SIN, COS, EXP and LOG of Liouville numbers

Abstract

For any Liouville number $\alpha$, all of the following are transcendental numbers: ${\text{e}}^{\alpha }$, ${\log }_{\text{e}}\alpha$, $\sin \alpha$, $\cos \alpha$, $\tan \alpha$, $\sinh \alpha$, $\cosh \alpha$, $\tanh \alpha$, $\arcsin \alpha$ and the inverse functions evaluated at $\alpha$ of the listed trigonometric and hyperbolic functions, noting that wherever multiple values are involved, every such value is transcendental. This remains true if "Liouville number" is replaced by "$U$-number", where $U$ is one of Mahler's classes of transcendental numbers.