B. Solving Problems Using the Cosine Law

The cosine law can be used to solve a variety of problems, much like the sine law.

A video demonstration of the solution for Example 1 is provided.

Example 1

When an earthquake occurs, different kinds of seismic waves are produced. Two types of seismic waves are S-waves and P-waves, which travel at different velocities. A seismic station can estimate the distance to an earthquake's epicentre by comparing the amount of time that elapses between the two types of waves reaching the station.

An earthquake's epicentre occurs 48 km away from seismic station 1, and 91 km away from seismic station 2. If the seismic stations are known to be 112 km apart, what is the angle formed at seismic station 2 between seismic station 1 and epicentre 1?

There are three known sides and an unknown angle involved, so you can use the cosine law. It may help to draw a diagram.

The goal is to determine the measure of angle C, so it makes sense to use the version of the cosine law.

The angle between possible epicentre 1 and seismic station 1 at seismic station 2 is approximately 24.7°.

For further information about the cosine law see pp. 146–149 of Principles of Mathematics 11.