Law of Cosines

Law of cosines and sines are the Trigonometric laws, which are used to find the angles of a triangle. Once we are given all the three sides of the triangle(SSS), we can easily compute the angles. Similarly, these laws can be used to calculate the sides of the triangle, where the values of angles will be given already.

Here you will learn the definition, formula, proof of the law of cosines with solved questions. Student’s can also reach us to learn about the law of sines. Keep reading the article below for further knowledge and to understand the concepts of cosine rule.

Law of Cosines Definition

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

Law of Cosine Formula

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

a2 = b2 + c22 – 2bc cos(x)

b2 = a2 + c2 – 2ac cos(y)

c2 = a2 + b2 – 2bc cos(z)

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

cos x = b2 + c2 – a2/2bc

cos y = a2 + c2– b2/2ac

cos z = a2 + b2– c2/2ab

Proof of Law of Cosines

Now let us learn the law of cosines proof here;

Proof of Law of Cosines

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

cos C = CD/a

or
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=\(\frac{BD}{a}\)

or

BD=asinC ……(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 aas 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 rule proved.

Law of Cosine Problems and Solutions

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

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

Solution: Consider the below triangle as triangle ABC, where,

a=10cm

b=7cm

&c=5cm

Law of Cosine problems

By using cosines law or the law of cosines,

a2 = b2 + c2 – 2bc cos(x)

Or

cos x = (b2 + c2 – a2)/2bc

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

cos(x) = (72 + 52 – 102)/(2 × 7 × 5)

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

cos(x)= -0.37 (Answer)

Try to solve more problems based on cosines formula by changing the values of sides a, b & c and cross-check law of cosines calculator available here with us.

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Practise This Question

Choose the pair of equations which satisfy the point (1,-1)