Cosine Rule

In this chapter we shall be exploring the cosine rule. In trigonometry we can use the cosine rule to find the missing sides and angles in a triangle. You must have some basic knowledge of trigonometry and bearings to appreciate this chapter.

Deriving the cosine rule

Let us have a look at how the cosine rule was found or derived. Here is a triangle;

The problem with this triangle is that we cannot solve x using the sine rule because we have one pair or x/sin62°, we have no other pair to equal this to. We can’t use Pythagoras because this is not a right angled triangle. But we can take advantage of Pythagoras.
First draw a line in between the triangle perpendicular to the line AB from C; and create another point D as shown below;

Let’s label DA x. Sine DA is x that must mean that BD is equal to c-x. We have a right angle triangle ACD so we can use Pythagoras here.

and

We also have another right angled triangle BCD. So we can use Pythagoras on that as well.

and

Notice we have found two equations for CD² so let’s take these two equations and make them equal to each other.

Expand the brackets and simplify to find b².

And do the same for a² to get’

Let’s look at the triangle ACD again, notice that;

…which means;

Substitute this into the a² equation we found above and we shall get;


The above is the first version rule of the cosine rule.
To find the second rule we just replace the x value in the b² equation as we have just done to get;

So the cosine rules corresponding to the following triangle shape are;


Let’s try finding sides using the above rules below;

Finding sides

In the following examples we shall be finding the value of x when it is opposite the angle and when x is adjacent to the angle;

Example 1

Find the value of x (Opposite the angle)

Let’s start by labelling the triangle then use the cosine rule;

The cosine rule as we discovered above is;

We substitute in the know values;

Now we solve to find the value of x. Remember you have to take into account x². We only want the value of x.

Example 2

Find the length of x (adjacent to the angle)

First we label the triangle as we did in the first example. Make good practice of it.

The cosine rule formula to use is;

We substitute in the known values into the formula;

Here we can use the quadratic formula to find the value of x.


We could have also found the value of x by using the sine rule to find the angle C and then find B using angles in a triangle and then finally with B as the angle;

Finding sides

Above we have been finding the sides on the triangle. Now we shall explore finding the angles of a triangle.
If we know the 3 sides of the triangle we can use the cosine rule to find any of the angles. Here is a triangle with 3 know sides.

Here we can rearrange the cosine rule to make cosA the subject and then undo the cosine. This is the cosine formula;

Rearrange to find cosA;

Now we substitute in the known values in our example to solve;


Now we undo the cosine to find the angle at A.

The angle at A as we have found is 43.5.

Example2

Here is another example for finding the angles using the cosine rule.
Find the θ in this triangle.

First we label the triangle to correspond to the cosine rule;

We use the cosine formula but we need to rearrange it such that cosA is the subject.

Now we substitute in the known values;

The angle at A is 125.1° which is an obtuse angle;