We are able to use the cosine rule to find the length of a side or a particular angle. We label the triangle up in exactly the same way as we did for the sine rule; sides are labelled with lower case letters and the angle opposite the side will be labelled with the same letter, but in capital form.

Here is the cosine rule:

The form above is perfect if we are looking for the length of a side. However, we may be asked to find the length of an angle, which would be the size of A. Whenever we are looking for an angle, it is easier to rearrange the formula above to make cosA the subject. To make cosA the subject, we need to isolate 2

*bc*cosA and the best way to isolate it is to move it to the other side of the equation so that we have positive 2*bc*cosA, rather than negative 2*bc*cosA. We move the 2*bc*cosA to the other side by adding 2*bc*cosA to both sides of the equation.The next step is to get rid of the *a*^{2}, which we do by taking *a*^{2} from both sides of the equation.

The final step is to divide by 2

*bc*.We now have our formula if we are working out the size of an angle.

Let’s now have a few examples.

Let’s now have a few examples.

**Example 1 – Finding a Length**

Find the length of

*x*for the triangle below. Give your answer to 1 decimal place.

The first step in answering this question is to label all of the sides and angles that we are either given or want to find. The best way to start labelling the triangle is with the angle. This is because the cosine rule only involves one angle. Therefore, the angle involved is going to be labelled as A and the side opposite the angle will be labelled as a; remember that we label angles in capital letters and sides in lower case letters. The other two sides that we are given in the question will be labelled as b and c; it does not matter which side is labelled a and which side is labelled b, so I am going to label the side that has a length of 9 b and the side that has a length of 6 c.

We are now able to sub these values into the cosine rule and work out what the value of

*x*is.We want to find the value of *x* and not *x*^{2}. Therefore, we need to square root both sides of the equation.

The question asks us to give the answer to one decimal place.

Therefore,

*x*is 7.2 units.