cos 2x is just one of the important trigonometric identities provided in trigonometry to find the worth of the cosine trigonometric role for double angles. That is additionally called a twin angle identity of the cosine function. The identity of cos 2x helps in representing the cosine of a link angle 2x in regards to sine and also cosine trigonometric functions, in terms of cosine role only, in regards to sine role only and in regards to tangent function only.

You are watching: Cos(x/2)

cos 2x identity have the right to be derived using various trigonometric identities. Permit us know the cos 2x identification in terms of various trigonometric functions and also its source in information in the complying with sections.

1.What is Cos 2x identification in Trigonometry?
2.Derivation the cos 2x using Angle addition Formula
3.Cos 2x In regards to sin x
4.Cos 2x In terms of cos x
5.Cos 2x In terms of tan x
6.Derivative and also Integral of cos 2x
7.How to apply cos 2x Identity?
8.FAQs top top cos 2x

What is Cos 2x identity in Trigonometry?


Cos 2x is an essential identity in trigonometry which can be express in different ways. It have the right to be express in terms of different trigonometric attributes such as sine, cosine, and also tangent. Cos 2x is among the twin angle trigonometric identities together the edge in factor to consider is a many of 2, that is, the twin of x. Let united state write the cos 2x identity in various forms:

cos 2x = cos2x - sin2xcos 2x = 2cos2x - 1cos 2x = 1 - 2sin2x

*


Derivation that cos 2x using Angle enhancement Formula


We know that cos 2x can be expressed in four various forms. Us will usage the angle enhancement formula because that the cosine function to derive the cos 2x identity. Note that the edge 2x can be created as 2x = x + x. Also, we know that cos (a + b) = cos a cos b - sin a sin b. Us will usage this come prove the identity for cos 2x. Making use of the angle addition formula for cosine function, we have

cos 2x = cos (x + x)

= cos x cos x - sin x sin x

= cos2x - sin2x

Hence, we have actually cos 2x = cos2x - sin2x


Cos 2x In regards to sin x


Now, that we have obtained cos 2x = cos2x - sin2x, we will derive the formula because that cos 2x in terms of sine duty only. Us will usage the trigonometry identification cos2x + sin2x = 1 come prove that cos 2x = 1 - 2sin2x. We have,

cos 2x = cos2x - sin2x

= (1 - sin2x) - sin2x

= 1 - sin2x - sin2x

= 1 - 2sin2x

Hence , we have cos 2x = 1 - 2sin2x in regards to sin x.


Cos 2x In terms of cos x


Just prefer we acquired cos 2x = 1 - 2sin2x, we will derive cos 2x in regards to cos x, the is, cos 2x = 2cos2x - 1. We will use the trigonometry identities cos 2x = cos2x - sin2x and cos2x + sin2x = 1 to prove the cos 2x = 2cos2x - 1. We have,

cos 2x = cos2x - sin2x

= cos2x - (1 - cos2x)

= cos2x - 1 + cos2x

= 2cos2x - 1

Hence , we have cos 2x = 2cos2x - 1


Cos 2x In terms of tan x


Now, that we have obtained cos 2x = cos2x - sin2x, we will derive cos 2x in terms of tan x. We will use a few trigonometric identities and also trigonometric formulas such together cos 2x = cos2x - sin2x, cos2x + sin2x = 1, and tan x = sin x/ cos x. Us have,

cos 2x = cos2x - sin2x

= (cos2x - sin2x)/1

= (cos2x - sin2x)/( cos2x + sin2x)

Divide the numerator and also denominator that (cos2x - sin2x)/( cos2x + sin2x) by cos2x.

(cos2x - sin2x)/(cos2x + sin2x) = (cos2x/cos2x - sin2x/cos2x)/( cos2x/cos2x + sin2x/cos2x)

= (1 - tan2x)/(1 + tan2x)

Hence, we have cos 2x = (1 - tan2x)/(1 + tan2x) in regards to tan x


Derivative and Integral the cos 2x


Derivative the cos 2x can quickly be calculated utilizing the formula d/dx = -asin(ax + b). Similarly, integral of cos 2x deserve to be easily acquired using the formula ∫cos(ax + b) dx = (1/a) sin(ax + b) + C. Therefore, we have d(cos 2x)/dx = -2sin 2x and also ∫cos 2x dx = (1/2) sin 2x + C.

Hence the derivative that cos 2x is -2 sin 2x and also the integral of cos 2x is (1/2) sin 2x + C.


How to use cos 2x Identity?


Cos 2x identity deserve to be provided for solving various math problems. Let united state consider an instance to recognize the application of cos 2x identity. We will determine the value of cos 120° utilizing the cos 2x identity. We know that cos 2x = cos2x - sin2x and sin 60° = √3/2, cos 60° = 1/2. Due to the fact that 2x = 120°, x = 60°. Therefore, we have

cos 120° = cos260° - sin260°

= (1/2)2 - (√3/2)2

= 1/4 - 3/4

= -1/2

Important note on cos 2x

cos 2x = cos2x - sin2xcos 2x = 2cos2x - 1cos 2x = 1 - 2sin2xcos 2x = (1 - tan2x)/(1 + tan2x)The derivative of cos 2x is -2 sin 2x and the integral of cos 2x is (1/2) sin 2x + C.

Related topics on cos 2x


Examples using cos 2x


Example 1: Prove the triple angle identification of cosine role using cos 2x formula.

Solution: The triple angle identification of the cosine duty is cos 3x = 4 cos3x - 3 cos x

To begin with, us will usage the angle enhancement formula of the cosine function.

cos 3x = cos (2x + x) = cos 2x cos x - sin 2x sin x

= (2cos2x - 1) cos x - 2 sin x cos x sin x

= 2 cos3x - cos x - 2 sin2x cos x

= 2 cos3x - cos x - 2 cos x (1 - cos2x)

= 2 cos3x - cos x - 2 cos x + 2 cos3x

= 4 cos3x - 3 cos x

Answer: Hence, we have actually proved cos 3x = 4 cos3x - 3 cos x making use of the cos 2x formula.


Example 2: Express the cos 2x formula in terms of cot x.

Solution: We understand that cos 2x = (1 - tan2x)/(1 + tan2x) and also tan x = 1/cot x

cos 2x = (1 - tan2x)/(1 + tan2x)

= (1 - 1/cot2x)/(1 + 1/cot2x)

= (cot2x - 1)/(cot2x + 1)

Answer: Hence, cos 2x = (cot2x - 1)/(cot2x + 1) in regards to cotangent function.


View prize >

go to slidego to slide



Have questions on straightforward mathematical concepts?
Become a problem-solving champ utilizing logic, no rules. Find out the why behind math with our certified experts

Book a free Trial Class


Practice concerns on cos 2x


Check price >

go come slidego to slide


FAQs top top cos 2x


What is cos 2x identity in Trigonometry?

Cos 2x is one of the dual angle trigonometric identities as the edge in factor to consider is a lot of of 2, that is, the double of x. It deserve to be express in state of various trigonometric functions such together sine, cosine, and also tangent.

What is cos 2x Formula?

It can be express in state of different trigonometric attributes such together sine, cosine, and tangent. It have the right to be express as:

cos 2x = cos2x - sin2xcos 2x = 2cos2x - 1cos 2x = 1 - 2sin2x

What is the Derivative the cos 2x?

The derivative that cos 2x is -2 sin 2x. Derivative the cos 2x can easilty it is in calculated using the formula d/dx = -asin(ax + b)

What is the Integral the cos 2x?

The integral that cos 2x have the right to be easilty acquired using the formula ∫cos(ax + b) dx = (1/a) sin(ax + b) + C. Therefore, we have actually ∫cos 2x dx = (1/2) sin 2x + C.

What is cos 2x In regards to sin x?

We have cos 2x = 1 - 2sin2x in terms of sin x.

What is cos 2x In regards to tan x?

We have cos 2x = (1 - tan2x)/(1 + tan2x) in regards to tan x.

See more: Fresh Turkey In Fridge For A Week, Did You Buy Your Fresh Turkey Too Soon

How to have cos 2x Identity?

Cos 2x identity can be acquired using different identities such together angle sum identity of cosine function, cos2x + sin2x = 1, tan x = sin x/ cos x, etc.