# Law of cosines

### From Math Images

(→example triangulation) |
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==Example Triangulation== | ==Example Triangulation== | ||

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

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

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

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

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+ | <math> c^{2} = 6^{2} + (6 \sqrt{2})^{2} -2 (6) (6 \sqrt{2}) \cos 45^\circ </math> | ||

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+ | Simplify for | ||

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+ | <math> c^{2} =36 + 36 (2) - 72 \sqrt{2}) \cos 45^\circ </math> | ||

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+ | Since <math> \cos 45^\circ = \frac{1}{\sqrt{2}}</math>, substitution gives us | ||

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+ | <math> c^{2} =36 + 36 (2) - 72 \sqrt{2} (\frac{1}{\sqrt{2}}) </math> | ||

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+ | Simplify for | ||

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+ | <math> c^{2} =36 + 72 - 72 </math> | ||

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+ | <math> c^{2} =36 </math> | ||

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+ | Taking the square root of both sides gives us | ||

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+ | <math> c =6 </math> | ||

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+ | ---- | ||

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+ | Now we can orient the triangle differently to get get a new version of the law of cosines so we can find angle measure <math>B</math>, | ||

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

## Revision as of 09:54, 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 ,

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 ,