Trigonometry (SOHCAHTOA) (Years 9–11) — Maths

Trigonometry uses the angles of a right-angled triangle to find missing sides — and works the other way to find missing angles. It is how surveyors measure the height of a hill they cannot climb, and how a ramp's steepness is worked out. The whole topic rests on one memory aid: SOH CAH TOA.

Why this matters: where Pythagoras needs two sides, trigonometry lets you get in with only one side and an angle — a much more common situation in the real world. Ramps, roofs, staircases, flight paths and the angle of the sun all rely on it, which is why SOH CAH TOA is one of the most-used tools in the whole GCSE.

From the angle θ: O = opposite, A = adjacent, H = hypotenuse

Labelling the sides

From angle θ: O is across, A is beside, H is the longest side.

Everything starts from the angle you are working with (call it θ). Relative to that angle:

Opposite (O) — the side directly across from the angle.
Adjacent (A) — the side right next to the angle (but not the hypotenuse).
Hypotenuse (H) — the longest side, always opposite the right angle.

Only O and A swap if you move to the other angle; H never changes. Label these three first, every single time.

SOH CAH TOA

Match the ratio to the two sides you have.

Sin = O/H · Cos = A/H · Tan = O/A

Pick whichever ratio uses the two sides you are dealing with — the one you know and the one you want. Think of it as choosing the right spanner: look at which two sides are in play, then reach for sin, cos or tan to match.

Sides involved	Ratio	Memory
Opposite & Hypotenuse	sin	SOH
Adjacent & Hypotenuse	cos	CAH
Opposite & Adjacent	tan	TOA

Angle 30°, hypotenuse 10, find the opposite: we have O and H → use sin. sin 30° = O/10, so O = 10 × sin 30° = 10 × 0.5 = 5.

Have O and H, so use sin: O = 10 × sin 30° = 5.

Finding an angle

If instead you know two sides and want the angle, choose the ratio as before, then undo it with the inverse buttons: sin⁻¹, cos⁻¹ or tan⁻¹ (the "second function" keys on a calculator).

Opposite 5, adjacent 5: O and A → tan. tan θ = 5/5 = 1, so θ = tan⁻¹(1) = 45°.

Common mistake: forgetting to use the inverse when hunting an angle. To find a side you type sin/cos/tan of the angle; to find the angle you must press sin⁻¹/cos⁻¹/tan⁻¹. Also check your calculator is in degrees mode, or the answers will be nonsense.

Memory hook: "Some Old Hippie Caught Another Hippie Tripping On Acid" gives SOH-CAH-TOA. Label O, A, H first; match the ratio to your two sides; for an angle, reach for the ⁻¹ button.