College Algebra & Trigonometry, 2018a

11.3. THE LAW OF COSINES 489
Factoring out the a 2 and replacing sin 2 C +cos 2 C with 1, we come out with one
of the most common forms of the Law of Cosines:
c 2 = a 2 sin 2 C + a 2 cos 2 C + b 2 − 2ab cos C
c 2 = a 2 (sin 2 C +cos 2 C)+b 2 − 2ab cos C
c 2 = a 2 (1) + b 2 − 2ab cos C
c 2 = a 2 + b 2 − 2ab cos C
Any letter may be used to represent each of the sides, but the order that the letters
are used in is very important. The side of the triangle isolated on the left hand
side must correspond to the angle used on the right hand side.
The Law of Cosines
a 2 = b 2 + c 2 − 2bc cos A
b 2 = a 2 + c 2 − 2ac cos B
c 2 = a 2 + b 2 − 2ab cos C
We’ll look at three examples - two in which two sides and the included angle are
given and one in which the three sides of the triangle are given.