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MathidiaMathematics Academy
Non-right-angled trigonometry
Lesson 39 of 75
Challenge

Cosine rule for angles

Objective. Rearrange the cosine rule to determine an angle.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

The cosine rule for angles rearranges a² = b² + c² − 2bc cos A to find an unknown angle when all three sides of a triangle are known, giving cos A = (b² + c² − a²)/(2bc). Substitute all three side lengths, calculate the right-hand side, then take the inverse cosine to find the angle. For sides a = 7, b = 8, c = 9, cos A = (64 + 81 − 49)/(2×8×9) = 96/144 = 0.667, giving A ≈ 48.2°.

Worked example

A triangle has sides 9 cm, 10 cm and 11 cm. Use the cosine rule for angles to find the angle opposite the 11 cm side.

Diagram of a triangle
  1. 1Let a = 11 (opposite the required angle), b = 9, c = 10
  2. 2Write the formula: cos A = (b² + c² − a²)/(2bc)
  3. 3Substitute: cos A = (81 + 100 − 121)/(2×9×10) = 60/180 = 1/3
  4. 4Find A: A = cos⁻¹(1/3) ≈ 70.5°

Answer: ≈ 70.5°

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Common mistake. A common mistake is substituting the sides into the formula with the wrong side labelled as a, since a must be the side opposite the angle being found. Always label the side opposite the required angle as a before substituting.

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