Answer to Question #117387 in Complex Analysis for Amoah Henry

Question

Simplify the following expressions: (a) (cos π/4 + i sin π/4)(cos 3π/4 + i sin 3π/4), (b)
√ 3, (b) w = −2 1 √
(cos π/4 + i sin π/4)2 (cos π/6 + i sin π/6)

Answer 1

a) $(cos (π/4) + i sin (π/4))(cos (3π/4) + i sin (3π/4))=$

$=(cos (π/4) + i sin (π/4))(-cos (π/4) + i sin (π/4))=$

$=-sin^2(\pi/4)-cos^2(\pi/4)=-1$

b) $(cos (π/4) + i sin (π/4))^2 (cos (π/6) + i sin (π/6))=$

$=(cos (2π/4) + i sin (2π/4)) (cos (π/6) + i sin (π/6))=$

$= i (cos (π/6) + i sin (π/6))=-sin (π/6)+icos (π/6)=$

$=-1/2+i\sqrt{3}/2$