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Geometry & Measures · Higher

Sine rule

The sine rule relates the sides and angles of any triangle. It is used when you know two angles and one side, or two sides and a non-included angle.

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Key facts to remember

  • 1Sine rule: a/sin A = b/sin B = c/sin C (or its reciprocal form).
  • 2Use to find a side: a = b × sin A / sin B.
  • 3Use to find an angle: sin A = a × sin B / b.
  • 4The ambiguous case occurs when finding an angle — there may be two possible solutions.
  • 5Label sides a, b, c opposite to angles A, B, C respectively.
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Formulas

Sine rule (finding a side)
a / sin A = b / sin B
Sine rule (finding an angle)
sin A / a = sin B / b
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Worked examples

Example 1

In triangle ABC, angle A = 42°, angle B = 68°, and side a = 9 cm. Find side b.

Working

  1. b / sin B = a / sin A
  2. b / sin 68° = 9 / sin 42°
  3. b = 9 × sin 68° / sin 42° = 9 × 0.9272 / 0.6691
  4. b ≈ 12.47 cm
Answerb ≈ 12.47 cm
Example 2

In triangle PQR, PQ = 11 cm, QR = 8 cm, and angle P = 35°. Find angle R.

Working

  1. sin R / PQ = sin P / QR
  2. sin R = 11 × sin 35° / 8 = 11 × 0.5736 / 8 = 0.7887
  3. R = sin⁻¹(0.7887) ≈ 52.0°
AnswerAngle R ≈ 52.0°
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Common mistakes

✗Using the sine rule when the cosine rule is needed (e.g. when two sides and the included angle are given).
✗Pairing the wrong side with the wrong angle — side a must be opposite angle A.
✗Not considering the obtuse angle solution in the ambiguous case.
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Exam tips

✓When finding an angle, always check if the obtuse angle is also a valid solution.
✓Label the triangle with sides a, b, c and angles A, B, C before starting.

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