What is the Cos 2x Formula?
In a right triangle, the trigonometric ratio of an angle explains the relationship that exists between the angle and the length of its sides. But then what is cos 2x? Cosine 2x or Cos 2x formula is also one such trigonometric formula, which is also known as double angle formula. It is called a double angle formula because it has a double angle in it. This is the reason why it is driven by the expressions for trigonometric functions of the sum and difference of two numbers (angles) and related expressions. Now that we know what the cos 2x formula is, we can move ahead and learn some more important things about trigonometry and also know what the formula of cos2x is.
Trigonometry and Right-Angled Triangle
In a right-angled triangle, the hypotenuse, the base(adjacent), and the perpendicular(opposite), i.e., the three sides of a right-angled triangle are from where the trigonometric ratios are derived. There are three primary trigonometric ratios in maths, which are also known as trigonometric identities. We can find the missing angles and missing sides of a right-angled triangle with the help of trigonometric ratios. In a right-angled triangle, one angle is 90 degrees, and the other two angles are 45 degrees each. The three sides of a right-angled triangle are
Hypotenuse: Hypotenuse is opposite to 90 degrees and is the longest side of the triangle.
Perpendicular (opposite): It is the side that is opposite to the unknown angle represented as θ and perpendicular to the base (that is, the angle between the base and the perpendicular is 90 degrees).
Base (adjacent): It is the base on which the triangle rests and it also contains both the angles, i.e., 90 degrees as well as the unknown angle θ.
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Pythagoras Theorem
According to the Pythagoras theorem,
In a right-angled triangle,
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What Cosine Function is?
The ratio of the side that is adjacent to the angle (θ) to the hypotenuse (longest side) in the triangle is defined as the Cosine of an angle. Now the question is what is the formula of cos2x?
Cos θ = Adjacent/Hypotenuse
The Trigonometric Formula Of Cos2x
Now if you are wondering what the formula of cos2x is, let me tell you that we have five cos x formulas.
The trigonometric formula of cos2x = Cos²x - Sin²x
The trigonometric formula of \[cos2x = 1 - 2Sin^{2}x \]
The trigonometric formula of \[cos2x = 2Cos^{2}x - 1 \]
The trigonometric formula of \[cos2x =\frac{1-tan^{2}x}{1+tan^{2}x}\]
The trigonometric formula of \[cos2x =\frac{Cos^{2}x - Sin^{2}x}{Cos^{2}x + Sin^{2}x}\]
Solved Examples
Prove Cos3x = 4 Cos³x - 3 Cos x
Solution: We have Cos3x = Cos(2x + x)
Cos 3x = Cos2x Cosx - Sin 2x Sin x
\[ Cos 3x = (2 Cos^{2}x - 1) Cos x - 2Sinx Cos x Sin x\]
\[ Cos 3x = (2 Cos^{2}x - 1) Cos x - 2Cos x (1-Cos^{2}x)\]
\[Cos3x = 2Cos^{3} x - Cos x - 2Cosx + 2Cos^{3}x \]
\[Cos3x = 4 Cos^{3}x - 3Cosx \]
Hence it is proved.
Solve cos 2a = sin a, for \[-\pi \leq a < \pi \]
Solution: We can use the double angle formula \[ cos 2 a = 1 - 2 sin^{2} a \],
Therefore, it turns out to \[ 1 - 2 sin^{2} a = sin a\]
\[ 2 sin^{2} a + sin a - 1 = 0\],
Factorizing this quadratic equation with variable sin a
(2 sin a − 1)(sin a + 1) = 0
2 sin a − 1 = 0 or sin a + 1 = 0
sin a = 1/2 or sin a = −1
The cos 2x formula is the double angle formula because it is obtained by the trigonometric functional expressions of the sum, as well as of the difference of two numbers, and also the related expression. The article will develop strong basics of trigonometry.
FAQs on Cos 2x Formula
1. What are Trigonometric Functions?
The primary trigonometric functions that we use are cosine, sine, and tangent. Trigonometric ratios such as Cos 60 degree value and others are used for common angles like 0⁰, 30⁰, 45⁰, 60⁰, 90⁰. They are used in trigonometric equations and calculations. Most of the trigonometry calculations that we perform are done by using the 6 trigonometric ratios that are present in trigonometry. Almost every important trigonometry formula can be derived with the help of these ratios. Some of the important trigonometric ratios are named Sin, Cos, and Tan, where Sin and cos are the fundamental or the basic ratios and Tan, sec, cot, and sec are the derived functions. Here are the ratios of Sin, Cos, and Tan.
\[Sine Function (sin), sin \theta = \frac{opposite}{hypotenuse}\]
\[Cosine Function (cos) , cos \theta = \frac{adjacent}{hypotenuse}\]
\[Tangent Function (tan), tan \theta = \frac{sin (\theta)}{cos(\theta)}\]
2. What is a Double Angle Formula?
Formulas that express the trigonometric functions of an angle 2x in terms of functions of an angle x at trigonometric formulae are known as the double angle formulae. They are called ‘double angles’ because they consist of trigonometric functions of double angles, i.e., sin 2A, cos 2A, and tan2A. We can start with the additional formulae of the double angle formulae for sin 2A, cos 2A, and tan 2A.
sin(A + B) = sin A cos B + cos A sin B
cos(A + B) = cos A cos B − sin A sin B
\[tan(A + B) = \frac{tan A + tan B}{1 - tan A . tan B}\]
If we let B equal to A, then the first of these formulae will become:
sin(A + A) = sin A cos A + cos A sin A
So that sin 2A = 2 sin A cos A
This is the first double-angle formula. In the same way, if we put B equal to A in the second addition formula we will have:
cos(A + A) = cos A cos A − sin A sin A
So that cos 2A = cos2 A − sin2 A
This is the second double angle formula.
Again, \[tan(A + A) = \frac{tan A + tan A}{1 - tan A . tan A}\]
So that tan \[ 2A = \frac{2 tan A}{1 - tan^{2}A}\]
These three are the double angle formulae that we should learn.
3. How to derive the Cos2x formula by using the Angle Addition Formula?
There are four different forms in which the cos2x formula can be expressed. For deriving the formula by using the angle addition, we have to use the angle addition formula of the cosine function. Now as you already know the angle 2x can be written as 2x = x + x, and also that cos (a + b) = cos a cos b - sin a sin b. Now, this can be used to substitute a = b = x into the formula for cos (a + b),
Therefore,
cos2x = cos (x + x)
= cos x cos x - sin x sin x
= cos2x - sin2x
And hence, cos2x = cos2x - sin2x
4. What are trigonometric identities?
In Mathematics, any equalities that deal with the trigonometric functions, and which are correct statements of these functions are regarded as trigonometric identities. To put it simply, you can think of trigonometric identities as a definition of the trigonometric functions, because just like the definition of the word explains the word, trigonometric identities explain the trigonometric functions in a similar manner. There are many trigonometric identities and each of them defines a different trigonometric function, and these trigonometric identities can be tangent.