SIN, COS, EXP AND LOG OF LIOUVILLE NUMBERS

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