How to solve cos⁡(2nx)=cos⁡(x) I am given the function F:R→R , x↦4x(1−x) and I have...

Jay Mckay

Jay Mckay



How to solve cos(2nx)=cos(x)
I am given the function F:RR , x4x(1x) and I have to find all n-cycles of the function. I have already reduced this to the problem of solving cos(2nx)=cos(x) for x[0,π]. Using Mathematica and n{1,2,3,4}, I concluded that the solutions must be {2kπ2n1,2kπ2n+1}[0,π]. Since the equation obviously has 2n solutions and the solution set has 2n elements, it only remains to verify that they are indeed solutions to the equation. I tried using cos(2x)=2cos2(x)1 and proving this by induction on n, but I was not able to.

Answer & Explanation




2022-01-30Added 13 answers

Using the identity
cosAcosB=2sin(BA2)sin(B+A2) ,
we have
which holds iff
sin(2n12x)=0  OR  sin(2n+12x)=0
2n12x=kπ  OR  2n+12x=kπ
x=2kπ2n1  OR  x=2kπ2n+1
for some kZ

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get your answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?