[Maths Class Notes] on Cos 2x Formula Pdf for Exam

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.

Pythagoras Theorem

According to the Pythagoras theorem, 

In a right-angled triangle,

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. 

1. The trigonometric formula of cos2x = Cos²x – Sin²x

2. The trigonometric formula of [cos2x = 1 – 2Sin^{2}x ]

3. The trigonometric formula of [cos2x = 2Cos^{2}x – 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

1. 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. 

2. 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.