Hi Ariana,
In principle the manipulation of functions within cadabra is made through SymPy. Hence, one could use (supposedly) any function recognised by SymPy, e.g. \sinh, \cosh, \tanh, etc.
However, cadabra does not necessarily know how to handle the output returned by SymPy, particularly because cadabra expressions are rendered as LaTeX expressions... translated by a "dictionary". If the "dictionary" does not have the function you want, you'll get errors in your notebook.
So, the manipulation of functions is limited by the size of the "dictionary".
I recommend you to see the GitHub page of DisplaySympy, in particular the definition DisplaySympy::DisplaySympy
Dox.