Cos 2x Formula

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What is Cos 2x Formula? 

In a right triangle, the trigonometric ratios of an angle explain to us 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 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 rest 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,

(Opposite)2+(Adjacent)2= (Hypotenuse)2


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 Cosine of an angle. Now the question is what is the formula of cos2x?

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Cos θ = \[\frac{Adjacent}{Hypotenuse}\]                                                 


The Trigonometric Formula Of Cos2x

Now if you are wondering what the formula of cos2x is, let me tell you that we have 5 cos x formula. 

  1. The trigonometric formula of cos2x = Cos²x - Sin²x

  2. The trigonometric formula of cos2x = 1 - 2Sin²x

  3. The trigonometric formula of cos2x = 2Cos²x - 1  

  4. The trigonometric formula of cos2x = \[\frac{1-tan^{2}x}{1+tan^{2}x}\] 

  5. The trigonometric formula of cos2x = \[\frac{Cos^{2}x-Sin^{2}x}{Cos^{2}x+Sin^{2}x}\] 


Solved Examples

Example 1) Prove Cos3x = 4 Cos³x - 3 Cos x  

Solution 1) We have Cos3x = Cos(2x + x)

Cos 3x =  Cos2x Cosx - Sin 2x Sin x 

Cos 3x = (2Cos²x - 1) Cos x - 2Sinx Cosx Sinx               

Cos3x = (2Cos²x - 1)Cosx - 2Cosx(1-Cos²x) 

Cos3x = 2Cos³x - Cosx - 2Cosx + Cos³x 

Cos3x = 4 Cos³x - 3Cosx 

Hence it is proved.


Example 2) Solve cos 2a = sin a, for – Π ≤  a< Π

Solution 2) We can use  the double angle formula cos 2a = 1 − 2 sin2 a

Therefore, it turns out to  1 − 2 sin2 a = sin a

 2 sin2 a + sin a − 1=0,

factorising this quadratic equation with variable sin x

(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

FAQ (Frequently Asked Questions)

Question 1: What are Trigonometric Functions? 

Answer: 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 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.


(i) Sine Function (sin) sin θ = opposite/hypotenuse


(ii) Cosine Function (cos) cosθ = adjacent/hypotenuse

     

(iii) Tangent Function (tan) tan θ = sin (θ)/cos(θ)    

Question 2: What is a Double Angle Formula?

Answer: 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 angle’ 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) 

                 = 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) = tan A + tan A 1 − tan A tan A 

So that tan 2A = 2 tan A 1 − tan2 A 

These three are the double angle formulae that we should learn.