The real hyperbolic sine function is a function defined using the real exponential function by the formula
\[\sinh(x):=\frac 12(\exp(x)-\exp(-x))\]
for all \(x\in\mathbb R\). Note that $\cosh$ can be interpreted as the "half the difference" of $\exp(x)$ and $\exp(-x)$.
The following graph visualizes the hyperbolic sine function: