Trigonometry — An Examiner's Guide

Trigonometry shows up on every GCSE Higher Maths paper, on most Foundation papers, and on virtually every A-Level Maths paper for the next two years if your child carries the subject forward. Get it solid at GCSE and you've smoothed the road for everything that comes after. This guide walks through SOH CAH TOA, how to pick the right ratio for the question in front of you, and the four mistakes I see most often when marking exam scripts.

The setup — labelling the triangle

Trigonometry only works on right-angled triangles at GCSE. Before you do anything else, label the three sides relative to the angle you're working with (not the right angle):

• Hypotenuse (H) — the long side, opposite the right angle. Always.
• Opposite (O) — the side opposite the angle you care about.
• Adjacent (A) — the side next to the angle (and not the hypotenuse).

Re-labelling for a different angle gives you different O and A sides, but the hypotenuse never changes. Get the labels right and the rest of the question is mechanical.

SOH CAH TOA — the three ratios

That mnemonic encodes:

• sin θ = O / H — Sine is Opposite over Hypotenuse
• cos θ = A / H — Cosine is Adjacent over Hypotenuse
• tan θ = O / A — Tangent is Opposite over Adjacent

The whole skill in trigonometry is choosing which of the three to use. Look at what you've got and what you want, and pick the ratio that contains those two sides.

Finding a missing side

Finding a missing angle

When the angle is the unknown, you use the inverse functions: sin⁻¹, cos⁻¹, tan⁻¹ (the keys above sin, cos, tan on your calculator).

Worded questions — angle of elevation and depression

These are the trigonometry questions that examiners love because they catch the students who only learned the formula and didn't think about the geometry.

• Angle of elevation — the angle up from the horizontal.
• Angle of depression — the angle down from the horizontal.

The mistakes that cost the most marks

Exact values you should memorise

For Higher tier, the values of sin, cos and tan at 0°, 30°, 45°, 60° and 90° come up routinely — and questions sometimes appear on the non-calculator paper, so you can't look them up. Worth committing to memory:

• sin 30° = ½, cos 30° = √3/2, tan 30° = 1/√3
• sin 45° = cos 45° = 1/√2, tan 45° = 1
• sin 60° = √3/2, cos 60° = ½, tan 60° = √3

For more on this, see the exact trigonometric values guide.

Where this leads next

Once SOH CAH TOA is solid, the natural next steps are 3D trigonometry (where you apply the same ratios but inside cuboids and pyramids), the sine rule and cosine rule (which extend trigonometry to non-right-angled triangles), and the area-of-a-triangle formula ½absinC. All three appear on Higher papers with high frequency.