Trigonometry
Trigonometry in this option centres on the right triangle, the sine and cosine laws, and the area formula that uses an angle - the tools the exam expects you to choose correctly from a diagram or a word problem involving elevations, bearings or distances that cannot be measured directly.
Right triangles
Use SOH-CAH-TOA: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent. Inverse ratios recover an angle. Know the exact values for 30°, 45° and 60° (sin 30° = 1/2, cos 30° = √3/2, tan 45° = 1). An angle of elevation is measured upward from the horizontal; an angle of depression, downward.
Oblique triangles
- Sine law: a/sin A = b/sin B = c/sin C. It requires at least one side–angle pair that are opposite each other.
- Cosine law: a² = b² + c² − 2bc·cos A. Use it for SAS (two sides and the included angle) or SSS.
- Area: A = ½·b·c·sin A.
Worked example
In triangle ABC, b = 9 cm, c = 7 cm and angle A = 52°. Find side a and the area. Cosine law (SAS, because the known angle lies between the two known sides): a² = 9² + 7² − 2(9)(7)cos 52° = 130 − 126(0.6157) = 130 − 77.58 = 52.42, so a ≈ 7.2 cm. Area = ½(9)(7)sin 52° = 31.5(0.7880) ≈ 24.8 cm². Reasoning note: the sine law is not used first because no side is paired with its opposite angle; it could be applied afterwards to find angle B.
Traps that cost marks
- Calculator left in radian mode, or inverse-trig answers reported without checking the quadrant.
- Applying the sine law when no side–angle opposite pair exists.
- Using a side that is not included between the two known sides in the cosine law.
- The ambiguous case: sin θ = sin(180° − θ), so an obtuse triangle may be missed.
- Mislabelling side a - it must be opposite angle A.