁡ ( − π / 3 )+i \sin ⁡ ( − π / 3 ) = 1/2+i (-\sqrt 3 /2)$

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

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