Cosine Rules

Laws of Cosine:

Cosine of an angle is a fundamental trigonometric ratio which relates the sides and angles of a right triangle. Cosine of an angle is defined as the ratio of the side adjacent to the reference angle and the length of the hypotenuse. There are certain laws or rules that relate the sides and angles of a triangle in terms of cosine trigonometric function. These rules are called Cosine rule formula or Cosine law. If a, b and c are the sides of the triangle and A, B and C are the angles of the triangle. Then the cosine rule formula states that:


a2 = b2 + c2 - 2 . bc . Cos A

b2 = c2 + a2 - 2 . ca . Cos B

c2 = a2 + b2 - 2 . ab . Cos C


What is the Cosine Rule Formula?

Statement:

The cosine rule states that the square on any one side of a triangle is equal to the difference between the sum of the squares on the other two sides and twice the product of the other two sides and cosine of the angle opposite to the first side. 


Explanation:

Consider the triangle shown in the above figure. In this triangle, ABC, a, b and c are the sides. According to the cosine rule proof, the square of the side BC i.e. ‘a’ is given as the product of the side ‘b’, side ‘c’ and the cosine of the angle A subtracted from the sum of the squares of the sides ‘b’ and ‘c’. It is Mathematically written as:

a2 = b2 + c2 - 2 . bc. Cos A

Laws of Cosine Trigonometric Function:

According to the laws of cosine, any one side of the triangle can be found when the other two sides and the angle opposite to the unknown side is given. The equations for the sides k, l and m of a triangle taken in order when ∠K, ∠L and ∠M are the angles opposite to the sides k, l, and m respectively  are given as:

k2 = l2 + m2 - 2 . lm. Cos K

l2 = m2 + k2 - 2 . mk. Cos L

m2 = k2 + l2 - 2 . kl. Cos M


Similarly, the cosine rule for angles of a triangle when all the three sides are given can be found as


Cos K = (l2 + m2 - k2) / 2lm

Cos L = (m2 + k2 - l2) / 2mk

Cos M = (k2 + l2 - m2) / 2kl

Cosine Rule Proof:

Data: 

Let us consider the triangle ABC as shown in the figure below. The sides of the triangle AB, BC and AC measure ‘c’, ‘a’ and ‘b’ units respectively. 


To Prove: 

c2 = a2 + b2 - 2 . ab . Cos C


Construction:

Draw BD perpendicular to AC at the point D

Cosine Rule Proof:

Statement

Reason

△BCD is a right triangle

Cos C =\[\frac{CD}{BC}\]

Cos C = \[\frac{CD}{a}\]

CD = a Cos C → (1)

Cosine of an angle is the ratio of adjacent side and hypotenuse in a right triangle.

b - CD = b - a Cos C 

Subtracting both sides of the equation from b

AD = b - a Cos C → (2)

From the figure, b - CD = AC - CD = AD

In the right triangle BCD, 

Sin C = \[\frac{BD}{BC}\]

Sin C = \[\frac{BD}{a}\]

BD = a Sin C → (3)

Sine of an angle is the ratio of its opposite side to the hypotenuse in a right triangle.

AB2 = BD2 + AD2 → (4)

Applying Pythagoras Theorem to △BAD

c2 = (a Sin C)2 + (b - a Cos C)2

c2 = a2 Sin2C + b2 + a2 Cos2C - 2ab Cos C

c2 = a2 (Sin2C + Cos2C) + b2 - 2ab Cos C

Substituting (2) and (3) in (4)

c2 = a2 (1) + b2 - 2ab Cos C

c2 = a2 + b2 - 2ab Cos C

Sin2C + Cos2C = 1



Cosine Rule Examples:

  1. In the triangle shown below, find the length of c.


Solution:

Given data: a = 8 cm, b = 11 cm and ∠C = 370

Cosine rule states that   

c2 = a2 + b2 - 2ab Cos C

c2 = 82 + 112 - 2 x 8 x 11 Cos 370

c2 = 64 + 121 - 176 x 0.8

c2 = 185 - 140.8

c2 = 44.2

c = ±√44.2

c = ± 6.65

However, length cannot have a negative value. So, c = 6.65 units.

Fun Facts:

  • Though the concept of cosine rule did not exist in 3rd century BC, there are evidence of concepts similar to cosine rule examples in the Mathematical works of Euclid, the father of geometry.

  • Laws of cosine are used in solving triangles and circles by a Mathematical procedure called triangulation.

FAQ (Frequently Asked Questions)

1. How Can We Prove Cosine Rule?

Cosine rule is one of the laws in trigonometry which relates the sides and angles of any triangle. There are a number of popular methods by which the laws of cosine can be proved to be correct. The few most popular and most commonly used concepts to prove cosine rule examples are:

  • Proof for the cosine rule using formula for distance between two coordinate points (Distance formula)

  • Proof of cosine rule for angles and sides of a triangle can be obtained using the basic concepts of trigonometry

  • Cosine rule can be proved using Pythagorean theorem under different cases for obtuse and acute angles.

  • Ptolemy’s theorem can also be used to prove cosine rule.

  • Cosine rule can also be derived by comparing the areas and using the geometry of a circle.

  • Laws of cosine can also be deduced from the laws of sine is also possible.

2. How are the Sides and Angles of a Triangle Determined Using Cosine Rule?

Cosine rule states that in any triangle, the square on one side is equal to the product of its other two sides and Cosine of the angle between the other two sides subtracted from the sum of the squares of the other two sides. So, any side of a triangle can be determined when two sides and the angle between those two sides are given.

a2 = b2 + c2 - 2 . bc . Cos A

b2 = c2 + a2 - 2 . ca . Cos B

c2 = a2 + b2 - 2 . ab . Cos C

Similarly the angle between any two sides is equal to the inverse cosine function of the quotient obtained when the difference between the third side and the sum of the two sides is divided by twice the product of the two sides. Any of the angles of a triangle can be determined when all the three sides are given by using cosine rule for angles.

∠A = Cos-1 [(b2 + c2 - a2) / 2bc]

∠B = Cos-1 [(c2 + a2 - b2) / 2ca]

∠C = Cos-1 [(a2 + b2 - c2) / 2ab]