Trigonometry
Trigonometry tests right-angled triangle ratios, the sine and cosine rules, areas of triangles, bearings and angles of elevation and depression. Diagrams may be absent, so sketch and label the given information.
Rules
- Right triangles: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent (SOHCAHTOA).
- Sine rule: a/sin A = b/sin B = c/sin C. Cosine rule: a² = b² + c² − 2bc cos A.
- Area = ½ab sin C, where C is the included angle.
- The longest side is opposite the largest angle; check whether the ambiguous case of the sine rule is possible.
- Bearings are measured clockwise from north and written with three figures.
Worked example
In triangle ABC, AB = 7 cm, AC = 9 cm and angle A = 52°. Find BC. BC is opposite angle A, so use the cosine rule with b = 9, c = 7: BC² = 9² + 7² − 2×9×7×cos 52° = 81 + 49 − 126×0.6157 = 130 − 77.58 = 52.42. BC = √52.42 ≈ 7.24 cm. The included angle is essential; the sine rule would need a matching side and angle.
Traps
Calculator in radian mode; using the sine rule when the included angle is given; forgetting to square root; rounding the cosine value too early; measuring bearings anticlockwise or giving two figures.