I've been reading up on hyperbolic functions and was wondering if there was a geometric definition for the hyperbolic angle and hyperbolic function.
In particular I was reading this: Alternative definition of hyperbolic cosine without relying on exponential function and this: Proofs of Hyperbolic Functions
My question is this - when we consider trigonometric functions, we treat them as points on a unit circle where (conventionally) the angle is formed between the x-axis and a ray drawn to some other point. I believe that for hyperbolic functions, we take the ray dividing the hyperbola in half, and another ray, and the angle between them is the hyperbolic angle.
For the hyperbola $x^2 - y^2 = 1$, somehow, the area of the sector is half the angle. Is there a proof for this?
Also, for the hyperbola $y = 1/x$, the area of the sector is equal to the angle. Is there a proof here? Why are the two cases different?