Compound Angles

Video lesson · from 34:463 worked examples

One lesson video covers all of Trigonometry, so it opens at 34:46 for this topic — not from the beginning.

Theory

Compound angles are formed by adding or subtracting angles, for example $45^\circ + 30^\circ = 75^\circ$ or $45^\circ - 30^\circ = 15^\circ$ .

Note that $\sin(A + B) \neq \sin A + \sin B$ . You can verify this by checking that $\sin(45^\circ + 30^\circ)$ and $\sin 45^\circ + \sin 30^\circ$ give different answers.

Addition Formulae

\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B

\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B

Notice how the signs work: sine keeps the same sign ( $\pm \to \pm$ ) while cosine flips the sign ( $\pm \to \mp$ ).

⚠️ Common Examiner Traps

The signs behave differently: $\cos(A+B) = \cos A\cos B - \sin A\sin B$ — the sign flips for cosine but matches for sine. Mixing these up is the most common compound-angle error.
Do not distribute: $\sin(A+B)$ is not $\sin A + \sin B$ . The formula exists precisely because that shortcut is false.
Choose angles that give exact values: for $75^\circ$ use $45^\circ + 30^\circ$ — both have exact values you know.
Keep it exact: in a non-calculator question the answer should be a surd, not a decimal.

Worked examples

Example 1

Expand and simplify $\cos(x + 60^\circ)$ .

\begin{aligned} \cos(x + 60^\circ) &= \cos x \cos 60^\circ - \sin x \sin 60^\circ \\ &= \cos x \cdot \left(\frac{1}{2}\right) - \sin x \cdot \left(\frac{\sqrt{3}}{2}\right) \\ &= \frac{1}{2}\cos x - \frac{\sqrt{3}}{2}\sin x \end{aligned}

Example 2

Show that $\sin(a + b) = \sin a \cos b + \cos a \sin b$ for $a = \frac{\pi}{6}$ and $b = \frac{\pi}{3}$ .

LHS (Left Hand Side):

\begin{aligned} \sin\left(\frac{\pi}{6} + \frac{\pi}{3}\right) &= \sin\left(\frac{\pi}{6} + \frac{2\pi}{6}\right) \\ &= \sin\left(\frac{3\pi}{6}\right) \\ &= \sin\left(\frac{\pi}{2}\right) \\ &= 1 \end{aligned}

RHS (Right Hand Side):

\begin{aligned} &\quad \sin\left(\frac{\pi}{6}\right) \cos\left(\frac{\pi}{3}\right) + \cos\left(\frac{\pi}{6}\right) \sin\left(\frac{\pi}{3}\right) \\ &= \left(\frac{1}{2}\right)\left(\frac{1}{2}\right) + \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) \\ &= \frac{1}{4} + \frac{3}{4} \\ &= \frac{4}{4} \\ &= 1 \end{aligned}

LHS = RHS. As required.

Example 3

Find the exact value of $\sin 75^\circ$ .

We can write $75^\circ$ as $45^\circ + 30^\circ$ to use exact values we know.

\begin{aligned} \sin 75^\circ &= \sin(45^\circ + 30^\circ) \\ &= \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ \\ &= \left(\frac{1}{\sqrt{2}}\right)\left(\frac{\sqrt{3}}{2}\right) + \left(\frac{1}{\sqrt{2}}\right)\left(\frac{1}{2}\right) \\ &= \frac{\sqrt{3}}{2\sqrt{2}} + \frac{1}{2\sqrt{2}} \\ &= \frac{\sqrt{3} + 1}{2\sqrt{2}} \end{aligned}