Trigonometry links the angles of a right-angled triangle to the ratios of its sides. SOH CAH TOA is the mnemonic every student needs: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent.
Labelling a right triangle
Pick the reference angle (the angle you're working with: not the right angle). Then: • Hypotenuse: the side opposite the right angle (always the longest side) • Opposite: the side directly opposite your reference angle • Adjacent: the side next to your reference angle (that is not the hypotenuse)
Finding a missing side
Example: In triangle ABC, angle A = 35°, the hypotenuse = 10 cm. Find the side opposite angle A. Sin 35° = Opposite / 10 Opposite = 10 × Sin 35° Opposite = 10 × 0.5736 = 5.74 cm (to 2 d.p.)
Finding a missing angle
Example: A ladder leans against a wall. The foot is 3 m from the wall and reaches 4 m up. Find the angle the ladder makes with the ground. Tan θ = Opposite / Adjacent = 4 / 3 θ = tan⁻¹(4/3) = 53.1°
Special angles to memorise
Sin 30° = 0.5, Cos 30° = √3/2, Tan 30° = 1/√3 Sin 45° = 1/√2, Cos 45° = 1/√2, Tan 45° = 1 Sin 60° = √3/2, Cos 60° = 0.5, Tan 60° = √3
Key points
- SOH CAH TOA: sin, cos, tan in terms of opposite, adjacent, hypotenuse
- Label sides relative to the angle you're using
- To find an angle: use the inverse (sin⁻¹, cos⁻¹, tan⁻¹)
- 30°, 45°, 60° special angles often appear without a calculator
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WAEC tip
WAEC frequently presents trig problems as real-life scenarios (ladders, flagpoles, angles of elevation and depression). Draw the diagram first: always.
Practise this
A flagpole casts a shadow of 12 m when the sun is at 40° above the horizon. Find the height of the flagpole.
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