# How is the law of cosines related to the pythagorean theorem?

Apr 2, 2018

The Pythagorean Theorem can be stated as ${a}^{2} = {b}^{2} + {c}^{2}$ (where $a$ is the hypotenuse) and works only in right-angled triangles.

The Cosine Law is ${a}^{2} = {b}^{2} + {c}^{2} - 2 b c \cos A$ and works for all kinds of triangles.

#### Explanation:

The explanation is the really cool bit!

$A$ in the cosine rule is the angle opposite to the side $a$.

Now, if $A = {90}^{o}$ (which means the side opposite it, $a$, is the hypotenuse), we know that $\cos {90}^{o} = 0$, so all the stuff after the minus sign in the Cosine Rule becomes $0$... and you will see that you just have the Pythagorean Theorem back again!

Because now you're working in a right-angled triangle.

Isn't that cool?