Cos 3X Formula

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.


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

FAQs (Frequently Asked Questions)

Q1. What is Cosine Function?

Ans. 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.

Q2. What is the Law of Cosine?

Ans. 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

Q3. What is the Use of the Law of Cosine?

Ans. The law of cosine is useful for determining the third side of a triangle when two sides and 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.