For any right triangle, the sides always keep the same ratios for a given angle — SOH-CAH-TOA is the memory trick that names them. It's how surveyors find a hill's height without climbing it, just measuring an angle and a distance.
Three ratios, one triangle
In a right triangle, once you pick an angle theta, the three sides get names: the hypotenuse (longest, opposite the right angle), the opposite side (across from theta), and the adjacent side (next to theta). Sine, cosine, and tangent are ratios of these sides — and for a given angle, the ratios are always the same no matter how big the triangle is.
- SOH — Sine = Opposite / Hypotenuse
- CAH — Cosine = Adjacent / Hypotenuse
- TOA — Tangent = Opposite / Adjacent
A ladder leans at 60° and reaches 5 m up a wall. sin 60° = 5/L, so the ladder is L = 5 / sin 60° ≈ 5.8 m long.
All right triangles with the same angle are similar — scaled copies of each other. Scaling multiplies every side by the same amount, so the ratios never change. That is what makes trig tables possible.
- Stand a known distance from a tall object (say 20 paces from a flagpole).
- Sight the top of the object and estimate the angle of elevation with a protractor and a straw (or a phone's level app).
- You know the adjacent side (your distance) and the angle — pick the ratio that involves opposite and adjacent.
- Compute height = distance × tan(angle), then add your eye height.
What you should see: You measured something too tall to reach with a tape measure using tangent — the same method surveyors use.