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MathidiaMathematics Academy
Non-right-angled trigonometry
Lesson 38 of 75
Standard practice

Cosine rule for sides

Objective. Find an opposite side from two sides and an included angle.

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  2. 2. Worked example
  3. 3. Practice
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Learn: the key idea

The cosine rule for sides finds a missing side when two sides and the angle between them (the included angle) are known, using a² = b² + c² − 2bc cos A. Substitute the two known sides and included angle, calculate the cosine term, then take the square root to find the unknown side. For sides b = 7, c = 9 and included angle A = 60°, a² = 49 + 81 − 2(7)(9)cos60° = 130 − 63 = 67, so a ≈ 8.19.

Worked example

In triangle XYZ, side x = 6 cm, side y = 8 cm, and the included angle Z = 55°. Use the cosine rule for sides to find side z.

Diagram of a triangle
  1. 1Write the cosine rule: z² = x² + y² − 2xy cos Z
  2. 2Substitute values: z² = 36 + 64 − 2(6)(8)cos55°
  3. 3Calculate: z² = 100 − 96(0.5736) ≈ 100 − 55.07 = 44.93
  4. 4Take the square root: z ≈ 6.70 cm

Answer: z ≈ 6.70 cm

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Common mistake. Students sometimes use the cosine rule when the angle given is not actually included between the two known sides, which requires the sine rule instead. Always confirm the angle lies between the two sides being substituted before applying this formula.

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