The Law of Cosines | Moosa Academy

Pythagoras' theorem only speaks about right triangles. The law of cosines extends it to every triangle by adding one correction term built from the included angle.

Theorem Law of cosines

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

a: the unknown side, opposite angle A.

b and c: the two sides enclosing angle A.

−2bc cos A: the extra term Pythagoras does not have.

Each side has its own version — the pattern is always "square of the unknown side = sum of the squares of the other two, minus twice their product times the cosine of the angle between them":

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

b^2 = a^2 + c^2 - 2ac\cos B

c^2 = a^2 + b^2 - 2ab\cos C

Note Pythagoras is the special case

At $A = 90°$ we have $\cos A = 0$ , the extra term vanishes, and the law becomes $a^2 = b^2 + c^2$ — Pythagoras' theorem.

The term also explains how the third side reacts: a small angle gives a large positive cosine, so more is subtracted and $a$ comes out short. An obtuse angle makes the cosine negative, so the term is added and $a$ comes out long.

Example Finding a side with an obtuse angle

In triangle ABC: $b = 7$ , $c = 9$ and $\angle A = 120°$ . Find $a$ .

a^2 = 7^2 + 9^2 - 2(7)(9)\cos 120°

\cos 120° = -0.5

, so the last term is

-126 \times (-0.5) = +63

a^2 = 49 + 81 + 63 = 193

a = \sqrt{193}

⟹ a ≈ 13.89 cm

Reference Which form to use

Unknown	Formula	Given
Side a	a² = b² + c² − 2bc cos A	SAS
Side b	b² = a² + c² − 2ac cos B	SAS
Side c	c² = a² + b² − 2ab cos C	SAS
An angle	cos A = (b² + c² − a²) ÷ 2bc	SSS

SAS — two sides and the angle between them known: find the third side. SSS — all three sides known: rearrange to find any angle.

Summary

a² = b² + c² − 2bc cos A generalises Pythagoras to any triangle.
Each side has its own form, always using the angle opposite it.
At 90° the cosine term disappears and Pythagoras returns.
Use SAS to find a missing side, and the rearranged form with SSS to find a missing angle.