# What is the DeMoivre's theorem used for?

Jul 25, 2018

More of the cases, to find expresions for $\sin n x$ or $\cos n x$ as function of $\sin x$ and $\cos x$ and their powers. See below

#### Explanation:

Moivre's theorem says that ${\left(\cos x + i \sin x\right)}^{n} = \cos n x + i \sin n x$

An example ilustrates this. Imagine that we want to find an expresion for ${\cos}^{3} x$. Then

${\left(\cos x + i \sin x\right)}^{3} = \cos 3 x + i \sin 3 x$ by De Moivre's theorem

By other hand applying binomial Newton's theorem, we have

${\left(\cos x + i \sin x\right)}^{3} = {\cos}^{3} x + 3 i {\cos}^{2} x \sin x + 3 {i}^{2} \cos x {\sin}^{2} x + {i}^{3} {\sin}^{3} x = {\cos}^{3} x - 3 \cos x {\sin}^{2} x + \left(3 {\cos}^{2} x \sin x - {\sin}^{3} x\right) i$

Then, equalizing both expresions as conclusion we have

$\cos 3 x = {\cos}^{3} x - 3 \cos x {\sin}^{2} x$
$\sin 3 x = 3 {\cos}^{2} x \sin x - {\sin}^{3} x$