trigonometry math

Start Your Free Trial

how to write cos 2x in terms of cos x?

Expert Answers info

ishpiro eNotes educator | Certified Educator

calendarEducator since 2013

write600 answers

starTop subjects are Math, Science, and Literature

The most straightforward way to obtain the expression for cos(2x) is by using the "cosine of the sum" formula: cos(x + y) = cosx*cosy - sinx*siny.

To get cos(2x), write 2x = x + x. Then,

cos(2x) = cos(x + x) = cosx*cosx - sinx*sinx = cos² (x) - sin² (x)

Then, from Pythagorean Identity, we can get express sine in terms of cosine:

sin² (x) + cos² (x) = 1

sin² (x) = 1 - cos² (x)

Plugging this into the formula for cos(2x), we get

cos(2x) = cos² (x) - (1 - cos²(x)) = 2cos²(x) - 1.

Alternatively, we could get the expression for cos(2x) in terms of sin(x):

cos(2x) = (1 - sin²(x)) - sin²(x) = 1 - 2sin²(x)

So, there are three formulas for the cosine of the double angle:

cos(2x) = cos²(x) - sin²(x)

= 1 - 2sin²(x)

= 2cos²(x) - 1.

The last one writes the cosine of 2x in terms of cosine of x.

check Approved by eNotes Editorial
justaguide eNotes educator | Certified Educator

calendarEducator since 2010

write12,544 answers

starTop subjects are Math, Science, and Business

We know that the expression for cos ( x + y) is:

cos (x + y) = cos x * cos y - sin x* sin y

Now substituting x for both x and y we get

cos ( x + x) = cos x * cos x - sin x * sin x

=> cos 2x = (cos x)² - (sin x)²

Now we use the relation (cos x)² + (sin x)² = 1 which gives (sin x)² = 1 - (cos x)²

So we can eliminate (sin x)² and get

=> cos 2x = (cos x)² - 1 + (cos x)²

=> cos 2x = 2*( cos x)² - 1

Therefore in terms of cos x , cos 2x = 2*( cos x)² - 1

check Approved by eNotes Editorial

bouch96 | Student

cos2x = cos (x+x)

cosxcosx - sinxsinx

(cosx)^2-(sinx)^2

Since (sinx)^2 = 1-(cosx)^2

Therefore: = (cosx)^2 -(1-(cosx)^2)

                = (cosx)^2 -1+(cosx)^2

                = 2(cosx)^2 -1

 

giorgiana1976 | Student

We'll write cos 2x as the cosine of the sum of 2 like angles:

cos(x+x) = cos x*cos x - sin x*sin x

cos(x+x) = (cos x)^2 - (sin x)^2 (1)

We'll write sin x in terms of cos x, applyingthe fundamental formula of trigonometry:

(sin x)^2 + (cosx)^2 = 1

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

We'll substitute (2) in (1):

cos(x+x) = (cos x)^2 - [1 - (cos x)^2]]

We'll remove the brackets:

cos 2x = (cos x)^2 - 1+ (cos x)^2]

We'll combine like terms:

cos 2x = 2(cos x)^2 - 1

So,the expression of cos 2x, written in terms of cos x, is:

cos 2x = 2(cos x)^2 - 1

check Approved by eNotes Editorial