How do you use DeMoivre's theorem to simplify #(3(cos((5pi)/6)+isin((5pi)/6)))^4#?

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Answer 1 · 2016-08-25T12:53:35

#-81/2(1+isqrt3)#.

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

#z=rcostheta+isintheta, z^n=r^n(cos ntheta+isin ntheta)#

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

#=3^4{cos(4(5pi/6))+isin(4(5pi/6))}#.

#=81(cos(10pi/3)+isin(10pi/3))#.

#=81(cos(3pi+pi/3)+isin(3pi+pi/3))#.

#=81(-cos(pi/3)-isin(pi/3))#.

#=-81(1/2+isqrt3/2)#.

#=-81/2(1+isqrt3)#.