Law of Cosines

This is an extension on our previous lesson of law of cosines . We shall look at how the law of cosines is applicable when the angle is acute and work out an example.

Proving law of cosines when angle is acute

There are 3 scenarios when the angle C is acute.

First, we drop a perpendicular from A down to the base of the triangle.

In figure 1, the angle at B is acute when B is left of D. Similarly, in figure 2, the angle at B is obtuse when B is right of D. In the last scenario, the perpendicular is dropped to B, forming a right angle triangle. We shall now prove the law of cosines.

Let h denotes the height of the triangle, d denotes BD and e denotes CD.

ratios to prove the law of cosines

Scholar's Tip: We usually label the lengths in small letters and they are directly opposite their angles.

From the diagrams, we can make out the following relationships.

cosine C =

When D lies on BC in figure 1, then d = a e. When D lies on the left of B as seen

in figure 2, then d = e a. As such, we square both sides of the equation to obtain

formula to derive the law of cosines

= (d-e)(d+e) +

= a(a-e) +


= final result for the law of cosines


Practise applying the law of cosines to the problem below.

A triangle ABC has lengths AB = 4 , AC = 6 and BC = 5 . Find the angle ABC.

Substituting it into the law of cosines,

Hope you have understood the theories on law of cosines.

Find out about the law of sines.

Return to Trigonometry Help or Basic Trigonometry .