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Cosine Rule: Explained with Examples, Concepts and Examples

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The cosine rule is a formula that connects the three sides of a triangle with one of its angles: c² = a² + b² − 2ab cos C. The square of one side equals the sum of the squares of the other two sides, minus twice their product times the cosine of the angle between them.

This rule matters because Pythagoras' theorem only works for right-angled triangles, while the cosine rule works for every triangle. Reach for it when you know two sides and the angle between them (SAS), or all three sides and want an angle (SSS).

Key Concepts

  • The formula: c² = a² + b² − 2ab cos C. Side c is opposite angle C, and sides a and b are the ones forming angle C.
  • Standard naming: capitals A, B, C are the angles; small letters a, b, c are the opposite sides.
  • SAS case: two sides with the angle between them known — the rule gives the third side directly.
  • SSS case: all three sides known — rearrange to cos C = (a² + b² − c²) / (2ab) and use inverse cosine to find the angle.
  • Pythagoras as a special case: if C = 90°, cos 90° = 0, and the rule collapses to c² = a² + b².

Formulas

  • Side form: c² = a² + b² − 2ab cos C; likewise a² = b² + c² − 2bc cos A and b² = a² + c² − 2ac cos B. All sides share one length unit (cm, m, km); angles are in degrees.
  • Angle form: cos C = (a² + b² − c²) / (2ab), used when all three sides are known.
  • Derivation idea: place vertex C at the origin and side a along the x-axis. Vertex A is then at (b cos C, b sin C) and vertex B at (a, 0), and the distance AB gives c² = (a − b cos C)² + (b sin C)² = a² + b² − 2ab cos C.

Worked Example

A triangle has sides a = 7 cm and b = 5 cm with the included angle C = 60°. Find side c.

  1. Write the rule: c² = a² + b² − 2ab cos C.
  2. Substitute: c² = 7² + 5² − 2 × 7 × 5 × cos 60°.
  3. cos 60° = 0.5, so c² = 49 + 25 − 70 × 0.5 = 74 − 35 = 39.
  4. c = √39 ≈ 6.24 cm.

Common Misconceptions

  • Wrong data type: the cosine rule does not handle SSA (two sides and a non-included angle) — that ambiguous case needs the sine rule. Use cosine for SAS or SSS only.
  • Wrong angle: the angle in the formula must sit between the two known sides. A known angle elsewhere in the triangle does not fit this formula.
  • Right triangles excluded: some students think the rule is only for non-right triangles. It works for right triangles too — it just simplifies into Pythagoras.

Real-World Applications

  • Navigation: finding the distance between two ships from their distances to a lighthouse and the angle between them.
  • Surveying: measuring distances across land that cannot be walked directly, using two measured sides and an angle.
  • Physics: computing the resultant of two forces acting at an angle — the cosine rule hides inside the parallelogram law of forces.

The cosine rule is the universal version of Pythagoras: c² = a² + b² − 2ab cos C. Use the side form for SAS and the rearranged angle form for SSS, always putting the angle between the two known sides.

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