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.
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
= (d-e)(d+e) +
= a(a-e) +
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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