Word Lesson: The Law of Cosines

By M Ransom

In order to solve problems which require the application of the Law of Cosines, it is necessary to

use basic right triangle trigonometry
know the Law of Cosines
solve equations requiring the use of inverse cosine: arccos(x) or cos^-1(x)
solve equations involving the quadratic formula
use a calculator to get values for the lengths of sides or the measures of angles of a triangle

A typical problem that requires the use of the Law of Cosines in order to solve it involves a triangle in which there is no right angle. We are given some information about a triangle, but we have to find measurements of other sides and/or angles. The Law of Cosines for a triangle ABC is stated below, assuming that the side opposite angle A is a, the side opposite angle B is b, and the side opposite angle C is c:

This can also be written as

Suppose in triangle ABC that . Find the measure of side a.

First, we make a diagram. A diagram of this triangle is shown below.

In the diagram, known angles and lengths of sides are labeled:

The variable a is chosen to represent the unknown measurement of the side opposite angle A. This is the object of the question.

To relate the known measurements and the variable, an equation is written. In this case the equation involves the Law of Cosines, keeping side a on the opposite side of the equation from angle A.

We have

which is the same as

This equation give us

We finish solving for a by taking the square root of 12.197 and we get

A triangle ABC has a = 8, b = 9, and c = 7. What is the measure of angle C?

A triangle ABC has a = 7, b = 6, and angle A = 80º. Find the measure of side c.

A triangle ABC has a = 3, b = 6, and angle A = 80º. Find the measure of side c.

Two airplanes leave an airport, and the angle between their flight paths is 40º. An hour later, one plane has traveled 300 miles while the other has traveled 200 miles. How far apart are the planes at this time?

38074.667 miles
458.293 miles
no possible answer - calculator yields a non-real number
195.127 miles

In triangle ABC, angle A is 31º, b = 5, and a = 4. What are the possible lengths of side c?

1.225 and 7.347
11.632 and 5.511
no possible values - error read non-real answer

In triangle ABC, angle A is 30º, b = 5, and a = 2.5. How many such triangles can be drawn?

0
1
2

This type of problem requires a diagram of a carefully labeled triangle. The measurements of sides and any given angle should be labeled based on the information given. It is sometimes the case that only one, or maybe even none, of the angles is known. It is the goal of the problem to find missing measurements in the triangle. First, you should assign a variable to represent the missing measurement. We usually use a lower case English letter to represent the measure of a side of the triangle and an upper case letter to represent an angle.

Use of the Law of Cosines involves a simple equation, but the solution may involve the use of the quadratic formula. It is important to set a calculator for degrees if that is the manner in which the angles are measured. If one angle and two sides are known, it is best to use the Law of Cosines to find the measurements of missing parts of the triangle. It is often the case that the Law of Sines can be used if the measures of two angles are known.