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It is straightforward to show that $(2\pi)^{-1}\arcsin(\sqrt{1/3})$ is irrational. Indeed, if this equals a rational number $r\in\mathbb{Q}$, then $2\sin(2\pi r)=\sqrt{4/3}$. However, $2\sin(2\pi r)$ is a sum of two roots of unity, hence an algebraic integer, while $\sqrt{4/3}$ is not.