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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.

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    $\begingroup$ Geometrically, sinh and cosh describe the unit hyperbola just as sin and cos describe the unit circle. $\endgroup$ – user296602 Jul 9 '16 at 3:59
  • $\begingroup$ In the sense that they parametrize it naturally? $\endgroup$ – Antonios-Alexandros Robotis Jul 9 '16 at 4:00
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    $\begingroup$ Here's a geometric view: math.stackexchange.com/a/757241/409 $\endgroup$ – Blue Jul 9 '16 at 4:02
  • $\begingroup$ 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. $\endgroup$ – Rellek Jul 9 '16 at 4:02
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    $\begingroup$ 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. $\endgroup$ – MPW Jul 9 '16 at 4:47
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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.

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