Finding a Side

Here is where trigonometry earns its keep. Give it one side and one angle of a right-angled triangle, and it will hand you any other side you ask for. How long must a wheelchair ramp be to reach a doorstep? How far up a wall does a ladder tilted at 70^\circ reach? How tall is a roof? All of these are one side of a right-angled triangle — and SOH-CAH-TOA finds it.

The whole method is four short moves: choose the ratio that links what you know to what you want, write the equation, rearrange to get the unknown on its own, and solve. Master those and you can measure the world with a protractor.

The method

Once you know an angle \theta and one side of a right-angled triangle, a trig ratio lets you find another side. The trick is to pick the ratio that links the side you know to the side you want, then rearrange to solve for the unknown.

Each of SOH-CAH-TOA connects two sides, so you choose the one whose two letters are the side you have and the side you need:

Starting from the definitions and rearranging gives a side directly:

\text{opposite} = \text{hypotenuse}\times\sin\theta \text{adjacent} = \text{hypotenuse}\times\cos\theta \text{opposite} = \text{adjacent}\times\tan\theta

If the unknown is on the bottom of the ratio instead, you divide — for example \text{hypotenuse} = \dfrac{\text{opposite}}{\sin\theta}.

To find an unknown side from an angle \theta and one known side:

Worked example 1 — the opposite side, with sine

Step through the three moves: spot what you know, choose the matching ratio, then rearrange and evaluate. Here the hypotenuse is 10 and the angle is 35^\circ; the unknown x lies opposite the angle, so we pair O and H and reach for sine.

Worked example 2 — finding the hypotenuse (a division)

A support wire runs from the top of a pole to the ground. It meets the ground at 40^\circ, and the pole (the side opposite that angle) is 6\text{ m} tall. How long is the wire — the hypotenuse?

We have O and want H, so it is sine again — but this time the unknown is on the bottom:

\sin 40^\circ = \frac{6}{\text{wire}} \quad\Rightarrow\quad \text{wire} = \frac{6}{\sin 40^\circ} \approx 9.3\text{ m}.

Notice the flip: when the unknown sits underneath, you divide by the ratio rather than multiplying. Same sine, different rearrangement.

Worked example 3 — the height of a tree from its shadow

A tree casts a shadow 12\text{ m} long. At that moment the sun is 50^\circ above the horizon. How tall is the tree?

Picture the right-angled triangle: the shadow on the ground is the side adjacent to the 50^\circ angle, and the tree's height is the side opposite it. O and A together → tangent:

\tan 50^\circ = \frac{\text{height}}{12} \quad\Rightarrow\quad \text{height} = 12\times\tan 50^\circ \approx 14.3\text{ m}.

The tree is taller than its shadow is long — which fits, because the sun is more than 45^\circ up. One tape measure on the ground and one angle, and you have measured a tree without climbing it.

Three traps swallow marks on almost every trig exam:

Finding sides with trig is not a classroom trick — it is the daily arithmetic of surveyors, builders and navigators. Stand a known distance from a building, measure the angle up to its top, and one tangent gives you its height. No ladder, no drone.

Scale that idea up and you get triangulation: chain triangle after triangle across the land, each sharing a side with the next, and you can map an entire country to the metre — which is exactly how Britain, India and France were surveyed in the 1700s and 1800s, long before GPS. Scale it up further still and astronomers use the width of Earth's orbit as the known side to reach out and measure the distance to nearby stars — the method called parallax. The same four moves you just learned, from a doorstep ramp all the way to the stars. Next you'll turn the method around to find the angle instead of the side.

See it explained