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Cos 3X Formula

Last updated date: 29th Feb 2024
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Formula of Cos 3X

The formula of cos 3x is a trigonometric triple angle identity that provides the relationship between basic trigonometric functions applied to three times an angle with respect to the trigonometric functions of the angle itself.

What is Cos3x?

Double angle formulas work when it comes to computing the value of a trigonometric function in some cases. But sometimes, other multiples and angles are required. 

Cos3x is an important identity in trigonometry, which is used to determine the value of the cosine function for an angle that is thrice the measure of angle x. It can be expressed in terms of the cos x. The behavior of the function cos3x is similar to that of cos x. As the period of cos x is 2π, the period of cos3x is 2π/3, that is, the cycle of cos3x repeats itself after every 2π/3 radians.

The Cos3 X Formula is used in two Cases:

  1. The Cos of the triple angle is expanded as the subtraction of three times cos of the angle, from four times cos cubed of the angle.

  2. The subtraction of three times cos of angle from four times the angle’s cos cube, is called ‘cos of triple angle’.

Cos3x’s expansion is derived by using the angle addition identity of cosine. It includes the term cos cube x (cos^3x). Cos^3x gives the value of the cube of the cosine function. 

Cos3x and cos^3x formulas help in solving several trigonometric problems.

Cos 3X Formula

The cos 3x formula is expressed as:

Cos 3x = 4 cos³x - 3 cos x 

Cos 3X Formula Derivation

To derive the formula of cos 3x, we can write the formula as cos (2x + x) . Hence, we can use the sum formula and double angle identity to get the desired equation.

Cos 3x = cos ( 2x + x)

= cos 2x cos x - sin 2x sin x (Sum formula for cosine)

= ( 1- 2 sin² x ) cos x - 2sin²x cos x( double-angle identity)

= cos x - 2sin²x cos x - 2sin²x cos x

= cos x - 4sin²x cos x

Using sin² + cos² = 1 , sin² = 1 - cos², we get

cos x - 4sin²x cos x = cos x - 4(1 - cos²x) cos x

= cos x - 4 cos x + 4 cos³ x

= 4 cos³ x - 3 cos x

How to Apply Cos3x Formula?

Let’s consider an example to understand the application of the Cos3x formula. We generally study and memorise the value of the cosine function for some specific angles like 0°, 30°, 45°, 60°, 90°, etc. 

Let’s find the value of cos 135° by applying the cos3x formula. 

If 3x = 135°, x = 135°/3 = 45°. 

So, cos 135° = cos (3 × 45°) = 4 cos3(45°) - 3 cos 45°

= 4 × (1/√2)3 - 3 × (1/√2)

= 2/√2 - 3/√2

= -1/√2

= - √2/2

By using the cos3x identity we have determined that the value of cos 135° is: 

- √2/2.


Theorems and Solutions to all the mathematical questions based on the syllabus for Mathematics is a blessing in disguise. You can find all the theorems and solutions pertaining to trigonometric at one place. Students should go through and look at all the questions carefully to understand the concepts used to solve these questions. This will help the students while preparing for term exams as well as competitive ones. 

FAQs on Cos 3X Formula

1. What are the general functions of triple angle formulas?

The general functions for multiple angle formulas are sine, tangent and cosine. The trigonometric triple-angle identities offer a relationship between the basic trigonometric functions applied to three times an angle, in terms of trigonometric functions of the angle itself.

The triple angle identities for sine, cosine and tangent are:

sin3θ=3sinθ−4sin3 θ


tan(3θ)= 3tanθ−tan3θ/ 1−3tan2θ 

2. What is cosine function?

The cosine function, also known as cos x, cos theta is defined as the ratio of the length of the base to the length of the adjacent side in a right-angled triangle. 

3. What is the law of cosine?

The law of cosine (also known as cosine rule, cosine formula or Al Kashi's theorem) relates the angle of the length of the sides of the right triangle to the cosines of one angle. In other words, it states that if the length of the two sides of a triangle and the angle between them is known then we can find the length of the third sides of a triangle. It is represented by

Here, a, b, and c are the sides of a triangle and y is the angle between them.

4. What is the use of the law of cosine?

The law of cosine is useful for determining the third side of a triangle when two sides and the angle between them is known and is also used to calculate the angle of a triangle if all the three sides of a triangle are known. 

5. Who invented Calculus?

Calculus was developed by mathematicians, Gottfried Leibniz and Isaac Newton, during the second half of the 17th century. Newton was the first to develop calculus and apply it directly to the understanding and exploration of physical systems. Leibniz, on the other hand, developed the notations used in calculus. Mathematicians use calculus to calculate exact numbers related to the area, volume, center of mass, pressure, and arc length. 

Basic maths use operations like plus, minus, times, and division, but calculus uses operations such as functions and integrals to calculate the rates of change.