# Difference between revisions of "Law of Cosines"

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The '''Law of Cosines''' is a theorem which relates the side-[[length]]s and [[angle]]s of a [[triangle]]. For a triangle with [[edge]]s of length <math>a</math>, <math>b</math> and <math>c</math> opposite [[angle]]s of measure <math>A</math>, <math>B</math> and <math>C</math>, respectively, the Law of Cosines states: | The '''Law of Cosines''' is a theorem which relates the side-[[length]]s and [[angle]]s of a [[triangle]]. For a triangle with [[edge]]s of length <math>a</math>, <math>b</math> and <math>c</math> opposite [[angle]]s of measure <math>A</math>, <math>B</math> and <math>C</math>, respectively, the Law of Cosines states: | ||

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In the case that one of the angles has measure <math>90^\circ</math> (is a [[right angle]]), the corresponding statement reduces to the [[Pythagorean Theorem]]. | In the case that one of the angles has measure <math>90^\circ</math> (is a [[right angle]]), the corresponding statement reduces to the [[Pythagorean Theorem]]. | ||

+ | ==Proofs== | ||

+ | ===Acute Triangle=== | ||

+ | <asy> | ||

+ | pair A,B,C,D,E; | ||

+ | C=(30,70); | ||

+ | B=(0,0); | ||

+ | A=(100,0); | ||

+ | D=(30,0); | ||

+ | size(100); | ||

+ | draw(B--A--C--B); | ||

+ | draw(C--D); | ||

+ | label("A",A,(1,0)); | ||

+ | dot(A); | ||

+ | label("B",B,(-1,-1)); | ||

+ | dot(B); | ||

+ | label("C",C,(0,1)); | ||

+ | dot(C); | ||

+ | draw(D--(30,4)--(34,4)--(34,0)--D); | ||

+ | label("f",(30,35),(1,0)); | ||

+ | label("d",(15,0),(0,-1)); | ||

+ | label("e",(50,0),(0,-1.5)); | ||

+ | </asy> | ||

+ | |||

+ | |||

+ | Let <math>a</math>, <math>b</math>, and <math>c</math> be the side lengths, <math>C</math> is the angle measure opposite side <math>c</math>, <math>f</math> is the distance from angle <math>C</math> to side <math>c</math>, and <math>d</math> and <math>e</math> are the lengths that <math>c</math> is split into by <math>f</math>. | ||

+ | |||

+ | We use the Pythagorean theorem: | ||

+ | |||

+ | <cmath>a^2+b^2-2f^2=d^2+e^2</cmath> | ||

+ | |||

+ | We are trying to get <math>a^2+b^2-2f^2+2de</math> on the LHS, because then the RHS would be <math>c^2</math>. | ||

+ | |||

+ | We use the addition rule for cosines and get: | ||

+ | |||

+ | <cmath>\cos{C}=\dfrac{f}{a}*\dfrac{f}{b}-\dfrac{d}{a}*\dfrac{e}{b}=\dfrac{f^2-de}{ab}</cmath> | ||

+ | |||

+ | We multiply by -2ab and get: | ||

+ | |||

+ | <cmath>2de-2f^2=-2ab\cos{C}</cmath> | ||

+ | |||

+ | Now remember our equation? | ||

+ | |||

+ | <cmath>a^2+b^2-2f^2+2de=c^2</cmath> | ||

+ | |||

+ | We replace the <math>-2f^2+2de</math> by <math>-2ab\cos{C}</math> and get: | ||

+ | |||

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

+ | |||

+ | We can use the same argument on the other sides. | ||

+ | |||

+ | ===Right Triangle=== | ||

+ | Since <math>C=90^{\circ}</math>, <math>\cos C=0</math>, so the expression reduces to the Pythagorean Theorem. You can find several proofs of the Pythagorean Theorem [[Pythagorean Theorem#Proofs|here]]. | ||

+ | |||

+ | ===Obtuse Triangle=== | ||

+ | The argument for an obtuse triangle is the same as the proof for an acute triangle. | ||

==See also== | ==See also== | ||

* [[Law of Sines]] | * [[Law of Sines]] | ||

* [[Trigonometry]] | * [[Trigonometry]] | ||

+ | [[Category:Trigonometry]] |

## Revision as of 12:13, 11 October 2007

The **Law of Cosines** is a theorem which relates the side-lengths and angles of a triangle. For a triangle with edges of length , and opposite angles of measure , and , respectively, the Law of Cosines states:

In the case that one of the angles has measure (is a right angle), the corresponding statement reduces to the Pythagorean Theorem.

## Proofs

### Acute Triangle

Let , , and be the side lengths, is the angle measure opposite side , is the distance from angle to side , and and are the lengths that is split into by .

We use the Pythagorean theorem:

We are trying to get on the LHS, because then the RHS would be .

We use the addition rule for cosines and get:

We multiply by -2ab and get:

Now remember our equation?

We replace the by and get:

We can use the same argument on the other sides.

### Right Triangle

Since , , so the expression reduces to the Pythagorean Theorem. You can find several proofs of the Pythagorean Theorem here.

### Obtuse Triangle

The argument for an obtuse triangle is the same as the proof for an acute triangle.