Law of Cosines


In this article, we are going to out how the law of cosines came about.

Definition of law of cosines :

law of cosines in a different form

We have to look at the different scenarios where the angle in question C is obtuse, right angle or acute. Only by determining the law of cosines is valid in each of these cases, then can we prove that this law of cosines exists.

Proving law of cosines when angle is obtuse

Using the pythagoras theorem ,

d = a + e

cosine C =

In supplementary equations, the cosine of an obtuse angle is the negative of the cosine of its acute angle. This means that cos (p-?) = - cos ? . Angle BCA is the negation of Angle DCA. This explains for the negative sign in the last equation.

From the first equation,

formula to derive the law of cosines

First, we substitute d = a + e into the equation.

Then, we expand the quadratic function.

Next, we substitute .

Finally, we substitute the e in and obtain the law of cosines.

formula showing the law of cosines

We managed to prove that the cosine rule holds true in the case when the angle is obtuse.

Proving law of cosines when angle is right angle

When C is a right angle, its cosine would be zero. This will reduce the equation from equation showing  law of cosines to the Pythagoras identity ,which is true for all right angle triangles. Thus we have shown that the law of cosines holds true when the angle is a right angle.

Learn more about the law of cosines .

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