Show that the function f(x)=\cos(x)+\cos(x\sqrt{2}) is not periodic.

Nyah Conrad

Nyah Conrad

Answered question

2022-03-01

Show that the function
f(x)=cos(x)+cos(x2)
is not periodic.

Answer & Explanation

greikkar5bu

greikkar5bu

Beginner2022-03-02Added 5 answers

Assume cos(x)+cos(2x)=cos(x+T)+cos(2x+2T)
Let x=0. Then
2=cosT+cos2T
Since the cosine is at most one, this means that simultaneously cosT=1 and cos2T=1. This is equivalent to:
T=2πn, for some nZ
2T=2πm, for some mZ
Substitute T from the first into the second:
2(2πn)=2πm
2=mn
So 2Q which is a contradiction, or T=0. Either way, you're done.

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