### Application of De Moivre's theorem for rational index

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De Moivre's theorem for rational index

If n is the rational number , then the value of or one of the values of $(cos(\theta )+isin\theta )^n$ is $cos(n\theta)+isin(n\theta ).$

In fact ,if n=$\frac{p}{q}$ where p and q are integers , q > 0 and p ,q have no factor in comman. Then$(cos(\theta )+isin\theta )^n$ has q distinct values  , one of which is $cos(n\theta)+isin(n\theta )$

Boundary condtion -  $k=0,1,2,..........................,q-1$