Right-Angled Trigonometry

How tall is that mountain? How far away is that ship? How wide is the river you cannot cross? For thousands of years people answered questions like these without a tape measure — by standing still, measuring one angle, and letting a triangle do the rest.

The secret is that in a right-angled triangle the sides and the angles are locked together. Fix an angle, and the three sides are no longer free to be anything they like: their ratios are pinned down exactly. Three ratios do all the work — sine, cosine and tangent — and generations of students have carried them in their heads under one battered mnemonic: SOH-CAH-TOA. Learn to read a right-angled triangle and you can measure the unreachable.

Naming the three sides

In a right-angled triangle, the three sides have special names — but only once you fix your attention on one of the two non-right angles, call it \theta:

the hypotenuse (H) — the longest side, always opposite the right angle;
the opposite (O) — the side across the triangle from \theta;
the adjacent (A) — the side next to \theta (but not the hypotenuse).

The magic of trigonometry is that for a given angle, the ratios of these sides are always the same — no matter how big the triangle is. A doorway-sized triangle and a mountain-sized triangle with the same angle share the very same ratios. Those three ratios are sine, cosine and tangent.

For an angle \theta in a right-angled triangle:

\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}} — SOH;
\cos\theta = \dfrac{\text{adjacent}}{\text{hypotenuse}} — CAH;
\tan\theta = \dfrac{\text{opposite}}{\text{adjacent}} — TOA;
each ratio depends only on the angle, not on the triangle's size.

See the ratios change

Drag the slider to change the angle \theta. The triangle's sides change, but notice how each ratio is a single number tied to the angle. Push \theta towards 90^\circ and watch the opposite side stretch until it nearly is the hypotenuse (so \sin\theta climbs towards 1); pull it towards small angles and the opposite shrinks away (so \sin\theta drops towards 0).

Worked example 1 — labelling O, A and H

A right-angled triangle has its right angle at the bottom-right corner. The angle \theta sits at the bottom-left. Which side is which?

The hypotenuse is the slanted side facing the right angle — the longest side. Mark it H.
The opposite is the vertical side, right across the triangle from \theta. You could draw a straight line from the angle and it would point at this side. Mark it O.
The adjacent is the last one — the bottom side, running away from \theta to the right angle. Mark it A.

A handy check: the adjacent and the hypotenuse always touch the angle \theta; the opposite never does.

Worked example 2 — which ratio do I want?

You know the opposite side and you want to talk about the hypotenuse. Which ratio connects them? Scan SOH-CAH-TOA for the pair "O and H" — that is SOH, so the ratio is sine:

\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}

The trick every time: name the two sides you care about, then find the one letter-pair in SOH-CAH-TOA that matches. "A and H" → cosine. "O and A" → tangent. There is exactly one match, so you can never be stuck.

Worked example 3 — a first calculation

A ladder leans against a wall. It is 6\text{ m} long (that is the hypotenuse) and makes an angle of 65^\circ with the ground. How high up the wall does it reach?

The height is the side opposite the 65^\circ angle, and we know the hypotenuse. "O and H" → sine:

\sin 65^\circ = \frac{\text{height}}{6} \quad\Rightarrow\quad \text{height} = 6\times\sin 65^\circ \approx 5.4\text{ m}.

No ladder-climbing, no risk — one angle and a little arithmetic. That is the whole promise of trigonometry in miniature. (Rearranging to actually solve for an unknown side is the next skill, in finding a side.)

The single most common trig mistake is mislabelling the opposite and the adjacent. Here is why it happens: those two names are not fixed to the triangle — they are fixed to your chosen angle.

The opposite is the side across from the angle you are using. Switch to the other non-right angle and the side that was "opposite" is now "adjacent", and vice versa. They swap.
The hypotenuse is the only label that never moves — it is always the side facing the right angle, the longest side, whichever angle you look from.

So the rule is: mark your angle first, and only then label O and A relative to it. Do that, and the #1 trig error simply cannot happen.

These three little ratios are how humans measured the previously unmeasurable. The heights of mountains, the distance to a ship at sea, the height of the Great Pyramid from the length of its shadow — all worked out from the ground, with an angle and a triangle.

The grandest example: around 240 BC the librarian Eratosthenes measured the size of the whole Earth from two shadows and an angle, and got astonishingly close to the right answer — never leaving Egypt. The same three ratios still run underneath your everyday world: GPS locating your phone, the geometry engines in video games, the trusses in a roof, and even the smooth waves of sound and light — all lean on sine, cosine and tangent. SOH-CAH-TOA may be the most useful nine letters you ever memorise.

Naming the three sides

See the ratios change

Worked example 1 — labelling O, A and H

Worked example 2 — which ratio do I want?

Worked example 3 — a first calculation

See it explained