The law of cosines

For right-angled triangles we have Pythagoras’ theorem. But what happens in a general triangle? The answer is the law of cosines.

Theorem — The law of cosines

In a general triangle $ABC$ with sides $a=BC$, $b=CA$, $c=AB$ opposite the vertices $A,B,C$, we have: $a^2 = b^2 + c^2 - 2bc\cos\widehat{A}.$ (And likewise for $b^2$ and $c^2$, by cyclic permutation.)

The theorem generalises Pythagoras’: if $\widehat{A} = \pi/2$, then $\cos\widehat{A}=0$ and the formula reduces to $a^2 = b^2 + c^2$. The “correction term” $-2bc\cos\widehat{A}$ measures how much the angle differs from a right angle.

Links

Topics: Triangle trigonometry
Concepts: Solving triangles · The law of cosines · Carnot’s theorem · Pythagoras’ theorem
Functions: Cosine
Methods: Solving triangles
Skills: Using formulae