The sine rule

Statement

• a/sin A = b/sin B = c/sin C = 2R, where R is the circumradius of the triangle.
• Use it when you know a matching pair: a side and the angle opposite to it.

Worked examples

1. Given a = 7, A = 40°, B = 60°: b = a·sin B / sin A = 7·sin 60° / sin 40° = 7(0.866)/0.643 ≈ 9.43.
2. Given b = 10, B = 50°, C = 70°: c = 10·sin 70° / sin 50° = 10(0.940)/0.766 ≈ 12.27.
3. Finding a side with the circumradius: a = 2R sin A. If R = 6 and A = 30°, then a = 12 × 0.5 = 6.

The ambiguous case

• When you are given two sides and a non-included angle (SSA), two different triangles may satisfy the data.
• Because sin(180° − A) = sin A, both A and 180° − A are possible; check whether both angles leave a valid third angle.
• Example: a = 6, b = 8, A = 35° → sin B = 8 sin 35°/6 ≈ 0.765 → B ≈ 49.9° or B ≈ 130.1°. Both are valid because 49.9° + 35° < 180° and 130.1° + 35° < 180°.