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

\(\begin{array}{l}(\cos x+i\sin x)^{n}=\cos (nx)+i\sin (nx)\end{array} \)

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

De Moivre’s Theorem for Fractional Power

\(\begin{array}{l}\large (\cos\theta+i\sin\theta)^{\frac{1}{n}}=\cos \left ( \frac{2k\pi +\theta}{n} \right )+i\sin \left ( \frac{2k\pi +\theta }{n} \right )for\;k=0,1,2,…,n-1\end{array} \)

Solved Examples

Question 1 – Solve
\(\begin{array}{l}(1+i)^{7}\end{array} \)

Solution:

\(\begin{array}{l}(1+i)^{7}\end{array} \)
=
\(\begin{array}{l}\left [ \sqrt{2}\left ( \frac{\sqrt{2}}{2}+i\frac{\sqrt{2}}{2} \right ) \right ]^{7}\end{array} \)

=

\(\begin{array}{l}\left [ \sqrt{2} \left ( \cos \frac{\pi }{4}+i\sin \frac{\pi }{4} \right )\right ]^{7}\end{array} \)

=

\(\begin{array}{l}\sqrt{2}^{7}\left ( \cos \frac{7\pi }{4}+i\sin\frac{7\pi }{4} \right )\end{array} \)

\(\begin{array}{l}8\sqrt{2}\left ( \frac{\sqrt{2}}{2}-i\frac{\sqrt{2}}{2} \right )\end{array} \)

\(\begin{array}{l}8-8i\end{array} \)

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