Right-angled trigonometry (SOH-CAH-TOA) only works when there is a 90° angle to lean on. Real triangles — a field, a roof truss, a sailing course — often have no right angle at all. The sine rule and cosine rule handle any triangle, and a neat formula lets you find its area without ever measuring the height.
Why this matters: surveyors, navigators and engineers work almost entirely with non-right triangles. On the Extended paper these rules appear every series, often inside a bearings or 3D problem worth 5–6 marks. Choosing the right rule quickly is the whole skill.
1. Which rule? Match what you are given
- Sine rule — use when you have a matching pair (a side and its opposite angle): a/sinA = b/sinB = c/sinC.
- Cosine rule — use when you have two sides and the angle between them (to find the third side), or all three sides (to find an angle): a² = b² + c² − 2bc·cosA.
- Area = ½ab·sinC — use when you have two sides and the included angle C.
| You know… | You want… | Use |
|---|---|---|
| Side + opposite angle, plus one more | A side or an angle | Sine rule |
| Two sides + angle between them | The third side | Cosine rule |
| All three sides | An angle | Cosine rule (rearranged) |
| Two sides + included angle | The area | ½ab·sinC |
2. Worked non-right triangle (cosine rule)
Two sides and the included angle. A triangle has b = 8 cm, c = 5 cm and the angle between them A = 60°. Find side a. Cosine rule: a² = b² + c² − 2bc·cosA = 8² + 5² − 2(8)(5)cos60° = 64 + 25 − 80(0.5) = 89 − 40 = 49. So a = √49 = 7 cm.
Area. Same triangle: area = ½bc·sinA = ½(8)(5)sin60° = 20 × 0.8660 = 17.3 cm² (3 s.f.).
3. Angles of elevation & depression, and a taste of 3D
- The angle of elevation is measured up from the horizontal to an object; the angle of depression is measured down. They are equal (alternate angles) between two points.
- In 3D, the angle between a line and a plane is found by dropping the line's end straight onto the plane and joining the foot to the base point — this makes a right-angled triangle you can solve.
Sine rule worked: In triangle ABC, A = 40°, B = 75°, and side a = 10 cm. Find side b. b/sinB = a/sinA, so b = a·sinB/sinA = 10 × sin75°/sin40° = 10 × 0.9659/0.6428 = 15.0 cm (3 s.f.).
Common exam mistake: using the cosine rule when a sine-rule pair is available (slower and error-prone), or forgetting the calculator is in the wrong angle mode. Always check it reads DEG for these questions.
Exam technique: the command word Calculate demands full working. Write the rule, substitute, then evaluate — method marks are there even if the final rounding slips. Keep unrounded values in your calculator until the very last step.