The cosine rule — Mathematics, 14–17 years
How to find a side in a triangle when the triangle is not right-angled. Mathematics, 14–17 years.
A wider triangle relationship
The cosine rule connects two sides, the angle between them, and the opposite side. It says c² = a² + b² − 2ab cos C. When C is 90°, cos C is 0, so the rule becomes Pythagoras; it therefore extends that familiar idea to tilted triangles.
Why Pythagoras was not enough
Pythagoras works only when two sides meet at a right angle. Surveyors, navigators and astronomers often meet triangles with other angles, so they need the third side without forcing the shape to be right-angled. The cosine rule supplies exactly that missing link.
Two paths and their angle
Two paths from a point are 7 km and 10 km long, with an angle of 60° between them. The direct distance c obeys c² = 7² + 10² − 2×7×10×cos 60°. Since cos 60° = 0.5, c² = 49 + 100 − 70 = 79. Thus c = √79 ≈ 8.89 km.
The angle must fit the side
A common mistake is using an angle that is not opposite the side called c, or changing the minus sign to plus. The diagram can make the pairing seem obvious when it is not. In c² = a² + b² − 2ab cos C, C is between a and b, and c faces C.
Mapping without a straight line
A rescue team can know two routes from a shared point and the angle between them, then estimate the distance between the endpoints. The same geometry helps survey land and locate objects from measured directions. It is useful precisely because real layouts are rarely perfect right triangles.
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