# Law of cosines

### From Math Images

(Difference between revisions)

(→Distance Formula) |
(→example triangulation) |
||

Line 33: | Line 33: | ||

<math> c^{2} = (a^{2}+b^{2}-2ab \cos C+b^{2}</math> | <math> c^{2} = (a^{2}+b^{2}-2ab \cos C+b^{2}</math> | ||

- | == | + | ==Example Triangulation== |

+ | Complete the triangle using the law of cosines. | ||

+ | |||

+ | [[Image:SAS triangle.jpg]] | ||

+ | |||

+ | To find the side length <math>c</math>, |

## Revision as of 10:41, 30 May 2011

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.

## 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.

To find the side length ,