I found this proof on mathoverflow. It's about the irrationality of $(\arcsin 1/4)/\pi$.
My question is: Assuming that we don't know the value of $(\arcsin 1/2)$, and only knew that $\sin(\arcsin 1/2) = 1/2$, then this proof wouldn't work just the same to establish the irrationality of $(\arcsin 1/2)/\pi = 1/6?$
No. It would not work the same way. With $\theta=\arcsin(1/2)$ the argument would still lead to the conclusion that $2i\sin\theta$ is an algebraic integer. However, this time $$2i\sin\theta=2i\cdot\frac12=i$$ actually is an algebraic integer, a zero of the monic polynomial $x^2+1$ with integer coefficients. So we did not reach that same contradiction.