Trigonometry: Finding a Missing Side

Pythagoras' theorem connects the three sides of a right-angled triangle — but it needs you to know two of them already. Trigonometry goes further: given just one side and one angle, it can find any other side. That's what makes it possible to find the height of a tree from its shadow, without climbing it.

Naming the sides, relative to an angle

Pick one of the non-right angles in the triangle. Relative to it:

• the hypotenuse is the longest side, opposite the right angle,
• the opposite side is the one across from your chosen angle,
• the adjacent side is the one next to your chosen angle (that isn't the hypotenuse).

SOH CAH TOA

Three ratios connect that angle to the triangle's sides:

• SOH: sin(angle) = Opposite / Hypotenuse
• CAH: cos(angle) = Adjacent / Hypotenuse
• TOA: tan(angle) = Opposite / Adjacent

Which one you use depends entirely on which two sides are involved — the one you know, and the one you want.

Example 1 — finding the opposite side

A ladder leans against a wall, making a 30° angle with the ground. The ladder itself (the hypotenuse) is 10 metres. How high up the wall does it reach (the opposite side)?

You know the angle and the hypotenuse, and want the opposite side — that's SOH:

opposite = hypotenuse × sin(angle) = 10 × sin(30°) = 10 × 0.5 = 5

The ladder reaches 5 metres up the wall.

Example 2 — finding the adjacent side

Now say the same ladder makes a 60° angle with the ground, and is still 8 metres long. How far is its base from the wall (the adjacent side)?

Angle and hypotenuse known, adjacent wanted — that's CAH:

adjacent = hypotenuse × cos(angle) = 8 × cos(60°) = 8 × 0.5 = 4

The base of the ladder is 4 metres from the wall.

Example 3 — finding a side without the hypotenuse

If the hypotenuse isn't involved at all — say you know the adjacent side is 6 metres and the angle is 45°, and want the opposite side — use TOA:

opposite = adjacent × tan(angle) = 6 × tan(45°) = 6 × 1 = 6

In short

• Trigonometry relates an angle to the side lengths of a right-angled triangle.
• Sides are named relative to your chosen angle: opposite, adjacent, hypotenuse.
• SOH CAH TOA tells you which ratio to use: pick the one connecting the side you know to the side you want.

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