I found on this wikipedia page https://en.wikipedia.org/wiki/Transcendental_number
that $\sin a, \cos a, \tan a$ are transcendental numbers for $a \neq 0$ and algebraic. But there's no mention about their inverse, $\arcsin, \arccos, \arctan$.
I will also include $\operatorname{arcsinh}, \operatorname{arccosh}, \operatorname{arctanh}.$
Assume $x$ is algebraic and $y=\arcsin(x)$ is algebraic as well. Then because of $\sin(y)=x$ with algebraic $y$ and $x$ we can conclude $y=0$. Therefore $\arcsin(x)$ is either $0$ or transcendental for algebraic $x$. Analogue you can show this for the other inverse trigonometric functions.