The Law of Cosines is an extraordinary mathematical expression in Geometry that is used to calculate the length of a side opposite to a known angle made by two measured lines. The Pythagoras theorem itself is a particular case of the Cosine Law and only applies to right angle triangles. On the other hand, the Law of Cosines, as well as the Law of Sines, can be used in calculation even if the triangle isn’t a right angle triangle.

Hence, the Law of Cosines is a significant expression in Mathematics that can be used to obtain the length of a side when the angle opposite to it is known or the other way around. Hence, it is a handy and useful expression. The Law of Cosines is also known as the Cosine rule. Let’s see how it is applied in both of these cases.

When we have a known angle and a side of unknown length.

In this case, let’s assume that the three sides of the triangle are labelled as a, b and c. Now let’s assume further that the length of sides a and b are known to us. Also, the angle made by a and b is given.

Now our job is to calculate the length of the third side c which is opposite to the given angle. We all remember the famous Pythagoras theorem because it is one of the fundamental things taught in Primary Schools.

The Pythagoras theorem states that the square of the hypotenuse is equal to the sum of squares of the other two sides. However, it only applies to right angle triangles and is actually a special case of the Law of Cosines.

The following is the law of cosines:

a2=b2+c2-2*bc cosΘ

When the angle Θ is equal to 90°, the term cosΘ evaluates to 0 and hence we get the Pythagoras theorem itself. However, in case the angle is not a right angle, the Law of Cosines comes into play.

We simply need the length of the two known sides, add their squares and then subtract twice the product of b, c, and cosΘ from the sum. This gives us the value of a2.

Lastly, we find the square root of the resulting number, and we have found the length of the unknown third side a.

When we know the length of all three sides and need to find the value of angles

We can, in a serial manner find the value in degrees of all the angles of a triangle of any kind using the second variation of the Law of Cosines. Take a note that this variation is merely a rearrangement of the above form.

In this one, we have to find the value of Θ and are provided with the length values for a, b and c.

We can do so by just substituting the values of a, b and c in the following equation:

Cos Θ= (a2+b2c2)/2ab

In the above equation, c is the side opposite to angle Θ, and the equation should be accordingly rearranged for the other two sides. We just have to swap c with a if we need to find the angle opposite to the latter.

Quench your Curiosity

To prove the Law of Cosines, almost all methods invariably use the Pythagoras theorem. Hence there’s often a debate as to which of the two is dependent on the other. It is to be understood that the Law of Cosines only applies in Euclidean Geometry and therefore both are interdependent.  

Whenever we differ the plane from one that is confirming of Euclidean metric, the Law of Cosine can’t hold true and other expressions come into play.