Law of Cosines

The Law of Cosines (also known as the Cosine Rule) allows to find the length of the third side of a triangle knowing two other sides and the angle between them.

Theorem (Law of Cosines)

Consider a triangle with the sides a, b, c and angles α, β, γ.

The Law of Cosines states that

where $\gamma$ denotes the angle between the sides $a$ and $b.$

The sides $a$ and $b$ can also be expressed in terms of two other sides and angle between them:

When the angle $$ between the sides $$ and $$ is a right angle, the Law of Cosines reduces to the Pythagorean Identity:

So the law of cosines is a generalization of the Pythagorean Theorem for an arbitrary triangle.

Proof

We consider separately the cases of acute and obtuse triangles.

In the first case, assume that the angle $$ is an acute angle.

Draw the altitude $$ from the vertex $$ to the side $$ Its height $$ can be found from the trigonometric identity for the right triangle $$

The line segment $$ (the projection of side $$ onto the side $$) has the length

In the right traingle $$ the side $$ is given by

Using the Pythagorean Identity for $$, we get:

Notice that

Therefore

Let now the angle $$ be obtuse.

In this case, the altitude $$ is given by

By the reduction identity, $$ Hence, the height $$ is expressed by the same formula:

Again, by the reduction identity, the length of the line segment $$ is given by

From the right triangle $$ we have

Similarly, the Law of Cosines can be proved for two other sides of the triangle.