A fundamental equation of trigonometry is $x²+y² = 1$, where $x$ is the "adjacent side" and $y$ the "opposite side".
If you experiment plot $f(x)$ out of the real domain - for example to $x=1.5$ you obtain $y$ imaginary - you will get an imaginary shape situated in a plane perpendicular to the plane $x,y$ and containing the $x$-axis. This shape is a hyperbola.
So you have two planes, one for the circle, and one for the hyperbola.
The "$z$-axis" (imaginary) where the hyperbola is plotted correspond to the "$\sinh$" and $x$ is the "$\cosh$" once the $R = 1$. Note that the $\sinh$ is situated in a plane $90$ degrees of the $x,y$-plane.
Observe that $$\sin iy = i \sinh y$$ is in accord with was explained above.
The geometric interpretation is easy.
It's valuable remember that the angle of a circumference could be measured by the double of the area of the sector. The hyperbolic angle could be measured by the double of area limited by the radius and the arc of hyperbola.