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.
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.
