When I see a triangle problem that can be solved using the Law of Cosines, I follow these steps: Identify the information that is given, Draw a picture if one is not provided, Decide which Law of Cosines formula to use, and then set up and solve the formula. The three versions of the Law of Cosines are essentially the same, only slightly different to solve for appropriate pieces of a triangle. I’ll show you what I mean in a moment when we complete the following example.

Three islands are situated in a triangular shape. The distance between Islands A and C is 5 miles. The distance between Islands A and B is 6 miles. The distance between Islands B and C is 7 miles. If you draw straight lines between all three islands, what is the angle between the lines drawn from Island A to Islands B and C? Of course this angle is needed to help vessels navigate from one island to another. Let’s identify the information and draw a diagram for this problem… Notice that I added lowercase letters to identify the side lengths of each triangle. The letters used for each side correspond to the opposite vertex. This helps to identify which version of the Law of Cosines to use… Since we are looking for angle A, and we have side lengths a, b and c, we can use the first version of the Law of Cosines.

I will now set up and solve the Law of Cosines. Feel free to try this on you own and skip ahead, or simply follow along with the video… In order to find either of the remaining angles we can use the same method if we carefully choose the right version of the law of Cosines; version two for angle B, and version three for angle C. Feel free to practice by solving for angle B and C. When finished, you can check your work by adding the three angles together. They should sum to one hundred eighty degrees. Good Luck!