Question

What is the formula of $\cos 3\theta$?

Answer 1

We know that,

$\cos \left(A+B\right)=\cos A\cos B-\sin A\sin B$

Find the formula of $\cos 3\theta$

$\cos 3\theta =\cos \left(2\theta +\theta \right)$

$\Rightarrow \cos 3\theta =\cos 2\theta \cos \theta -\sin 2\theta \sin \theta$ $\left[\because \cos \left(A+B\right)=\cos A\cos B-\sin A\sin B\right]$

$\Rightarrow \cos 3\theta =\left(2{\cos }^2\theta -1\right)\cos \theta -2\sin \theta \cos \theta \sin \theta$ $\left[\because \sin 2\theta =2\sin \theta \cos \theta \mathrm{and}\cos 2\mathrm{\theta }=2{\cos }^2\mathrm{\theta }-1\right]$

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

$\Rightarrow \cos 3\theta =2{\cos }^3\theta -\cos \theta -2\left(1-{\cos }^2\theta \right)\cos \theta$ $\left[\because {\sin }^2\theta =1-{\cos }^2\theta \right]$

$\Rightarrow \cos 3\theta =2{\cos }^3\theta -\cos \theta -2\cos \theta +2{\cos }^3\theta$

$\Rightarrow \cos 3\theta =4{\cos }^3\theta -3\cos \theta$

Hence, the formula of $\cos 3\theta$ is $4{\cos }^3\theta -3\cos \theta$.