Cosine Formula: Proof, Applications, Quadrants & Laws

Cosine formula, also known as the law of cosines, cosine rule and al-Kashi’s theorem, denotes the lengths of the sides of a triangle with respect to the cosine of one of its angles.

• This law was developed in the 15th century by a Persian mathematician and astronomer, called Jamshid al-Kashi. The cosine function is one of the primary functions out of the six functions in trigonometry.
• The cosine formula is derived from this function. It is based on the Pythagoras theorem and states that it is the ratio of the adjacent to the hypotenuse. 
• The Cosine Rule states that, in a triangle, the square of the length of any of its sides is equivalent to the sum of the squares of the length of the other sides minus twice the product of the other two sides multiplied by the cosine of angle included between them. 

For example, consider that a, b and c are lengths of the side of a triangle ABC. Now, for that matter, we have:

• a² = b² + c² – 2bc cos ∠x
• b² = a² + c² – 2ac cos ∠y
• c² = a² + b² – 2ab cos ∠z

Here, ∠x, ∠y and ∠z are known to be the angles between the sides of the triangle.

Read More: NCERT Solutions for Class 11 Mathematics Trigonometric Functions

Key Terms: Cosine, Cos, Trigonometric Functions, Trigonometric Identities, Trigonometry, Cot, Sine, Tan, Cosec, Trigonometry Table, Hypotenuse, Adjacent Side, Pythagoras Theorem

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What is Cosine?

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The cosine function is commonly known as the cos function. The law of cosine, or the cosine rule states that: 

Thus,

→ c² = a² + b² + 2ab cos α

In a right-angled triangle, the cos function is denoted by the ratio of the adjacent side of the triangle to its hypotenuse. 

For example, let us consider a right-angled triangle PQR. 

• PR = hypotenuse
• QR = base
• Acute angle = a

Here, we can say that cos a = adjacent/hypotenuse (hypotenuse is the longest side opposite to the right angle of triangle)

Cos a = QR/PQ

Cos a = base/height

Cos a = b/h

The cosine rule is associated with the lengths of the sides of a triangle with any of its angles being a cosine angle. By means of this, the length of the side of a triangle can be evaluated or the angle between the sides can also be measured.

Trigonometric Functions Detailed Video Explanation

Also Read: Trigonometric Functions

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Laws of Cosine

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Cosine rules in order to determine the length of the sides a, b and c of the triangle ABC can be represented by:

• a² = b² + c² – 2bc cos x
• b² = a² + c² – 2ac cos y
• c² = a² + b² – 2ab cos z

Cosine Rule Triangle

Likewise, to determine the angles x, y and z, the formulae can also be shown as:

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

Cosine Law Proof

The law of cosine implies, for a given triangle, say ABC that has sides a, b and c, we have;

c² = a² + b² – 2ab cos C

Now, to prove the same, we have:

Consider that a triangle ABC is assigned. From the vertex of angle B, we can draw a perpendicular that touches the side AC at point D. It, thus, can be defined as the height of the triangle indicated by h.

In the triangle BCD, according to trigonometry ratio, 

cos C = CD/a       [\(\therefore\) cos θ = Base/Hypotenuse]

It can also be expressed as:

CD = a cos C … (1)

Now, after we subtract equation 1 from side b on both sides, 

b – CD = b – a cos C

or DA = b – a cos C

Now, in triangle BCD, we can see:

sin C = BD/a      [\(\therefore\)sin θ = Perpendicular/Hypotenuse]

It can also be expressed as:

BD = a sin C … (2)

By employing Pythagoras' theorem for the triangle ADB, we can obtain;

c² = BD² + DA²        [Hypotenuse² = Perpendicular² + Base² ]

After replacing the value of DA and BD from equations 1 and 2, we see:

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

As per trigonometric identities, we know;

sin²θ+ cos²θ = 1

Hence,

c² = a²  + b² – 2ab cos C (Hence, proved).

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Cosine Formula

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The formula to determine the sides of the triangle by using the cosine rule:

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

What is Sine Formula?

According to sine law, 

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

Here, a,b and c are the sides of a triangle and A, B and C are their respective angles. 

It can also be expressed as:

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

Cosine Formula Using Cofunction Identities

The cofunction identities express the relation between the cofunctions which are sin, cos; sec, cosec, tan, and cot. By using the cofunction identities,

• cos x = sin (90^o - x) (OR)
• cos x = sin (π/2 - x)

Cosine Formulas Using Sum or Difference Formulas

The Cosine function of the sum/difference formulas include:

• cos(x + y) = cos (x) cos(y) – sin (x) sin (y)
• cos (x – y) = cos (x) cos (y) + sin (x) sin (y)

Cosine Formula of Double Angle

Cosine Formula of Double Angles can be illustrated as,

• cos 2x = cos²(x) – sin²(x)
• cos 2x = 2 cos²(x) − 1 
• cos 2x = 1 – 2 sin²(x)
• cos 2x = [(1 - tan²x)/(1 + tan²x)]

Cosine Formula of Triple Angle

The triple angle formula of the cosine function can be shown as,

→ cos 3x = 4 cos³x - 3 cos x

Cosine Formula of Half Angle

In trigonometry, we also use half-angle formulas which are responsible to deal with half of the angles (x/2). Thus,

→ cos (x/2) =± √[ (1 + cos x) / 2]

Also Check: Right Angle Formula

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Law of Cosines Applications

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There are several applications of the Cosile Formula, including:

In Trigonometry 

The Cosine Rule states that the square of the length of a side of a given triangle is equal to the sum of the squares of the length of the other sides minus twice the product of the other two sides multiplied by the cosine of the angle present between them.

According to cos law, it is used to find the missing sides of a right-angled triangle.

• a² = b² + c² − 2bc.cosA
• b² = a² +c² − 2ac.cosB
• c² = a² + b² − 2ab.cosC

In Pythagorean Identity

One of the trigonometric identities establishes the relationship between sin and cos. It says, sin2x + cos2x = 1, for any x. We can use this method to solve for cos x. 

Consider sin2x + cos2x = 1

Subtracting sin2x from both sides,

cos2x = 1 - sin2x

Taking square root on both sides,

cos x = ± √(1 - sin2x) 

There are other important identities of cos as well. These are, 

• cos2 (x) + sin2 (x) = 1
• cos θ = 1/sec θ
• cos (−θ) = cos (θ)
• arccos (cos (x)) = x + 2kπ  [where k=integer]
• Cos (2x) = cos2 (x) − sin2 (x)
• cos (θ) = sin (π/2 − θ)

Cosine Table in Trigonometry

The cosine table lays down various values associated with different degrees of cos. It helps in arithmetic and geometric calculations.

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Inverse Cosine Formula

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This function is also known as Arccos and is denoted by cos⁻¹. 

For a right triangle, if the given sides are 1, 2, and √3, then the cos function can be effectively used to measure the angle.

cos of angle A, cos(a)= adjacent/hypotenuse.

So, cos(a) = √3/2

Now, the angle “a” will be cos⁻¹(√3/2)

Or, a = π/6 = 30°

Cosine Properties With Respect to the Quadrants

The value of the cos function changes accordingly as the degree varies from 0 to 360. The values of cos are positive in the first and fourth quadrant, whereas they are negative in the second and third quadrant.

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Things to Remember

• Cosine formula Indicates the lengths of the sides of a triangle with respect to the cosine of one of its angles.
• Cosine formula is also known as the Law of Cosines, the Cosine Rule and Al-Kashi’s Theorem.
• The Cosine formulas are, \(\begin{array}{l}a = \sqrt{b^2 + c^2 – 2~b~c~ cos x}\end{array}\), \(\begin{array}{l}b = \sqrt{a^2 + c^2 – 2~a~c~ cos y}\end{array}\) and \(\begin{array}{l}c = \sqrt{a^2 + b^2 – 2~a~b~cos z}\end{array}\).
• Cosine graphs help in finding out the average daily temperature of any location.
• The cosine graph is similar to that of the sine graph. It goes up and then goes down throughout. 

Check Also:

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Previous Year Questions

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• Let P={θ:sinθ−cosθ=2​cosθ} and Q={θ:sinθ+cosθ=2​sinθ} be two sets. Then:….[JEE MAIN 2016]
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