TITLE: Can hyperbolic functions be defined in terms of trigonometric functions?
QUESTION [29 upvotes]: For example, can $\sinh x$ be written as a function of $\sin x$? 
Another question, are hyperbolic functions dependent of their trigonometric correspondence in any way?

REPLY [12 votes]: This may not be what you mean, but the phrase "as a function of" sometimes has a narrow and precise meaning. Specifically, "$x$ can be written as a function of $y$" means that there is a function $f$ such that $x = f(y)$.
To answer that narrow question, $\sinh x$ cannot be written as a function of $\sin x$ because $\sin$ is periodic along the real axis but $\sinh$ is not.
More specifically:

Assume $\forall{x \in R},~ \sinh(x) = f(\sin(x))$
Therefore, $\sinh(0) = f(\sin(0))$ and $\sinh(2\pi) = f(\sin(2\pi))$
Therefore, $\sinh(0) = f(0) = \sinh(2\pi)$

The last equation is not true.