Law of Cosines
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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 ,
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
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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