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Double Angle Formula for Cosine

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A trigonometric ratio is the ratio of the lengths of any two sides of a right triangle. These ratios may be used to compute the sides of a right triangle as well as the angles created between them. The cosine ratio is calculated by computing the ratio of the length of the adjacent side of an angle divided by the length of hypotenuse. It is denoted by the abbreviation cos.

 

If θ is the angle that lies between the base and hypotenuse of a right-angled triangle then,

cos θ = Base/Hypotenuse = BC/AC

Cos Double angle formula

In trigonometry, cos 2x is a double angle identity. Because the cos function is a reciprocal of secant function, it may also be represented as cos 2x = 1/sec 2x. It’s a significant trigonometric identity that may be used to a variety of trigonometric and integration problems. The value of cos 2x repeats after every Ï€ radians, cos 2x = cos (2x + Ï€). It has a considerably narrower graph than cos x. It’s a trigonometric function that returns the cos function value of a double angle.

cos 2x = cos2 x – sin2 x

The above formula can be simplified further by using sine cosine identity.

Putting sin2 x = 1 – cos2 x, the formula becomes,

cos 2x = cos2 x – (1 – cos2 x)

cos 2x = 2 cos2 x – 1 

Putting cos2 x = 1 – sin2 x, the formula becomes,

cos 2x = (1 – sin2 x) – sin2 x

cos 2x = 1 – 2 sin2 x

Derivation

The formula for cos 2x can be derived by using the sum angle formula for cosine function.

We already know, cos (A + B) = cos A cos B – sin A sin B

To calculate the value of cosine double angle, the angle A must be equal to angle B.

Putting A = B we get,

cos (A + A) = cos A cos A – sin A sin A

cos 2A = cos2 A – sin2 A

This derives the formula for double angle of cosine ratio.

Sample Problems

Problem 1. If cos x = 3/5, find the value of cos 2x using the formula.

Solution:

We have, cos x = 3/5.

Clearly, sin x = 4/5.

Using the formula we get,

cos 2x = cos2 x – sin2 x

= (3/5)2 – (4/5)2

= 9/25 – 16/25

= -7/25

Problem 2. If cos x = 12/13, find the value of cos 2x using the formula.

Solution:

We have, cos x = 12/13.

Clearly, sin x = 5/13.

Using the formula we get,

cos 2x = cos2 x – sin2 x

= (12/13)2 – (5/13)2

= 144/169 – 25/169

= 119/169

Problem 3. If sin x = 3/5, find the value of cos 2x using the formula.

Solution:

We have, sin x = 3/5.

Clearly cos x = 4/5.

Using the formula we get,

cos 2x = cos2 x – sin2 x

= (4/5)2 – (3/5)2

= 16/25 – 9/25

= 7/25

Problem 4. If tan x = 12/5, find the value of cos 2x using the formula.

Solution:

We have, tan x = 12/5.

Clearly sin x = 12/13 and cos x = 5/13.

Using the formula we get,

cos 2x = cos2 x – sin2 x

= (5/13)2 – (12/13)2

= 25/169 – 144/169

= -119/169

Problem 5. If sec x = 17/8, find the value of cos 2x using the formula.

Solution:

We have, sec x = 17/8.

Clearly cos x = 8/17 and sin x = 15/17.

Using the formula we get,

cos 2x = cos2 x – sin2 x

= (8/17)2 – (15/17)2

= 64/289 – 225/289

= -161/225

Problem 6. If cot x = 15/8, find the value of cos 2x using the formula.

Solution:

We have, cot x = 15/8.

Clearly cos x = 15/17 and sin x = 8/17.

Using the formula we get,

cos 2x = cos2 x – sin2 x

= (15/17)2 – (8/17)2

= 225/289 – 64/289

= 161/225

Problem 7. If cos2 x = 5/8, find the value of cos 2x using the formula.

Solution:

We have,

cos2 x = 5/8

Using the formula we get,

cos 2x = 2 cos2 x – 1

= 2 (5/8) – 1

= 5/4 – 1

= 1/4

Problem 8. If sin2 x = 6/7, find the value of cos 2x using the formula.

Solution:

We have,

sin2 x = 6/7

Using the formula we get,

cos 2x = 1 – 2 sin2 x 

= 1 – 2 (6/7)

= 1 – 12/7

= -5/7



Last Updated : 10 Jan, 2024
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