Hyperbolic "trig" functions such as $\sinh$, $\cosh$, have close analogies with regular trig functions such as $\sin$ and $\cos$. Yet the hyperbolic versions seem to be encountered relatively rarely. (My frame of reference is that of someone with college freshman/sophomore, but not advanced math.)
Why is that? Is it because the hyperbolic versions of these functions are less common/useful than the circular versions?
Can you do the "usual" applications (Taylor series, Fourier series) with hyperbolic functions as you can with trigonometric?
I'm not a professional mathematician. I've had three semesters of calculus and one of linear algebra/differential equations, and "barely" know about hyperbolic functions. The question is with that frame of reference.
Best Answer
I can think of two reasons.
When we do geometry, we usually work in Euclidean space, where the intrinsic property of a line segment between two points is its length, given (in two dimensions) by $\ell^2 = \Delta x^2 + \Delta y^2$. We are allowed to change our reference frame as long as we preserve lengths, which means that the transformation of $(\Delta x,\Delta y)$ is a rotation, and the transformed vector must lie on the circle $\Delta x^2 + \Delta y^2 = \operatorname{const}$, which is naturally parametrized by $\sin$ and $\cos$. The hyperbolic variants don't have much to do with circles or with rotation, so are not relevant here (unless you supply them with imaginary arguments, at which point you're really working just with $\sin$ and $\cos$ in disguise).
On the other hand, the natural setting for special relativity is Minkowski space, where the invariant property of the interval between two points in spacetime is of the form $s^2=\Delta x^2-\Delta t^2$, with a minus sign. Here the allowed changes of reference frame are given by Lorentz transformations, the transformed interval lies on $\Delta x^2-\Delta t^2=\operatorname{const}$ which is a hyperbola, and indeed one finds hyperbolic trigonometric functions to be quite useful in special relativity.
The usual trigonometric functions $\sin$ and $\cos$ are two real solutions to the differential equation $y'' = -y$, which describes a simple harmonic oscillator (a conservative system in a parabolic potential well) which is a central example in much of classical physics. The hyperbolic versions $\sinh$ and $\cosh$ are solutions to $y'' = y$ instead. This equation is rarely useful to model physical systems in real life because all its solutions are unbounded and gain infinite amounts of kinetic energy. Even taken locally, the equation describes an unstable equilibrium, so any real system will not spend most of its time there without additional forcing.