# Question #81d1e

May 4, 2017

By deMoivre:

$\textcolor{red}{\left(\cos 3 \theta + i \sin 3 \theta\right)} = {\left(\cos \theta + i \sin \theta\right)}^{3}$

And the RHS has binomial expansion:

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

Collecting the real parts of the equivalent red expressions:

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

$= {\cos}^{3} \theta - 3 \cos \theta \left(1 - {\cos}^{2} \theta\right)$

$\implies \cos 3 \theta = 4 {\cos}^{3} \theta - 3 \cos \theta$ and $A = 4 , B = - 3$