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Pure · Trigonometry

Chapter 1 · 3

The idea

Addition formulae

The compound-angle formulae for sin(A±B), cos(A±B) and tan(A±B) — the sign patterns to get right, and using them for exact values of non-standard angles, expanding, proving identities and solving equations.

A full journey — read it, play with it, work it, then earn real exam marks. Everything stays on the timeline below.

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Pure · Trigonometry

Addition formulae

The compound-angle formulae for sin(A±B), cos(A±B) and tan(A±B) — the sign patterns to get right, and using them for exact values of non-standard angles, expanding, proving identities and solving equations.

Why it works

The shortcut that doesn't exist

You know the exact values of sin⁡45∘\sin 45^\circ and sin⁡30∘\sin 30^\circ. So what is sin⁡75∘\sin 75^\circ? The tempting move is to add:

sin⁡75∘=?sin⁡45∘+sin⁡30∘=22+12≈1.21.\sin 75^\circ \stackrel{?}{=} \sin 45^\circ + \sin 30^\circ = \frac{\sqrt2}{2} + \frac12 \approx 1.21.

That answer is impossible — a sine can never exceed 11. The trig functions are not linear: sin⁡(A+B)\sin(A+B) is not sin⁡A+sin⁡B\sin A + \sin B, and this page exists because the true expansion is something better — an exact formula that turns angles you don't know into angles you do.

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The rest of the explanation, plus 3 worked examples you step through move by move.

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