Law of cosines

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(New page: 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 o...)
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When to use it: SAS, SSS.
When to use it: SAS, SSS.
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==Proof==
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Let <math> \vartriangle ABC </math> be oriented so that <math> C</math> is at the origin, and <math> B</math> is at the point<math> (a,0)</math>.
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[[Image:Law_of_cosines_proof.jpg]]
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===Distance Formula===
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<math> distance = \sqrt {(x_{2}-x_{1})^{2} + (y_{2}-y_{1})^{2}}</math>
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<math>c</math> is the distance from <math> A</math> to <math> B</math>.
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Substituting the appropriate points into the distance formula gives us
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<math> c = \sqrt {(a-b \cos C)^{2} + (0-b \sin C)^{2}}</math>
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Squaring the inner terms, we have
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<math> c = \sqrt {(a^{2}-2ab \cos C+b^{2} \cos^{2} C) + (b^{2} \sin^{2} C)}</math>
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Since <math> \cos^{2} C + \sin^{2} C = 1</math>,
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<math> c = \sqrt {(a^{2}+b^{2}-2ab \cos C+b^{2}}</math>
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Square both sides for
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<math> c^{2} = (a^{2}+b^{2}-2ab \cos C+b^{2}</math>

Revision as of 15:40, 26 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.

 c^{2} = a^{2} + b^{2} - 2ab \cos C

When to use it: SAS, SSS.

Proof

Let  \vartriangle ABC be oriented so that  C is at the origin, and  B is at the point (a,0).

Image:Law_of_cosines_proof.jpg

Distance Formula

 distance = \sqrt {(x_{2}-x_{1})^{2} + (y_{2}-y_{1})^{2}}

c is the distance from  A to  B.

Substituting the appropriate points into the distance formula gives us

 c = \sqrt {(a-b \cos C)^{2} + (0-b \sin C)^{2}}

Squaring the inner terms, we have

 c = \sqrt {(a^{2}-2ab \cos C+b^{2} \cos^{2} C) + (b^{2} \sin^{2} C)}

Since  \cos^{2} C +  \sin^{2} C = 1,

 c = \sqrt {(a^{2}+b^{2}-2ab \cos C+b^{2}}

Square both sides for

 c^{2} = (a^{2}+b^{2}-2ab \cos C+b^{2}

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