this is from Digital Signal Processing, 4th ed, Sanjit K Mitra, problem 2.39b.
The question is: Determine the fundamental period of
$x[n] = cos(0.6n\pi + 0.3\pi)$
Since x[n], is in square brackets, this means discrete time, not continuos time, thus the answer must be an integer.
The test is: $cos(0.6n\pi + 0.3%pi)$ = $cos(0.6(n+Nk)\pi + 0.3\pi)$
Where N is a positive integer (and the value we seek) and k is any integer.
Where N is the answer.
Step 1: Inverse cos() both sides
$0.6n\pi + 0.3\pi$ = $0.6(n+Nk)\pi + 0.3\pi$
step 2: Simplify, remove common term $0.3\pi$ from both sides And expand the () on the LHS
$0.6n\pi$ = $0.6n\pi + 0.6Nk\pi$
step 3: Remove $0.6\pi$, it is a common common terms:
$0$ = $0.6Nk\pi$
Step 4: The 0 on the LHS is really problematic, but we can also write it as $2\pi$ because $0$ equals $2\pi$
Thus $2\pi$ = $0.6Nk\pi$
step 5: Remove $\pi$ from both sides
$2$ = $0.6Nk$
$Nk$= $2/0.6$, or $3.33333$ repeating
One of these two (N or k) must be a non-integer.
Or is this problem misleading, in that this signal is periodic in the continuous domain, and aperiodic in the discrete domain.