Difference between revisions of "Law of Cosines"

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{{WotWAnnounce|week=Oct 4-Oct 10}}
 
 
 
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]].
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==Proofs==
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===Acute Triangle===
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<asy>
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pair A,B,C,D,E;
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C=(30,70);
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B=(0,0);
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A=(100,0);
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D=(30,0);
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size(100);
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draw(B--A--C--B);
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draw(C--D);
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label("A",A,(1,0));
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dot(A);
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label("B",B,(-1,-1));
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dot(B);
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label("C",C,(0,1));
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dot(C);
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draw(D--(30,4)--(34,4)--(34,0)--D);
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label("f",(30,35),(1,0));
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label("d",(15,0),(0,-1));
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label("e",(50,0),(0,-1.5));
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</asy>
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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>.
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We use the Pythagorean theorem:
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<cmath>a^2+b^2-2f^2=d^2+e^2</cmath>
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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>.
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We use the addition rule for cosines and get:
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<cmath>\cos{C}=\dfrac{f}{a}*\dfrac{f}{b}-\dfrac{d}{a}*\dfrac{e}{b}=\dfrac{f^2-de}{ab}</cmath>
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We multiply by -2ab and get:
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<cmath>2de-2f^2=-2ab\cos{C}</cmath>
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Now remember our equation?
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<cmath>a^2+b^2-2f^2+2de=c^2</cmath>
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We replace the <math>-2f^2+2de</math> by <math>-2ab\cos{C}</math> and get:
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<cmath>c^2=a^2+b^2-2ab\cos{C}</cmath>
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We can use the same argument on the other sides.
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===Right Triangle===
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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]].
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===Obtuse Triangle===
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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]]
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[[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 $a$, $b$ and $c$ opposite angles of measure $A$, $B$ and $C$, respectively, the Law of Cosines states:

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

$b^2 = a^2 + c^2 - 2ac\cos B$

$a^2 = b^2 + c^2 - 2bc\cos A$

In the case that one of the angles has measure $90^\circ$ (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 $a$, $b$, and $c$ be the side lengths, $C$ is the angle measure opposite side $c$, $f$ is the distance from angle $C$ to side $c$, and $d$ and $e$ are the lengths that $c$ is split into by $f$.

We use the Pythagorean theorem:

\[a^2+b^2-2f^2=d^2+e^2\]

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

We use the addition rule for cosines and get:

\[\cos{C}=\dfrac{f}{a}*\dfrac{f}{b}-\dfrac{d}{a}*\dfrac{e}{b}=\dfrac{f^2-de}{ab}\]

We multiply by -2ab and get:

\[2de-2f^2=-2ab\cos{C}\]

Now remember our equation?

\[a^2+b^2-2f^2+2de=c^2\]

We replace the $-2f^2+2de$ by $-2ab\cos{C}$ and get:

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

We can use the same argument on the other sides.

Right Triangle

Since $C=90^{\circ}$, $\cos C=0$, 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.

See also

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