Is cos (Öx) periodic ?

Graph

The graph seems to show that y = cos (Öx) is not periodic. Can we prove that?

Proof

The proof uses the “Method of contradiction”.

Assume that cos (Öx) is periodic,

$ T > 0 s.t. "x ³ 0,

In particular, we choose x = 0 and x = T,

Solve (1) and (2),

where m, n are integers.

In (5), the left hand side is irrational and right hand side is rational (note : m, n Î Z)

\ There is contradiction and the given function is not periodic.