Hyperbolic Trig Functions (in the works)
Sinh is called Hyperbolic Sine. Sinh and 5 other functions are analogous to the trig functions. They are defined this way:
Sinh[x]:= ( E^2-E^(-x) )/2 or -I*Sin[I*x] Cosh[x]:= ( E^2+E^(-x) )/2 or Cos[I*x] Tanh[x]:= Sinh[x]/Cosh[x] or -I*Tan[I*x] Sech[x]:= 1/Cosh[x] Csch[x]:= 1/Sinh[x] Coth[x]:= Cosh[x]/Sinh[x]
The E is the number Limit[(1 + 1/x)^x, x -> ∞] ~= 2.71828. The capital I denotes the complex number Sqrt[-1].
They are called hyperbolic trig functions because they bear strong similarities to the trig functions. Trig functions relate to a circle, while hyperbolic trig functions relate to a rectangular hyperbola x^-y^==1 in a similar way. And all identies of trig functions have a analogue in the hyperbolic ones of the same form.
The hyperbolic cosine Cosh is famously known as catenary.
Related Web Sites
Robert Yates: Curves and Their Properties.