Cosine Rule: Laws, Formula & Proof

Cosine Rule is a square of the length of any side of a triangle, equal to the sum of the squares of the length of the other sides and minus twice the product of the other two sides multiplied by the cosine of the angle.

• These rules are called Cosine law or Cosine rule formula.
• However, cosine rules can be used when either three sides of the triangle are given or two sides of angles are given.
• The cosine rule, also known as the law of cosines, helps to determine the length of one side of a triangle when the length of the other two sides and the angle between them are known.

The Cosine rule further claims that:

Cosine rule is a useful tool that has wider applications in trigonometry, navigation, and other fields where triangles are used. The formula can be written in a number of ways, but a common form is:

c² = a² + b² – 2ab cos(C)

Here,

• c = length of the unknown side
• a and b = lengths of the other two sides
• C = angle between a and b.

Also Read: Trigonometric Functions

Keyterms: Cosine, Traingle, Angle,  Pythagoras theorem, Trigonometry ratio

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What is Cosine Rule?

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The cosine rule is a formula which helps in relating the sides and angles present in a triangle. It further also claims that “the square of one side of a triangle is equal to the sum of the squares of the other two sides minus twice the product of the two sides and the cosine of the angle between them.”

Suppose if a, b and c are lengths of the side of a triangle ABC, then the cosine rule formula states that:

Where,

• ∠x is the angle between sides b and c.
• Similarly, ∠y and ∠z is the angle between ca and ab

Hence, by using the cosine rule we can calculate the length of the side of a triangle.

What is Cosine Rule?

Cosine Formula

The formula required in order to determine the sides of the triangle using cosine rule is given below:

• \(\begin{array}{l}a = \sqrt{b^2 + c^2 – 2~b~c~ cos x}\end{array}\)
• \(\begin{array}{l}b = \sqrt{a^2 + c^2 – 2~a~c~ cos y}\end{array}\)
• \(\begin{array}{l}c = \sqrt{a^2 + b^2 – 2~a~b~cos z}\end{array}\)

Sine Formula

According to the sine law, 

a / Sin A= b/ Sin B= c / Sin C

Here,

a,b and c are the triangle’s sides and A, B and C are its respective angles. 

It can also be written as:

→ a : b: c = Sin A: Sin B: Sin C

Also check:

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Laws of Cosine

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Let's take the example: 

Suppose we have the a,b, & c as the length of the side of triangle ABC. Hence, as per the formula:

a² = b² + c² – 2bc cos x

b² = a² + c² – 2ac cos y

c² = a² + b² – 2ab cos z

However to find the angles x, y and z, these formulae can be written as :

• cos x = (b² + c² -a²)/2bc
• cos y = (a² + c² -b²)/2ac
• cos z = (a² + b² – c²)/2ab

Cosine Rule Proof

As per the Cosine Rule, Triangle ABC with side a,b, & c we have,

c² = a² + b² – 2ab cos C

Now let us prove this law.

Let's take a triangle ABC with the vertex of angle B, we draw a perpendicular touching the side AC at point D. This is the height of the triangle denoted by h.

After that, as per the trigonometry ratio, we know the triangle BCD;

→ cos θ = Base/Hypotenuse

i.e.

cos C = CD/a

or we can write;

CD = a cos C ………… (1)

Let's subtract equation 1 from side b on both sides, and then we get;

→ b – CD = b – a cos C

or

→ DA = b – a cos C

Again, as per the trigonometry ratio, we know the triangle BCD:

sin θ Perpendicular/Hypotenuse

i.e.

→ sin C = BD/a

or we can write;

BD = a sin C ……….(2)

By considering Pythagoras' theorem in triangle ADB, we get;

→ Hypotenuse² = Perpendicular² + Base²

i.e.

c² = BD² + DA²

Let's substitute the value of DA and BD from equations 1 and 2;

→ c² = (a sin C)² + (b – a cos C)²

→ c² = a² sin²C + b² – 2ab cos C + a² cos² C

→ c² = a² (sin²C + cos² C) + b² – 2ab cos C

By trigonometric identities, we know;

→ sin²θ + cos²θ = 1

Therefore,

c² = a² + b² – 2ab cos C

Hence, proved.

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Solved Example of Cosine Rule

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Example: What will be the angle of triangle ABC if AB = 42cm, BC = 37cm and AC = 26cm?

Solution: As per the question we have following given things:

a = 37

b = 26 and

c = 42 

Formula of cosine rule: a² = b² + c² − 2bc cos A

So,

37² = 26² + 42² − 2(26)(42) cos A 

cos A = 26² + 42² − 37² /(2)(26)(42)

After solving the cos A we get the value of cos A as

cosA= 1071/2184 

Which is equal to

Cos A= 0.4904

Thus,

A = cos − 1 0.4904 = 60.63^o

Trigonometric Functions Video Explanation

Also check:

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Things to Remember

• To find the length of sides of triangle ABC, we can write as; a² = b² + c² – 2bc cos α, b² = a_² + c² – 2ac cos β and c² = b² + a² – 2ba cos γ
• And if we want to find the angles of ABC, then the cosine rule is applied as; cos α = [b² + c² – a²]/2bc, cos β = [a² + c² – b²]/2ac and cos γ = [b² + a² – c²]/2ab
• The sine rule: a/sin A = b/sin B = c/sin C
• The Cosine rule is also known as the Law of Cosines.
• Cosine Rule is a formula which helps relate the sides and angles of a triangle.

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