What do sinh and cosh have to do with exp?

Question

My friend told me that $\sinh$ and $\cosh$ result from an exponential function, but I can't figure out why

Answer 1

$$\cosh(x) = \frac{1}{2} (\exp(x)+\exp(-x))$$ $$\sinh(x) = \frac{1}{2} (\exp(x)-\exp(-x))$$

$\exp(x)$ is the natural exponential function $\exp(x)=e^x$