What are hyperbolic trig functions, and how do you solve them? For example, sinh(-1)

We can compare/contrast areas of a semi-circular region (involving the arcsin function) to areas under a hyperbola. The hyperbolic functions arise from this second application. 
Other applications include: 
(1) The curve created by a chain hung from fixed points is called a catenary which is the graph of the hyperbolic cosine. The St. Louis Arch is based on an inverted catenary. Catenaries were studied when mathematicians tried to find the curve through which a falling body would move the fastest between two points separated both horizontally and vertically.
(2) Another common application is solving problems involving a curve known as the tractrix. This is the curve that would occur if you were to pull a boat in towards shore by walking down the shore. These curves can also arise when looking at chase curves -- as the prey runs away at an angle, the best path for the predator is the tractrix.
 
The definitions of the hyperbolic functions involve the exponential function. In particular,sinh(x)=1/2(e^x-e^(-x)) and cosh(x)=1/2(e^x+e^(-x))
So,  sinh(-1)=1/2(1/e-e) which is approximately -1.1752
http://mathworld.wolfram.com/HyperbolicSine.html