Cosine rule
- Olympiad · IOQM
The cosine rule gives the third side of any triangle from the other two sides and the angle between them.
Name the triangle ABC, with side a opposite angle A, b opposite B and c opposite C. Then:
- c² = a² + b² - 2ab cos C.
The rule can also be turned round to find an angle when all three sides are known.
It is Pythagoras' theorem made to work for every triangle. When C = 90°, cos C = 0. The last term drops out and you get c² = a² + b². The term 2ab cos C is the correction for an angle that is not a right angle.
A small example
Two sides of a triangle are 3 cm and 5 cm, and the angle between them is 60°. Find the third side.
- Take a = 3, b = 5 and C = 60°, with cos 60° = 1/2.
- c² = 3² + 5² - 2 × 3 × 5 × 1/2.
- c² = 9 + 25 - 15 = 19.
- So c = √19 cm, which is about 4.36 cm.
A quick check: the answer lies between 5 - 3 = 2 and 5 + 3 = 8, as any third side must. When you know two angles and a side instead, the sine rule is quicker.
Where it fits
This goes beyond the Class 10 board syllabus. HBCSE's syllabus for the Mathematical Olympiad names it under plane geometry. The Mathematics Teachers' Association (India), MTA(I), conducts IOQM, the first stage of the Mathematical Olympiad Programme that HBCSE organises for NBHM.
Sources
Facts last checked against these sources on 30 September 2026.
- Syllabus for Mathematical Olympiad (Plane Geometry) · HBCSE, TIFR
- Mathematical Olympiad 2026-2027: stages of selection · HBCSE, TIFR
- Brochure: Mathematical Olympiads 2026-2027 · HBCSE, TIFR
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