$$\sinh x = \frac{e^x-e^{-x}}{2}, \quad \cosh x = \frac{e^x+e^{-x}}{2}$$
Hyperbolic functions are solutions to ODE $$\ddot{y} - y=0$$ , with general solution $y = A \sinh x + B \cosh x$. An alternative way to write the general solutions is $y = c_1 e^x + c_2 e^{-x}$.
The hyperbolic functions are analogs of circular functions (a.k.a. trigonometric functions).
Differentiation: $$(\sinh x)' = \cosh x$$ $$(\cosh x)' = \sinh x$$
Relations to circular functions: $$\cos(ix) = \cosh x$$ $$\sin(ix) = i \sinh x$$
Complex functions: ($z = x+iy$) $$\cos z = \cosh y \cos x - i \sinh y \sin x$$ $$\sin z = \cosh y \sin x + i \sinh y \cos x$$