The Cosine Rule

The sine rule needs a side paired with its opposite angle. When you don't have such a pair — say you know two sides and the angle between them — you reach for the cosine rule instead. It is a generalised Pythagoras that works in any triangle:

a^2 = b^2 + c^2 - 2bc\cos A

Use it in two situations:

you know two sides and the included angle (b, c and A) and want the third side a;
you know all three sides and want an angle — rearrange to \cos A = \dfrac{b^2 + c^2 - a^2}{2bc} then take the inverse cosine.

Notice that when A = 90^\circ we have \cos A = 0, so the last term vanishes and the rule collapses to a^2 = b^2 + c^2 — Pythagoras' theorem.

In any triangle, with each side opposite the angle of the same letter:

to find a side from two sides and the included angle, use a^2 = b^2 + c^2 - 2bc\cos A;
to find an angle from all three sides, rearrange to \cos A = \dfrac{b^2 + c^2 - a^2}{2bc};
when the included angle is 90^\circ, \cos A = 0 and it reduces to Pythagoras' theorem a^2 = b^2 + c^2.

The included angle

Here is a scalene triangle. Step through it to see how sides b and c meet at vertex A — the angle A between them is the one the cosine rule uses to reach the opposite side a.