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I need some assistance in calculating the domain of this function:

f(x)=ln(sin(pi/x))

I started with sin(pi/x)>0 due to the ln function.

From here 0<(pi/x)<pi. That lead me to some calculations giving x>1, but obviously I have periods of 2*pi to include.

The answer is: x>1, 1/(2k+1)<x<1/2k for k=1,2,3,... and 1/2k<x<1/(2k+1) for k=-1,-2,-3,...

I don't understand why or how to get to this solution.

thanks !