How to find the angle subtended to the origin by the unit hyperbola through the point (1,0)?

Question

I'm trying to find the angle subtended by the unit hyperbola through the point (1,0). I think that I should be integrating something, but I'm not sure how to set it up. I've been trying to think of this as it would be related to a unit circle, where we would have $R=1$ and then the following $$ \int_0^{\theta_{0}} R^2(\sin^2\theta+\cos^2\theta)d\theta=R^2\int_0^{2\pi}d\theta=(1)(2\pi)=2\pi $$ So the angle subtended would just be $2\pi$. I know that $1=\cosh^2 x-\sinh^2 x,$ but as I'm only interested in the right hyperbola, I'm not sure I can use the same trick. Beyond this, I'm stuck. Any ideas? Thank you!

Answer 1

Read $\;$G.A.Jennings Modern Geometry with Applications (1994) pp.178-179.

There you find a definition of hyperbolic angle (p.178) and a formula (p.179).