Intuition Behind the Hyperbolic Sine and Hyperbolic Cosine Functions

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.

• Geometrically, sinh and cosh describe the unit hyperbola just as sin and cos describe the unit circle. – user296602 Jul 9 '16 at 3:59
• In the sense that they parametrize it naturally? – Antonios-Alexandros Robotis Jul 9 '16 at 4:00
• Here's a geometric view: math.stackexchange.com/a/757241/409 – Blue Jul 9 '16 at 4:02
• I always thought hyperbolic trig functions made the most sense in the context of complex analysis. They are kind of natural consequence of making the change of variable $x \to ix$ in the trig functions. – Rellek Jul 9 '16 at 4:02
• The identities $\cos ix=\cosh x$ and $\sin i x=i\sinh x$ are quite pleasing because they are completely analogous to the familiar formulas $\cos(-x)= \cos x$ and $\sin (-x)=-\sin x$. This analogy makes them easily memorized. – MPW Jul 9 '16 at 4:47