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

Examples

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.

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