Geometric meaning of hyperbolic functions. Trigonometric functions have strong geometric meanings, which make them useful for dealing with complex number (which can be thought as point on the complex plane). In my point of view, by the e-definition, hyperbolic functions are defined more naturally than trigonometric functions [only involve exp(real number)]. Usually simple things have more applications, so is there geometric meaning of hyperbolic functions (if possible, I'd like to focus on plane geometry)? Thank you.
 A: The catenary is a hyperbolic function, coshx. Catenaries have a lot of applications in physics. They were studied extensively by the bernoullies, Huygens and Euler, so we are talking here about the 17th century era. Catenaries were the answer to some complicated math/physics problems of that time. The arch in St Louis, MO is an upside down catenary (and not a parabola), a classical example of a hyperbolic application in physics and engineering. Hope this answers a bit of your questions
A: Not plane geometry, but spherical, uses the trig functions even more thoroughly than Euclidean geometry does. For instance, the equivalent formula to Pythagoras on the surface of a sphere is $\cos c=\cos a\cos b$. On the hyperbolic plane, you get trigonometry very easily: take any spherical trig formula, and replace the trig functions when applied to sides of triangles with the corresponding hyperbolic formulas, but salt with minus signs occasionally. For instance, the Pythagorean formula is $\cosh c=\cosh a\cosh b$. For another analogous pair, the Law of Cosines on the sphere is $\cos c=\cos a\cos b + \sin a\sin b\cos C$, where as usual, $C$ is the angle opposite the side $c\,$; the corresponding hyperbolic-trig formula is $\cosh c=\cosh a\cosh b-\sinh a\sinh b\cos C$.
