Confusion surrounding the geometric definitions of $\sinh$ and $\cosh$

Question

$\cos x$ and $\sin x$ can be defined by rotating $x$ units around a unit circle: $\cos x$ is the $x-$coordinate and $\sin x$ the $y$-coordinate. However, I am struggling to understand the analogous definition for $\sinh x$ and $\cosh x$. I understand that a hyperbola can be defined as the set of all points satisfying $(\cosh t,\sinh t)$. However, this still begs the question as to how one can work out what $\cosh t$ and $\sinh t$ are in the first place. I know that the hyperbolic functions can be defined using exponentials, but I think a geometric interpretation would be nice.