After enough time studying mathematics, we develop an instinct for the sine and cosine functions and their relationship to our standard Euclidean Geometry. I have come across the functions $\sinh(x)$ and $\cosh(x)$ multiple times while studying math including:
$(1)$ Lorentz Transformations
$(2)$ Integrals and Identities
$(3)$ Complex Analysis.
Taken at face value, I understand these functions and their definitions $-$ but I feel like I'm missing the point. What is a natural way for me to understand these functions as intuitively as I understand $\sin(x)$ and $\cos(x).$
Note: I have consulted other answers looking for the answer to this question. I am searching for a more fundamental explanation of how these functions came about analogous to the natural representations of $\sin$ and $\cos$ in terms of angles on the unit circle. Of course If I overlooked such an explanation, please simply point me to it.
There is an absolutely fascinating little booklet called "Hyperbolic Functions" by V. G. Shervatov in which the author develops circular and hyperbolic functions in parallel from a purely geometric viewpoint.
It is from the "Russian Series In Mathematics" and was written decades ago (1950s, I think) and is out of print, but is still out there if you search for it. Google is your friend in this regard.
I bought a copy of this as a kid and I think it changed my life. It may well be the reason I became a mathematician.