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Sine rule for a non-right triangle

2026-09-10Plane GeometryMr Wong

a/sin A = b/sin B = c/sin C = 2R.

Key formulas

a/sin A = b/sin B = c/sin C

When it fits

Use sine rule when you know a side and its opposite angle, plus one more side or angle.

Example

A=40°, a=7, B=60°. Then 7/sin40° = b/sin60° → b = 7 sin60°/sin40°.

Ambiguous case

SSA can give two triangles. Check whether angle B could also be 180°−B₁.

Worked example

Worked example

Problem. In △ABC, ∠A = 40°, a = 7, ∠B = 60°. Find side b. Leave in exact calculator-ready form.

Solution. a/sin A = b/sin B → 7/sin40° = b/sin60° → b = 7 sin60° / sin40° = 7(√3/2)/sin40°.

2R

The common ratio equals the circumdiameter 2R.

Review quiz

In △ABC, a/sin A equals:

Practice questions

  1. A=30°, a=5, B=45°. Find b.
  2. What extra check does the ambiguous case need?

Study notes

Sine or cosine rule?

Cosine rule when you know two sides and the included angle, or all three sides.

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