Question

prove and explain by using demoivres theorem
let n be a negative integer and equal to -m ,say , where m is a positive integer .
(cosΘ+i sinΘ) to the power n is equal to (cosΘ+i sinΘ) to the power -m.

Answer 1

De Moivre's formula reads(cosθ+isinθ)n=cos(nθ)+isin(nθ)Of course this identity implies the real part should be also equality. That iscos(nθ)=R{(cosθ+isinθ)n}Hence we havecos(3θ)=R{cos3θ+3icos2θsinθ−3cosθsin2θ−isin3θ}=cos3θ−3cosθsin2θ