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
picture+pic; 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)); (Error making remote request. Unknown error_msg)
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