What connection do the hyperbolic trig functions have to the actual trig functions?

Question

As far as my understanding goes, trigonometry is the math of right triangles. Sine is the opposite side over the hypotenuse, cosine is the adjacent side over the hypotenuse, etc. The unit circle definition comes as a result of this fact.

What connection do the the hyperbolic trig functions have to the actual trig functions?

From Wikipedia: "Just as the points ($\cos t, \sin t$) form a circle with a unit radius, the points ($\cosh t, \sinh t$) form the right half of the equilateral hyperbola. The hyperbolic functions take a real argument called a hyperbolic angle. The size of a hyperbolic angle is twice the area of its hyperbolic sector. The hyperbolic functions may be defined in terms of the legs of a right triangle covering this sector." — njuffa, Aug 21 '19 at 04:32
Related (duplicate?): "Geometric construction of hyperbolic trigonometric functions". Also, "Alternative definition of hyperbolic cosine without relying on exponential function"; in particular, my answer. — Blue, Aug 21 '19 at 04:38
I think this question is addressing the direct relationship more than the similarities in their connection the either the unit circle or the parabola. — Burt, Aug 21 '19 at 04:40
@burt: You think? It's your question; don't you know? :) ... In any case, since comments are easily overlooked (and sometimes hidden), please edit your question to make any clarifications. — Blue, Aug 21 '19 at 04:43
@burt In that case Wikipedia: "The Gudermannian function, named after Christoph Gudermann (1798–1852), relates the circular functions and hyperbolic functions without explicitly using complex numbers." — njuffa, Aug 21 '19 at 04:48

score 1 · Accepted Answer · 2019-08-21T12:02:47.187

1

$\cos(ix)=\cosh(x)$ and $\sin(ix)=i\sinh(x)$. Beyond that, most of the hyperbolic trig identities have a similar form compared to their "circle" trig counterparts. For instance,

$$\sinh(x+y)=\sinh x\cosh y+\cosh x+\sinh y$$ $$\cosh^2x-\sinh^2=1 $$

Also, as noted in the comments and on Wikipedia, there is a connection to the hyperbola $x^2-y^2=1$ analogous to the unit circle.

edited Aug 21 '19 at 12:02

answered Aug 21 '19 at 04:35

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That is: $\sin$ and $\sinh$ are the same complex function, up to some rotations of the complex plane. Same for $\cos$ and $\cosh$. – GEdgar Aug 21 '19 at 12:11

What connection do the hyperbolic trig functions have to the actual trig functions?

1 Answers1