Apply the addition formulae for sine, cosine and tangent and the double angle formulae to expand, simplify and evaluate trigonometric expressions

What this dot point is asking

SEAB wants you to use the addition formulae, which expand $\sin(A \pm B)$, $\cos(A \pm B)$ and $\tan(A \pm B)$, and the double angle formulae that follow by setting $B = A$. With these you can find exact values of unusual angles, expand combined-angle expressions, and prepare equations for solving.

The answer

The addition formulae

For any two angles:

Note the sign reversal in the cosine formula: the right side takes the opposite sign to the left.

The double angle formulae

Setting $B = A$ gives:

The three forms of $\cos 2A$ are interchangeable; choose the one that matches what you know.

Using them for exact values

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.

Choosing the right cosine form

If a problem gives $\sin A$, use $\cos 2A = 1 - 2\sin^2 A$; if it gives $\cos A$, use $\cos 2A = 2\cos^2 A - 1$. Matching the form to the data avoids extra work.

Using double angle formulae in reverse for proofs

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.

Combining the formulae to reach a triple angle

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.

Examples in context

Example 1. Combining oscillations. Adding two waves of the same frequency but different phase, $\sin(\omega t + \phi)$, expands with the addition formula into a single sine and cosine, the first step in seeing the combined wave, which connects to the R-formula.

Example 2. Halving an angle in optics. A reflection angle that is twice another appears in mirror and lens geometry; the double angle formula relates the two, letting a single measured angle determine the other.

Try this

Q1. Expand $\sin(\theta + 30^\circ)$. [2 marks]

• Cue. $\sin\theta\cos 30^\circ + \cos\theta\sin 30^\circ = \dfrac{\sqrt{3}}{2}\sin\theta + \dfrac{1}{2}\cos\theta$.

Q2. Given $\cos A = \dfrac{3}{5}$ with $A$ acute, find $\cos 2A$. [2 marks]

• Cue. $\cos 2A = 2\cos^2 A - 1 = 2\cdot\tfrac{9}{25} - 1 = -\dfrac{7}{25}$.

Q3. Write $2\sin 3\theta\cos 3\theta$ as a single trigonometric ratio. [2 marks]

• Cue. This is $\sin 2(3\theta) = \sin 6\theta$.