Trigonometry
Trigonometry
Trigonometry covers right-angled triangles, the sine and cosine rules for triangles without a right angle, the area formula, angles of elevation and depression, and the graphs of the sine, cosine and tangent functions. It links directly to bearings questions, where a triangle is usually hidden inside the diagram.
Right-angled triangles
- SOHCAHTOA: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
- Exact values to know: sin 30° = ½, cos 60° = ½, tan 45° = 1, sin 45° = cos 45° = 1/√2.
Non-right triangles
- Sine rule: a/sin A = b/sin B = c/sin C. Use it with a matching side-and-angle pair.
- Cosine rule: a² = b² + c² − 2bc cos A. Use it with two sides and the included angle, or three sides.
- Area = ½ab sin C, where C is the angle between sides a and b.
Graphs and equations
- y = sin x and y = cos x have period 360° and amplitude 1; y = tan x has period 180° and asymptotes at 90°, 270°, and so on.
Worked example. In triangle ABC, b = 9 cm, c = 7 cm and angle A = 40°. Find side a. a² = b² + c² − 2bc cos A = 81 + 49 − 2(9)(7)cos 40° = 130 − 126(0.7660) = 130 − 96.52 = 33.48. a = √33.48 ≈ 5.79 cm.
Traps that cost marks.
- Leaving the calculator in radian mode.
- Reaching for the sine rule when you are given two sides and the included angle - that needs the cosine rule.
- Using area = ½ab sin C with a side that is not part of the included angle.
- Rounding at an intermediate line and carrying the error into the final answer.