Law of Cosines: More Examples

Let’s look at a couple more examples before you practice on your own.

Example 1:
In triangle ABC, angle A measures 77 degrees, side b measures 100 and side c measures 95.  Find the measure of side a.

Solution:
It is important to notice that we are given the information about two sides (b and c) and the included angle.  We do not have an angle and a side that are opposite each other such that we could use the Law of Sines.  Instead, we must use the Law of Cosines.

a² = 100² + 95² – 2(100)(95) cos 77°
a² ≈ 10,000 + 9025 – 19,000(0.2250)
a² ≈ 14750
a ≈
a ≈ 121.45

Example 2:
Find the measure of the angle opposite the longest side of a triangle with side lengths 15, 21, 16.

Solution:
The longest side is 21.  If that is side ‘a’, we are looking for angle A.

21² = 15² + 16² – 2(15)(16) cos A
441 = 225 + 256 – 480 cos A
-40 = -480 cos A
0.0833 = cos A
cos-1(0.0833) = cos-1(cos A)
85.22 = A