Proving S_n will be negative infinitely often in Wolfram|Alpha?

Questions
1

Let

S_ n=\sin^{-1}\left(\sin\left(2\pi^{2}\left(n!\right)\right)\right)+\frac{\pi}{5}

where n is a positive integer.

I need to prove that S_n<0 for infinitely many positive integral values of n. When I entered S_n in Wolfram Alpha, I got its graph, which is available [here] (arcsin(sin(2π^2*n!)) + π/5 - Wolfram|Alpha).

How can I prove that S_n will be negative infinitely often? What are the ways to prove it? Please help me proving this. Thank You.

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