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incl. ambiguous case

04The Sine Rule

Key idea

For any triangle, each side is proportional to the sine of the angle opposite it. Use the sine rule when you have a matching side–angle pair plus one more piece. When you use it to find an angle, always check whether the obtuse alternative also works — the ambiguous case.

When to use it

Use the sine rule when you know an angle and the side opposite it, plus one other side or angle. If you only have two sides and the angle between them, you need the cosine rule instead.

Side up or angle up

To find a side, write the rule with sides on top: a / sin A = b / sin B. To find an angle, flip it: sin A / a = sin B / b.

The ambiguous case

sin θ = sin(180° − θ), so a given sine has two possible angles. When finding an angle, check whether the obtuse option still gives angles summing under 180° — if so, both are valid answers.

a / sin A = b / sin B = c / sin C
sin A / a = sin B / b (for angles)
Common error — Giving only the acute angle when the obtuse angle is also valid. Whenever you use the sine rule to find an angle, test 180° − θ.
Worked examples
Example 1 · Finding a side

In triangle ABC, a = 8 cm, angle A = 42° and angle B = 65°. Find side b.

  1. b / sin B = a / sin A
  2. b / sin 65° = 8 / sin 42°
  3. b = 8 × sin 65° / sin 42°
  4. b = 10.8 cm (3 s.f.)
Example 2 · Finding an angle

In triangle ABC, b = 12 cm, angle B = 80° and a = 9 cm. Find angle A.

  1. sin A / a = sin B / b
  2. sin A = 9 × sin 80° / 12 = 0.7386
  3. A = sin⁻¹(0.7386)
  4. A = 47.6° (3 s.f.)

Exam-style question bank

5 of 5 shown
Foundations[3 marks]

In triangle ABC, a = 7 cm, angle A = 45° and angle B = 60°. Find side b.

Standard[4 marks]

In triangle ABC, a = 15 cm, angle A = 100° and c = 9 cm. Find angle C.

Standard[4 marks]

In triangle ABC, angle A = 50°, angle B = 60° and a = 10 cm. Find side b.

Exam[5 marks]

In triangle ABC, angle B = 72°, angle C = 43° and b = 14 cm. Find side c.

Challenge[5 marks]

In triangle PQR, p = 9 cm, q = 7 cm and angle Q = 38°. Find the two possible values of angle P.