What is the importance of $\sinh(x)$? I stumbled across $\sinh(x)$. I am only a calculus uno student, but was wondering when this function comes into play, and what is its purpose? Last, does it have world applications, or is it a human-made concept?
 A: The hyperbolic sine (and the hyperbolic cosine) arises naturally in the solution to the differential equation
$$f''=f$$
which is
$$f(x)=a\sinh(x)+b\cosh(x)$$
A: This point of view uses the linear algebra: 
The set of functions $f:\Bbb R\to \Bbb R$ is a linear space usually  denoted $\mathscr F(\Bbb R,\Bbb R)$ or simply $\mathscr F(\Bbb R)$ and it's a direct sum of the subspace of even functions $\mathscr E(\Bbb R)$ and the subspace of odd functions $\mathscr O(\Bbb R)$:
$$\mathscr F(\Bbb R)= \mathscr E(\Bbb R)\oplus \mathscr O(\Bbb R)$$
which means that any function $f$ is a sum of an even function and an odd function and this writing is unique
$$f(x)=\underbrace{\frac{f(x)+f(-x)}{2}}_{\text{even part}}+\underbrace{\frac{f(x)-f(-x)}{2}}_{\text{odd part}}$$
Now for the $\sinh$ function we can see easily that it's the odd part of the exponential function.
A: The functions $\cosh x = \cfrac {e^x+e^{-x}}2$ and $\sinh x = \cfrac {e^x-e^{-x}}2$ arise when we decompose the exponential function $e^x$ ito its even and odd parts.
I've never seen an essay or article which really explores the significance of this fact, though, but think it is perhaps more significant than a simple observation.
A: One definition of the hyperbolic sine is 
$$
\sinh(x) = \frac{e^x - e^{-x}}{2}.
$$ 
The function comes into play in partial differential equations which are used in many real world applications. One example of a PDE is the Laplace equation on a square $\nabla^2u = 0$ where $\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}$. On one side we will have periodic boundary conditions and the other will consist of hyperbolic trig functions namely $\sinh$ and $\cosh$.
A: Also, $\displaystyle \tanh x=\frac{\sinh x}{\cosh x}$ describe the geometry of Special Theory of Relativity, as composition of parallel velocities works like
$$
\displaystyle\tanh(\alpha+\beta)=\frac{\tanh\alpha+\tanh\beta}{1+\tanh\alpha\tanh\beta}
$$
$$
\displaystyle\frac{u}{c}\oplus\frac{v}{c}=\frac{\frac{u}{c}+\frac{v}{c}}{1+\frac{u}{c}\frac{v}{c}}
$$
A: The other answers are spot-on, but I'll take the opportunity to point out yet another real-world application of hyperbolic trig functions.
A catenary is the curve you get by hanging a chain of uniform density by its two endpoints:  
$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ 
Catenaries will arise, for example, in suspension bridges and power cables hung from transmission towers.  These curves and their properties are nicely described using the hyperbolic trig functions.  As an interesting aside, square wheels will "roll" smoothly over a surface of inverted catenaries$^\dagger$, provided that the arclength of each catenary is equal to the side length of the square.  

$\text{ }$
This is all to say: one can imagine that engineers and architects would find interest in studying these curves in certain situations.  See this paper as an example, where one repeatedly encounters the hyperbolic sine and cosine functions.

$^\dagger$The intrigued reader can see this in action here.  The particularly intrigued reader might be asking whether a suitable road exists for other wheel shapes, and this question is addressed extensively by Hall & Wagon in their 1992 paper Roads and Wheels, again making heavy use of these hyperbolic functions.
A: The hyperbolic sine (and cosine) is a linear combination of two inverse exponentials. Exponentials are more fundamental functions and one could do only with them. They appear in many problems governed by linear differential equations, i.e. a great deal of physics.
By contrast, the circular functions (which are also imaginary exponentials), playing a similar role in linear systems, cannot be disregarded. As they enjoy numerous transformation properties, there was some influence on the survival of the hyperbolic functions.
But to be honest, $\sinh$ and $\cosh$ are really secondary functions.
