Law of Cosines: Overview, Questions, Preparation

The cosine of an angle is the ratio between its length to the length of the opposite side. It is known as the cosine law. If ABC is a triangle, then because of the cosine theorem, we can assume that: 

a2 = b2 + c2 – 2bc cos α, where b, c, and an are the sides of the triangle, and α is the angle between b and c. 

The Law of Cosines, also known as Cosine Rule or Cosine Formula, compares the length of the triangle to the cosines of one of its three included angles. If the length of the two sides and the angle between them can be known, so the third side can be known. It is defined by: 

c2 = a2 + b2 – 2ab cos γ

The field of triangle ABCD and the degree estimate of the angle γ between BC and BD. 

To find the lengths of the three sides of the triangle, we can write as (△ABC). 

a2 = b2 + c2 – 2bc cos α

b2 = a2 + c2 – 2ac cos β

c2 = b2 + a2 – 2ba cos γ

Similarly, in order to find the angles of △ABC, the cosine rule is used. 

cos α = [b2 + c2 – a2]/2bc

cos β = [a2 + c2 – b2]/2ac

cos γ = [b2 + a2 – c2]/2ab

In a triangle, a, b, and c are lengths of one of the three sides.

Solving SSS congruence

Knowing the lengths of the three sides of a triangle, we need to calculate the length of the triangle whose undefined measure is constant. Therefore, the law of cosines may be used to locate the missing angle. 

If we try to find the right angle, we will first find cos α = [b2 + c2 – a2]/2bc. 

Since finding the second angle, we will use the same rule to find the second angle, cos β = [a2 + c2 – b2]/2ac. 

Another angle that you can locate using the sum of the angles of the triangle's angles. This implies that the sum of the angles of a triangle would exceed 90 degrees.

Law of Cosines in Class 11

This concept is taught under chapter Trigonometric Functions. You will learn about the laws and formulas of the cosine function. The weightage of this chapter is 14 marks in the final exam.

 

A triangle ABC has sides a=12cm, b=8cm and c=6cm. Now, find its angle ‘y’.

 

Solution:

Consider the below triangle as triangle ABC, where

a=12cm

b=8cm

c=6cm

By using cosines law,

a2 = b2 + c2 – 2bc cos(y)

Or

cos y = (b2 + c2 – a2)/2bc

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

cos(y) = (82 + 62 – 122)/(2 × 8 × 6)

cos(y)= (64 + 36 -144)/96

cos(y)= -0.4

Q: What is the cosine law? 

A: The cosine of an angle is the ratio between its length to the length of the opposite side. 

Q: What does the cosine law state? 

A: The cosine law states that if γ, β and α are the angles of a triangle, then γ = β * α. 

a2 = b2 + c2 – 2bc cos α

b2 = a2 + c2 – 2ac cos β

c2 = b2 + a2 – 2ba cos γ

where a, b and c are the salient dimensions of the triangle. 

Q: When and how do we use the rule of cosines? 

A: The cosine law is used to help decide the third side of an equilateral triangle where the lengths of the opposite sides and the angle formed between them are known.   

Q: Can the cosine rule be used on all the triangles? 

A: The Law of cosines was not only limited to right triangles, and can be used to solve a number of problems with unknown sides and unknown angles. 

Q: How can you find the angle using the cosine law? 

A: The straight-line triangle formula for finding unknown angles using cosine law is given by: 

cos α = [b2 + c2 – a2]/2bc

cos β = [a2 + c2 – b2]/2ac

cos γ = [b2 + a2 – c2]/2ab