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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.

Alekos Robotis
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    Geometrically, sinh and cosh describe the unit hyperbola just as sin and cos describe the unit circle. –  Jul 09 '16 at 03:59
  • In the sense that they parametrize it naturally? – Alekos Robotis Jul 09 '16 at 04:00
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    Here's a geometric view: http://math.stackexchange.com/a/757241/409 – Blue Jul 09 '16 at 04:02
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    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 09 '16 at 04:02
  • It's pretty similar. Points on the unit circle are all of the form $(\cos t, \sin t)$ and points on the right branch of the unit hyperbola are all of the form $(\cosh t, \sinh t)$. If you prefer a complex analytic interpretation, $\cosh(ix) = \cos x$ and a similar result for sinh. –  Jul 09 '16 at 04:04
  • This isn't a fundamental relationship, but it does tie the two families of functions together. Have you ever heard of the Gudermannian function.? – Steven Alexis Gregory Jul 09 '16 at 04:13
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    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 09 '16 at 04:47
  • Thank you all for the useful comments - I am exploring all of these suggestions. – Alekos Robotis Jul 09 '16 at 04:53

1 Answers1

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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.

MPW
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