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:
- opposite and hypotenuse \to sine;
- adjacent and hypotenuse \to cosine;
- opposite and adjacent \to tangent.
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:
- label the sides O, A, H relative to \theta;
- choose the ratio (sin, cos or tan) that uses the known side and the unknown side;
- multiply (when the unknown is on top) or divide (when it is on the bottom) to solve.