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:
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.
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
= - √2/2
By using the cos3x identity we have determined that the value of cos 135° is:
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