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The sine and cosine rules Cambridge IGCSE Mathematics revision

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In plain words

SOH CAH TOA only works when there's a right angle. For any other triangle there are two rules that do the same job. Which one you need depends on what you've been given.

Three things to know

  1. Sine rule: a/sin A = b/sin B = c/sin C. Use it when you have a side and the angle opposite it, plus one other thing.
  2. Cosine rule: a² = b² + c² − 2bc cos A. Use it when you have two sides and the angle between them, or all three sides.
  3. Area of any triangle = ½ab sin C, where C is the angle between sides a and b.

Worked example

A triangle has sides of 5 cm and 7 cm with an angle of 60° between them. Find the third side.

  1. Two sides and the angle between them: use the cosine rule.
  2. a² = 5² + 7² − 2 × 5 × 7 × cos 60° = 25 + 49 − 35 = 39.
  3. a = √39 = 6.24 cm.

Tips and tricks

  • Label the triangle first: side a is opposite angle A, b opposite B, c opposite C.
  • Got a side and its opposite angle as a pair? Sine rule. If not, cosine rule.
5 questions, about 2 minutes.

It lands in your notebook with its questions as flashcards.

The sine and cosine rules: 5 questions and answers

These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.

  1. Which rule do you use when you know two sides and the angle between them, and want the third side?
    • the sine rule
    • the cosine rule (the answer)
    • Pythagoras' theorem
    • SOH CAH TOA

    There is no side-and-opposite-angle pair to start the sine rule.

  2. What is the formula for the area of a triangle using two sides and the angle between them?
    • ab sin C
    • ½ab sin C (the answer)
    • ½ab cos C
    • a² + b²

    C must be the angle between a and b.

  3. In the sine rule, each side is divided by what?
    • the angle next to it
    • the sine of the angle opposite it (the answer)
    • the cosine of the angle opposite it
    • the longest side

    a/sin A = b/sin B.

  4. A triangle has sides 4 cm and 5 cm with an angle of 90° between them. What is its area?
    • 20 cm²
    • 10 cm² (the answer)
    • 9 cm²
    • 4.5 cm²

    ½ × 4 × 5 × sin 90°, and sin 90° = 1.

  5. You know all three sides of a triangle and want an angle. Which rule?
    • the sine rule
    • the cosine rule (the answer)
    • the area formula
    • none

    Rearranged: cos A = (b² + c² − a²) ÷ 2bc.

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