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Geometry & measures · Trigonometry (sine & cosine rules)

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The sine rule

Why a/sin A = b/sin B holds in any triangle, choosing the sine rule by spotting a matching side–angle pair, turning it upside down to find an angle, and the ambiguous case where two triangles fit.

Geometry & measures · Trigonometry (sine & cosine rules)

The sine rule

Why a/sin A = b/sin B holds in any triangle, choosing the sine rule by spotting a matching side–angle pair, turning it upside down to find an angle, and the ambiguous case where two triangles fit.

Why it works

Drop a perpendicular and the rule falls out. In any triangle ABCABC, drop a height hh from CC onto ABAB. In the left triangle h=bsin⁡Ah = b\sin A; in the right one h=asin⁡Bh = a\sin B. The same height, so

bsin⁡A=asin⁡B⟹asin⁡A=bsin⁡B.b\sin A = a\sin B \quad\Longrightarrow\quad \frac{a}{\sin A} = \frac{b}{\sin B}.

Repeating from another vertex extends it to all three:

asin⁡A=bsin⁡B=csin⁡C.\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}.

The lower-case letter is always the side opposite the capital-letter angle — aa faces AA. Getting that pairing wrong is the main source of wrong answers, so label the diagram before substituting.

Choose the sine rule when you can see a complete pair. If the question gives you a side and the angle opposite it, plus one more piece, the sine rule works. If instead you have two sides and the angle between them, or all three sides, no pair is complete and you need the cosine rule.

Turn it upside down for an angle. To find a side, use asin⁡A=bsin⁡B\frac{a}{\sin A} = \frac{b}{\sin B} and multiply. To find an angle, flip both sides first:

sin⁡Aa=sin⁡Bb,\frac{\sin A}{a} = \frac{\sin B}{b},

so the unknown sine ends up on top and one clean multiplication gives sin⁡A\sin A. Then take sin⁡−1\sin^{-1}.

The ambiguous case. Because sin⁡θ=sin⁡(180°−θ)\sin\theta = \sin(180° - \theta), an obtuse angle has the same sine as its acute partner. When you find an angle with the sine rule, there may be two possible triangles: θ\theta and 180°−θ180° - \theta. GCSE questions usually settle it for you — the diagram, or a phrase like "angle BB is obtuse". If a question says "find the obtuse angle", compute the acute one first and subtract from 180°180°.