What is the cosine rule?
The cosine rule states c² = a² + b² − 2ab·cos C for any triangle, where C is the angle opposite side c. It links all three sides with one angle.
// maths › Non-Right-Angle Trigonometry
Cosine rule
Use the cosine rule c squared equals a squared plus b squared minus 2ab cos C to find a third side from two sides and the included angle, or any angle from all three sides, in degrees or radians, with a triangle whose angle links back to Pythagoras at ninety degrees.
c² = a² + b² − 2ab·cos C
Angle unit
Frequently asked questions
When should I use the cosine rule instead of the sine rule?
Use the cosine rule when you have two sides and the included angle (SAS) to find the third side, or all three sides (SSS) to find an angle — cases the sine rule cannot start from.
Can you give a worked example?
With a = 5, b = 7, and the included angle C = 60°: c² = 25 + 49 − 2·5·7·cos 60° = 74 − 35 = 39, so c = √39 ≈ 6.24. The same angle in radians is 60° = π/3 ≈ 1.0472 rad; switch the Degrees/Radians toggle and the working updates to match.
How is it related to Pythagoras?
When C = 90° (π/2 ≈ 1.5708 rad), cos C = 0 and the formula reduces to c² = a² + b² — exactly Pythagoras' theorem. The cosine rule is its generalisation to any angle.
Where is this used in real life?
Navigators and surveyors compute distances across awkward terrain, engineers analyse forces and frameworks, and it is used in GPS, robotics arm geometry, and computer graphics.