TITLE: Intuition Behind the Hyperbolic Sine and Hyperbolic Cosine Functions
QUESTION [5 upvotes]: 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.

REPLY [7 votes]: 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.