Quick questions on Addition and double angle formulae explained: O-Level A-Maths

Split an awkward angle into a sum or difference of special angles, such as $75^\circ = 45^\circ + 30^\circ$ or $15^\circ = 45^\circ - 30^\circ$, then apply the addition formula with the known exact values.

Read backwards, the double angle formulae let you replace a single-angle expression with a double-angle one, which is the key to many identity proofs and to integrating squared trig functions. Rearranging $\cos 2A = 1 - 2\sin^2 A$ gives $\sin^2 A = \tfrac{1 - \cos 2A}{2}$, and rearranging $\cos 2A = 2\cos^2 A - 1$ gives $\cos^2 A = \tfrac{1 + \cos 2A}{2}$. These "power reduction" forms turn a squared ratio into a first-power expression in the double angle, which is exactly what is needed to simplify $\sin^2\theta$ in an identity or to integrate $\cos^2\theta$ later. Recognising a double angle formula in reverse is a frequently rewarded move in A-Maths proofs.

The addition and double angle formulae chain together to expand a triple angle such as $\sin 3A$. Write $3A = 2A + A$ and apply the addition formula: $\sin 3A = \sin(2A + A) = \sin 2A\cos A + \cos 2A\sin A$. Substituting $\sin 2A = 2\sin A\cos A$ and $\cos 2A = 1 - 2\sin^2 A$, then simplifying with $\cos^2 A = 1 - \sin^2 A$, yields $\sin 3A = 3\sin A - 4\sin^3 A$. The technique of splitting a multiple angle into a double plus a single, then expanding, shows how the basic formulae generate higher-angle identities and is a satisfying way to see them work together.

When recovering $\cos A$, choose the sign from the quadrant of $A$.

Expand $\sin(\theta + 30^\circ)$. [2 marks]

Given $\cos A = \dfrac{3}{5}$ with $A$ acute, find $\cos 2A$. [2 marks]

Write $2\sin 3\theta\cos 3\theta$ as a single trigonometric ratio. [2 marks]