Law of Cosines
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:
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.
Since , , so the expression reduces to the Pythagorean Theorem. You can find several proofs of the Pythagorean Theorem here.
The argument for an obtuse triangle is the same as the proof for an acute triangle.