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

[\LARGE (\cos x+i\sin x)^{n}=\cos (nx)+i\sin (nx)]

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

De Moivre’s Theorem for Fractional Power

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

Solved Examples

Question 1 – Solve $(1+i)^{7}$

Solution:

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

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

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

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

= $8-8i$

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