How do you use DeMoivre's theorem to simplify (3(cos(5π6)+isin(5π6)))4?

1 Answer
Aug 25, 2016

812(1+i3).

Explanation:

DeMoivre's Theorem states that, for a complex no.

z=rcosθ+isinθ,zn=rn(cosnθ+isinnθ)

Hence, {3(cos(5pi/6)+isin(5pi/6)}^4

=34{cos(4(5π6))+isin(4(5π6))}.

=81(cos(10π3)+isin(10π3)).

=81(cos(3π+π3)+isin(3π+π3)).

=81(cos(π3)isin(π3)).

=81(12+i32).

=812(1+i3).