What is important to know about Sine rule?

Sine rule: a/sin A = b/sin B = c/sin C (or its reciprocal form).

What is important to know about Sine rule?

Use to find a side: a = b × sin A / sin B.

What is important to know about Sine rule?

Use to find an angle: sin A = a × sin B / b.

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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.

Revising sine rule the night before an exam? These are condensed revision notes — the facts, formulas and mistakes examiners actually mark, with no long explanations. Prefer a full walkthrough with step-by-step reasoning instead? Search “Sine rule GCSE” on the site for the complete guide.

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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

b / sin B = a / sin A
b / sin 68° = 9 / sin 42°
b = 9 × sin 68° / sin 42° = 9 × 0.9272 / 0.6691
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

sin R / PQ = sin P / QR
sin R = 11 × sin 35° / 8 = 11 × 0.5736 / 8 = 0.7887
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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