Trigonometry
Trigonometry on this syllabus covers right-angled triangles, the sine and cosine rules, the area formula, angles of elevation and depression, and three-figure bearings.
Core rules
- Right triangles: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent.
- Sine rule: a ÷ sin A = b ÷ sin B = c ÷ sin C. Each side is paired with the angle opposite it.
- Cosine rule: a² = b² + c² − 2bc cos A. Use it with two sides and the included angle, or with all three sides to find an angle.
- Area of any triangle = ½ab sin C, where C is the included angle.
- Exact values: sin 30° = ½, cos 60° = ½, tan 45° = 1, sin 45° = cos 45° ≈ 0.707, sin 60° = cos 30° ≈ 0.866.
- Bearings are measured clockwise from north and written with three digits, such as 047°. A back-bearing differs by 180°.
Worked example
In triangle ABC, b = 7 cm, c = 9 cm and angle A = 52°. Find side a and the area. Reasoning: the included angle is given, so use the cosine rule. a² = 7² + 9² − 2(7)(9)cos 52° = 49 + 81 − 126(0.6157) = 130 − 77.58 = 52.42, so a ≈ 7.24 cm. Area = ½ × 7 × 9 × sin 52° = 31.5 × 0.7880 ≈ 24.8 cm². The sine rule would not work directly here, because no side and its opposite angle are both known.
Traps that cost marks
- Leaving the calculator in radian mode.
- Pairing a side with the wrong angle in the sine rule.
- Using the sine rule when the cosine rule is required.
- Taking the smaller angle when the triangle is obtuse, since sin θ = sin(180° − θ).
- Measuring a bearing anticlockwise or writing it with fewer than three digits.