WE USE THE LAW OF COSINES to solve triangles that are not right-angled. In particular, when we know two sides of a triangle and their included angle, then the Law of Cosines enables us to find the third side. Thus if we know sides
(The Law of Cosines is a extension of the Pythagorean
theorem; because if θ were a right angle,
we would have
Proof of the Law of Cosines Let
ABC be a triangle with sides
Draw BD perpendicular to CA, separating triangle ABC
into the two right triangles BDC, BDA. BD is the height Call CD Also, since
then
Now, in the right triangle BDC, according to the Pythagorean theorem,
so that
In the right triangle BDA,
For
Finally, for
That is,
This is what we wanted to prove. In the same way, we could prove that
and
This is the Law of Cosines. |