Double angle formulae

The double angle formulae allow us to calculate

\begin{aligned} \sin (2 A) & = 2 \sin (A) \cos (A) \\ \cos (2 A) & = \cos^{2} (A) - \sin^{2} (A) \\ \tan (2 A) & = \frac{2 \tan (A)}{1 - \tan^{2} (A)} \end{aligned}

This is really just an application of the addition formulae that we've already learned, as we'll see in the first example.

A brief note about $\cos (2 A)$ . Because $\sin^{2} A + \cos^{2} A = 1,$ we can get alternative forms of the double angle formula for $\cos$ as so

\begin{aligned} \cos (2 A) & = \cos^{2} (A) - \sin^{2} (A) \\ = 2 \cos^{2} (A) - 1 \\ = 1 - 2 \sin^{2} (A) \end{aligned}

These can be supremely useful (as will be explained in the first example).

Given that $\cos x = \frac{1}{2}$ for acute $x$ , find the value of $\sin (2 x)$ .
Given that $\cos θ = 0.9$ find the value of $\cos 2 θ$ .