http://www.ck12.org Chapter 4. Basic Triangle Trigonometry
a=a 1 +a 2 ( 1 )
b^2 =a^21 +h^2 ( 2 )
c^2 =a^22 +h^2 ( 3 )
cosC=ab^1 ( 4 )
Once you verify for yourself that you agree with each of these facts, check algebraically that these next two facts
must be true.
a 2 =a−a 1 ( 5 ,from 1)
a 1 =b·cosC ( 6 ,from 4)
Now the Law of Cosines is ready to be proved using substitution, FOIL, more substitution and rewriting to get the
order of terms right.
c^2 =a^22 +h^2 (3 again)
c^2 = (a−a 1 )^2 +h^2 (substitute using 5)
c^2 =a^2 − 2 a·a 1 +a^21 +h^2 (FOIL)
c^2 =a^2 − 2 a·b·cosC+a^21 +h^2 (substitute using 6)
c^2 =a^2 − 2 a·b·cosC+b^2 (substitute using 2)
c^2 =a^2 +b^2 − 2 ab·cosC (rearrange terms)There are only two types of problems in which it is appropriate to use the Law of Cosines. The first is when you
are given all three sides of a triangle and asked to find an unknown angle. This is called SSS like in geometry. The
second situation where you will use the Law of Cosines is when you are given two sides and the included angle and
you need to find the third side. This is called SAS.
Example A
Determine the measure of angleD.
Solution:It is necessary to set up the Law of Cosines equation very carefully withDcorresponding to the opposite
side of 230. The letters are notABClike in the proof, but those letters can always be changed to match the problem
as long as the angle in the cosine corresponds to the side used in the left side of the equation.