# What is know about the transcendence of inverse trigonometry and inverse hyperbolic functions?

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 $\operatorname{arcsinh}, \operatorname{arccosh}, \operatorname{arctanh}.$

• If $\arcsin x$ is algebraic for some $x$, then $\sin(\arcsin x)=x$ is trascendental. – Mariano Suárez-Álvarez Apr 3 '18 at 4:35

## 1 Answer

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.

• I think this can also be used for the hyperbolic functions, but I am not quite sure. – Peter Apr 3 '18 at 7:14
• Is the arcsin of a transcedental number algebraic? Since the the sin of a algebraic number is transcedental. – Pinteco Apr 3 '18 at 21:06
• I have no specific counterexample, but I am pretty sure that it is not in general. – Peter Apr 3 '18 at 21:10