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Pure · Trigonometry

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The sine and cosine rules

Solving any (non-right-angled) triangle — the sine rule, the cosine rule and the area formula ½ab·sinC — and how to choose the right one from what you're given.

Pure · Trigonometry

The sine and cosine rules

Solving any (non-right-angled) triangle — the sine rule, the cosine rule and the area formula ½ab·sinC — and how to choose the right one from what you're given.

Why it works

SOH-CAH-TOA only works in right-angled triangles. For any triangle, two rules take over. Label each side with the lower-case letter of the angle opposite it: side aa is opposite angle AA, and so on.

Sine rule: asin⁡A=bsin⁡B=csin⁡C.\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}. Each side over the sine of its opposite angle. Use it when your information comes in a side–opposite-angle pair plus one more piece (so you can match a known side to its angle).

Cosine rule: a2=b2+c2−2bccos⁡A.a^2 = b^2 + c^2 - 2bc\cos A. Use it when you have two sides and the angle between them (to find the third side), or all three sides (to find an angle, rearranged as cos⁡A=b2+c2−a22bc\cos A = \frac{b^2 + c^2 - a^2}{2bc}). It's really Pythagoras with a correction: if A=90°A = 90° then cos⁡A=0\cos A = 0 and it collapses to a2=b2+c2a^2 = b^2 + c^2.

Area of a triangle: Area=12absin⁡C\text{Area} = \tfrac{1}{2}ab\sin C — half the product of two sides times the sine of the angle between them.

Choosing the rule. Ask what you've got:
  • two sides + the included angle, or all three sides → cosine rule
  • a complete side–angle pair (and you want another side or angle) → sine rule
One subtlety — the ambiguous case: when you use the sine rule to find an angle, sin⁡θ=k\sin\theta = k has a second answer 180°−θ180° - \theta, so sometimes two different triangles fit the data. Check whether the obtuse option is possible before discarding it. And don't round mid-calculation — carry full accuracy and round only at the end.