TITLE: Geometric meanings of hyperbolic cosine and sine
QUESTION [10 upvotes]: In euclidean geometry, $\cos$ and $\sin$ are used for angles in trigonometry.
Is there an equivalent for $\cosh$ and $\sinh$ the hyperbolic cosine and sine, and not cosine and sine ?

REPLY [5 votes]: Copying from my own answer to a different question, and at the same time expanding on the answer by I Like Serena.
Take the unit circle, $x^2+y^2=1$. When describing a sector of this circle, you can talk about its angle $\theta$ in radians, or the arc length $r\theta$, or the doubled area $r^2\theta$, and always obtain the same number since $r=1$. But it is this last interpretation as a (doubled) area which translates to hyperbolas.
Take the unit hyperbola $x^2-y^2=1$, and consider the area enclosed by an arc of this hyperbola, starting at the point $(1,0)$, if both ends of this arc are connected to the origin $(0,0)$. Twice this area corresponds to the angle of circular trigonometric functions, and the sinh and cosh of that argument will be the $y$ and $x$ coordinate to the point where your arc on the hyperbola ends.
Using an image from Wikipedia to illustrate this: