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: | ||
Revision as of 11:13, 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.