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. 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.