# Law of cosines

### From Math Images

The law of cosines is a formula that helps in triangulation when two or three side lengths of a triangle are known. The formula relates all three side lengths of a triangle to the cosine of a particular angle.

When to use it: SAS, SSS.

## Contents |

## Proof

Let be oriented so that is at the origin, and is at the point.

### Distance Formula

is the distance from to .

Substituting the appropriate points into the distance formula gives us

Squaring the inner terms, we have

Since ,

Square both sides for

## Example Triangulation

Complete the triangle using the law of cosines.

### Solution

To find the side length ,

Simplify for

Since , substitution gives us

Simplify for

Taking the square root of both sides gives us

Now we can orient the triangle differently to get get a new version of the law of cosines so we can find angle measure ,

Substituting in the appropriate side lengths gives us

Simplify for

Subtracting from both sides gives us

Dividing both sides by gives us