Solving any triangle

SOH-CAH-TOA only works on right triangles. But most triangles in the real world have no right angle. Two laws extend trigonometry to any triangle.

The naming convention

First, the standard labels: each side is named with the lowercase of the angle opposite it — side a faces angle A, side b faces B, side c faces C.

The law of sines

In any triangle, each side and the sine of its opposite angle keep the same ratio:

a / sin A = b / sin B = c / sin C

Use it when you know an angle and its opposite side, plus one more piece — the AAS, ASA, or SSA cases. Set two of the ratios equal and solve for the unknown.

The law of cosines

When the law of sines doesn't have enough to start (no angle-and-its-opposite-side pair), the law of cosines fills the gap:

c² = a² + b² − 2ab·cos C

It's the Pythagorean theorem with a correction term. If C = 90°, then cos C = 0, the last term vanishes, and it collapses back to c² = a² + b² — Pythagoras is just the right-angle special case.

Use the law of cosines for the SAS case (two sides and the angle between them) and the SSS case (all three sides, solving for an angle).

Which law, when?

A quick rule: if you have an angle paired with its opposite side, reach for the law of sines; otherwise, start with the law of cosines.

Why this matters

These two laws are how surveying, navigation, and engineering measure distances no ruler can reach — the height of a mountain, the distance to a ship, the span of a bridge — from a couple of angles and a baseline. Trigonometry leaves the right triangle and meets the real world here.