Cosine Rule (Law of Cosines) Formula & Examples

Cosine Rule (Law of Cosines)

The cosine rule (also called the law of cosines) is a triangle formula used to find a missing side length or angle when you have enough information about the other sides/angles. It works for any triangle (not just right-angled triangles).

📌 Cosine Rule Formula

For a triangle with sides a, b, c opposite angles A, B, C:

c² = a² + b² − 2ab cos(C)

You can also write the other two versions:

  • a² = b² + c² − 2bc cos(A)
  • b² = a² + c² − 2ac cos(B)

When do you use the cosine rule?

  • SAS: You know two sides and the included angle (e.g. a, b, and angle C) and need the third side.
  • SSS: You know all three sides (a, b, c) and need an angle (A, B, or C).

Tip: If the triangle is right-angled and the included angle is 90°, the cosine rule reduces to Pythagoras because cos(90°) = 0.

🧮 Example 1: Find a missing side (SAS)

Given a = 7, b = 10, and angle C = 60°, find side c.

  1. Start with: c² = a² + b² − 2ab cos(C)
  2. Substitute values: c² = 7² + 10² − 2(7)(10)cos(60°)
  3. Compute: c² = 49 + 100 − 140(0.5) = 149 − 70 = 79
  4. Square root: c = √79 ≈ 8.89

🧮 Example 2: Find an angle (SSS)

Given a = 8, b = 11, c = 13, find angle C.

Rearrange the cosine rule to solve for the angle:

cos(C) = (a² + b² − c²) / (2ab)

  1. Substitute: cos(C) = (8² + 11² − 13²) / (2·8·11)
  2. Compute: cos(C) = (64 + 121 − 169) / 176 = 16 / 176 = 0.0909
  3. Angle: C = arccos(0.0909) ≈ 84.8°

Step-by-step method (quick)

  1. Label sides opposite their angles (a ↔ A, b ↔ B, c ↔ C).
  2. Choose the correct form of the cosine rule for the side/angle you need.
  3. Substitute known values carefully.
  4. Calculate, then round sensibly (e.g. 2 decimal places).
  5. Sanity-check: the largest side should be opposite the largest angle.

Common mistakes to avoid

  • Using the wrong angle (the cosine uses the included angle between the two known sides).
  • Mixing degrees and radians in your calculator settings.
  • Forgetting the minus sign: a² + b² − 2ab cos(C).
  • Rounding too early (keep more decimals until the final step).

🔁 Related Topics

  • Sine rule (law of sines)
  • Pythagoras theorem
  • Area of a triangle (½ab sin C)
  • Trigonometry basics (sin, cos, tan)

Disclaimer: This page is provided for general educational reference only. While care has been taken to present accurate formulas and examples, results may vary depending on rounding, calculator settings (degrees vs radians), and input precision. This content does not replace professional educational, engineering, or technical advice.