Law of Cosines

The law of cosines or Cos law is one of the trigonometric laws which is used to find the angles or sides of a triangle. Once all the three sides of the triangle(SSS) are given, the angles can be easily computed using the cos law. Similarly, cosine laws can be used to calculate the sides of the triangle, where the values of angles will be given already. Here, we are going to discuss the definition, formulas, proof, and examples of the law of cosines are given in detail.

Table of Contents:

Law of Cosines Definition

In Trigonometry, the law of Cosines, also known as Cosine Rule or Cosine Formula basically relates the length of the triangle to the cosines of one of its angles.

Law of cosines

Law of cosines

Law of Cosines Formula

As per the cosines law formula, to find the length of sides of triangle say △ABC, we can write as;

  • a2 = b2 + c2– 2bc cos (C)
  • b2 = a2 + c2 – 2ac cos (A)
  • c2 = a2 + b2 – 2bc cos (B)

And if we want to find the angles of △ABC, then the cosine rule is applied as;

  • cos C= (b2 + c2 – a2)/2bc
  • cos A = (a2 + c2– b2)/2ac
  • cos B = (a2 + b2– c2)/2ab

Where a, b and c are the lengths of legs of a triangle.

Summary for Cos Law Formulas:

To Find Length To Find Angles
a2 = b2 + c2– 2bc cos(C) cos C= (b2 + c2 – a2)/2bc
b2 = a2 + c2 – 2ac cos(A) cos A = (a2 + c2– b2)/2ac
c2 = a2 + b2 – 2bc cos(B) cos B = (a2 + b2– c2)/2ab

Also Check: Law of cosines calculator

Cosines Law Proof

Now let us learn the law of cosines proof here;

Law of Cosines Proof

Law of Cosines Proof

In the right triangle BCD, by the definition of cosine function:

cos C = CD/a


CD=a cos C

Subtracting above equation from side b, we get

DA = b − acosC ……(1)

In the triangle BCD, according to Sine definition

sin C = BD/a


BD = a sinC ……(2)

In the triangle ADB, if we apply the Pythagorean Theorem, then

c2 = BD2 + DA2

Substituting for BD and DA from equations (1) and (2)

c2 = (a sin C)2 + (b-acosC)2

By Cross Multiplication we get:

c2 = a2 sin2C + b2 – 2abcosC + a2 cos2C

Rearranging the above equation:

c2 = a2 sin2C + a2 cos2C + b2 – 2ab cosC

Taking out a2 as a common factor, we get;

c2 = a2(sin2C + cos2C) + b2 – 2ab cosC

Now from the above equation, you know that,

sin2θ + cos2θ = 1

∴ c2 = a2 + b2 – 2ab cosC

Hence, the cosine law is proved.

Law of Cosines Example

Let us understand the concept by solving one of the cosines law problems.

Problem: A triangle ABC has sides a=10cm, b=7cm and c=5cm. Now, find its angle ‘x’.


Consider the below triangle as triangle ABC, where,




Law of Cosine Problems

Law of Cosine Problems

By using cosines law,

a2 = b2 + c2 – 2bc cos(x)


cos x = (b2 + c2 – a2)/2bc

Substituting the value of the sides of the triangle i.e a,b and c, we get

cos(x) = (72 + 52 – 102)/(2 × 7 × 5)

cos(x)=(49 + 25 -100)/70

cos(x)= -0.37

It is important to solve more problems based on cosines law formula by changing the values of sides a, b & c and cross-check law of cosines calculator given above.

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