The sine and cosine rules

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What this lesson covers

  • Use the sine rule when you have a side and its opposite angle.
  • Use the cosine rule for two sides and the angle between them.
  • Use the cosine rule for three sides.
Key idea

Which rule

  • Sine rule: a/sin A = b/sin B = c/sin C.
  • Cosine rule: a² = b² + c² − 2bc cos A.
  • To find an angle from three sides: cos A = (b² + c² − a²) ÷ 2bc.
Worked example

Sine rule

  • A = 40°, B = 60°, a = 10. Find b.
  1. b/sin 60° = 10/sin 40°
  2. b = 10 × 0.8660 ÷ 0.6428
  3. b = 13.47
Worked example

Cosine rule for a side

  • b = 7, c = 9, A = 50°. Find a.
  1. a² = 49 + 81 − 126 cos 50°
  2. a² = 130 − 80.99 = 49.01
  3. a = 7.00
Worked example

Cosine rule for an angle

  • The sides are 5, 6 and 7. Find the largest angle, opposite the 7.
  1. cos C = (25 + 36 − 49) ÷ 60
  2. cos C = 0.2
  3. C = 78.5°

Common mistakes

  • Using a calculator set to radians.
    Check it shows degrees before you start.
  • Calculating (b² + c² − 2bc) × cos A.
    Only the last term carries cos A.

Your turn

  • 1. A = 70°, B = 45°, a = 12. Find b.
  • 2. Two sides are 8 and 10 with 60° between them. Find the third side.
Show answer

1. 9.03 2. √84 = 9.17

Summary

What to take away

  • A matching side and angle: sine rule.
  • Otherwise: cosine rule.

More in AS 91259

The sine and cosine rulesNotesArea of any triangleNotesBearings and 3D problemsNotesRadians, arcs and sectorsNotes

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