How do you use DeMoivre's theorem to simplify #(5(cos(pi/9)+isin(pi/9)))^3#?

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Answer 1 · 2016-08-06T08:56:08

#=125(1/2+ (sqrt(3))/2i)#

Could also write as #125e^((ipi)/3)# using Euler's formula if you so desired.

De Moivre's theorem states that for complex number

#z = r(costheta + isintheta)#

#z^n = r^n(cosntheta + isinntheta)#

So here,

#z = 5(cos(pi/9) + isin(pi/9))#

#z^3 = 5^3(cos(pi/3) + isin(pi/3))#

#=125(1/2+ (sqrt(3))/2i)#