The ratio of a sum of angles is not the sum of the ratios: \(\sin(60^\circ+30^\circ)=1\), but \(\sin 60^\circ+\sin 30^\circ\approx 1.37\). The compound-angle formulas give the correct expansions.
Compound-angle formulas
| Formula | Sign pattern |
|---|---|
| \(\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B\) | Same sign as inside the bracket. |
| \(\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B\) | Opposite sign to inside the bracket. |
| \(\tan(A\pm B)=\dfrac{\tan A\pm\tan B}{1\mp\tan A\tan B}\) | Same sign on top, opposite sign below. |
Read the upper signs together and the lower signs together: for example \(\cos(A-B)=\cos A\cos B+\sin A\sin B\). These formulas appear in the exam formula list — the skill is applying them with the correct signs.
Exact values from special angles
Angles such as \(15^\circ\), \(75^\circ\) and \(105^\circ\) are sums or differences of the special angles \(30^\circ\), \(45^\circ\) and \(60^\circ\), so their ratios have exact values.
| Split the angle | Working |
|---|---|
| \(105^\circ=60^\circ+45^\circ\) | \(\sin 105^\circ=\sin 60^\circ\cos 45^\circ+\cos 60^\circ\sin 45^\circ\) \(=\dfrac{\sqrt{3}}{2}\cdot\dfrac{1}{\sqrt{2}}+\dfrac{1}{2}\cdot\dfrac{1}{\sqrt{2}}\) \(=\dfrac{\sqrt{6}+\sqrt{2}}{4}\) |
| \(75^\circ=45^\circ+30^\circ\) | \(\cos 75^\circ=\cos 45^\circ\cos 30^\circ-\sin 45^\circ\sin 30^\circ\) \(=\dfrac{1}{\sqrt{2}}\cdot\dfrac{\sqrt{3}}{2}-\dfrac{1}{\sqrt{2}}\cdot\dfrac{1}{2}\) \(=\dfrac{\sqrt{6}-\sqrt{2}}{4}\) |
When a question gives the ratios of \(A\) and \(B\) together with their quadrants, first find the missing ratios with a right-angled triangle in the correct quadrant, then substitute the exact values into the formulas.
Never distribute a ratio over a bracket: \(\sin(A+B)\) is not \(\sin A+\sin B\). Always expand with the compound-angle formulas.
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