Difference between revisions of "Law of Cosines"

Revision as of 14:08, 7 October 2007

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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

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Info: a, b, and c are the side lengths, and C is the angle measure opposite side C. f is the height 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

@@ Line 45: / Line 45: @@
 ===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===

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Difference between revisions of "Law of Cosines"

Revision as of 14:08, 7 October 2007

Contents

Proofs

Acute Triangle

Right Triangle

Obtuse Triangle

See also