De Moivre Formula

De Moivre’s formula states that for any complex number (and, in particular, for any real number) x and integer n it holds that

((cos x+isin x)^{n}=cos (nx)+isin (nx))

This formula is named after Abraham de Moivre, a French mathematician.

De Moivre’s Theorem for Fractional Power

$large (costheta+isintheta)^{frac{1}{n}}=cos left ( frac{2kpi +theta}{n} right )+isin left ( frac{2kpi +theta }{n} right )for;k=0,1,2,…,n-1$

Solved Examples

Question 1 – Solve $(1+i)^{7}$


$(1+i)^{7}$ = $left [ sqrt{2}left ( frac{sqrt{2}}{2}+ifrac{sqrt{2}}{2} right ) right ]^{7}$

= $left [ sqrt{2} left ( cos frac{pi }{4}+isin frac{pi }{4} right )right ]^{7}$

= $sqrt{2}^{7}left ( cos frac{7pi }{4}+isinfrac{7pi }{4} right )$

= $8sqrt{2}left ( frac{sqrt{2}}{2}-ifrac{sqrt{2}}{2} right )$

= $8-8i$


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