A complete guide to solving any triangle — the sine rule, the cosine rule, the ambiguous case, and the ½ab sin C area formula.
SOH-CAH-TOA only works for right-angled triangles. For any other triangle — one with no right angle at all — a different pair of tools is needed: the sine rule and the cosine rule. Together with a third formula for area, these let you find any missing side, angle or area in any triangle, given enough starting information. This guide explains when to use each rule, works through every case in detail, and covers the one tricky exception — the ambiguous case of the sine rule.
SOH-CAH-TOA relies on having a 90° angle to define "opposite" and "adjacent" sides meaningfully. In a triangle with no right angle, these rules break down entirely. The sine rule and cosine rule instead work with a general labelling system that applies to any triangle, right-angled or not.
By convention, a triangle's angles are labelled A, B and C at each vertex, and the side opposite each angle is labelled with the matching lowercase letter: side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C.
The sine rule connects a side and its opposite angle. It is used whenever you know a matching side-angle pair, plus one more side or angle.
In triangle ABC, angle A = 40°, angle B = 65°, and side a = 8 cm. Find side b.
a/sin A = b/sin B, so 8/sin 40° = b/sin 65°. Rearranging: b = 8 × sin 65° / sin 40° = 8 × 0.906 / 0.643 = 11.3 cm (3 s.f.).
When finding an angle, the sine rule is flipped: sin A / a = sin B / b = sin C / c.
In triangle ABC, a = 7 cm, b = 9 cm, and angle B = 72°. Find angle A.
sin A / 7 = sin 72° / 9, so sin A = 7 × sin 72° / 9 = 7 × 0.951 / 9 = 0.740. So A = sin⁻¹(0.740) = 47.7° (3 s.f.).
When using the sine rule to find an angle from two sides and a non-included angle, there can sometimes be two valid answers, because sine is positive in both the first and second quadrants (sin θ = sin(180° − θ)). If a calculator gives an angle θ, the second possible answer is 180° − θ. Both should be checked: the second solution is only valid if adding it to the known angle still leaves less than 180° in total.
From Example 2, the calculator gives A = 47.7°. The second possible value is 180 − 47.7 = 132.3°. Checking: 132.3° + 72° (angle B) = 204.3°, which exceeds 180°, so this second solution is not valid here — A = 47.7° is the only answer.
The cosine rule is used when the sine rule cannot be applied directly — specifically when you know two sides and the included angle (SAS), or all three sides (SSS) with no angle at all yet.
In triangle ABC, b = 6 cm, c = 9 cm, and angle A = 55°. Find side a.
a² = 6² + 9² − 2(6)(9)cos 55° = 36 + 81 − 108 × 0.574 = 117 − 62.0 = 55.0. So a = √55.0 = 7.42 cm (3 s.f.).
When all three sides are known but no angle, the cosine rule is rearranged: cos A = (b² + c² − a²) / (2bc).
A triangle has sides a = 8 cm, b = 10 cm, c = 13 cm. Find angle C (opposite the longest side).
cos C = (a² + b² − c²) / (2ab) = (64 + 100 − 169) / (2 × 8 × 10) = −5 / 160 = −0.03125. So C = cos⁻¹(−0.03125) = 91.8° (3 s.f.). Note the negative value inside cos⁻¹ correctly signals an obtuse angle.
| Information given | Rule to use |
|---|---|
| Two angles and any side (AAS or ASA) | Sine rule |
| Two sides and a non-included angle (SSA) | Sine rule (check the ambiguous case) |
| Two sides and the included angle (SAS) | Cosine rule |
| All three sides, no angle (SSS) | Cosine rule |
A quick way to decide: if the unknown angle sits "between" the two known sides, or if you have three sides and no angle, reach for the cosine rule. Otherwise, if you already have one matching side-angle pair, use the sine rule.
This formula finds a triangle's area using two sides and the angle between them, without needing to know or calculate the triangle's height directly.
A triangle has sides of 7 cm and 10 cm with an included angle of 48°. Find its area.
Area = ½ × 7 × 10 × sin 48° = 35 × 0.743 = 26.0 cm² (3 s.f.).
Exam-style problems often require using one rule to find a missing piece of information, then a second rule (or the area formula) to complete the question.
A triangle has sides b = 5 cm, c = 8 cm, and angle A = 60°. Find side a, then the triangle's area.
a² = 5² + 8² − 2(5)(8)cos 60° = 25 + 64 − 80(0.5) = 89 − 40 = 49, so a = 7 cm.
Area = ½ × 5 × 8 × sin 60° = 20 × 0.866 = 17.3 cm² (3 s.f.).
Common errors include: applying the sine rule when two sides and the included angle are given (the cosine rule is needed instead); forgetting to check the ambiguous case when the sine rule is used to find an angle from an SSA setup; using the wrong pair of sides in the ½ab sin C formula (a and b must be the two sides forming the included angle C); and losing track of which angle is opposite which side when substituting into the cosine rule.
| Formula | Use |
|---|---|
| a/sin A = b/sin B = c/sin C | Sine rule — AAS, ASA, or SSA (check ambiguous case) |
| a² = b² + c² − 2bc cos A | Cosine rule — SAS or SSS |
| Area = ½ab sin C | Area from two sides and the included angle |
Every non-right-angled triangle problem reduces to identifying which of these three formulas matches the information given, then substituting carefully.
This article is original EDUSAMBAM educational writing. It is designed as a broad sine rule, cosine rule and triangle area resource covering the tools needed to solve any triangle. Exact examination requirements can vary between examination boards and syllabuses, so students should also compare their work with the specification and past-paper requirements of their own board.
Recommended study approach: learn the three formulas and the decision table in Section 10, practise identifying which rule a question requires before calculating, and always check for the ambiguous case whenever the sine rule is used to find an angle.
20 questions covering the sine rule, cosine rule, the ambiguous case, and the area formula for any triangle. Answer every question, then submit to see your score instantly.