 # 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:

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

=

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

=

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

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

$$\begin{array}{l}8-8i\end{array}$$