how to find the discrete time fourier series of sin function

Question

I have this discrete signal sin($0.1 \cdot \pi n$) and I want to find the DTFS of it.

first I have found its fundamental period N which is 20 but what am struggling with is how am I going to find 29 values of CK.

so my question is: is there any other simpler way to find its DTFS.

NOTE: I don't want its DFT!