One of the usual introductions to hyperbolic trig functions is through the analogy with circular functions. These introductions show that the hyperbolic "angle" can be identified with an _area_ under a hyperbola (y=1/x or one of its rotations), which then leads one to exponentials and logarithms. http://en.wikipedia.org/wiki/Hyperbolic_angle#Imaginary_circular_angle http://myyn.org/m/article/hyperbolic-angle/ Bjørn Felsager and Christina Sheets have also developed some wonderful teaching materials for hyperbolic trigonometry/geometry (do a Google search). These materials are simple enough to enable a high school student with only a traditional plane geometry background to start appreciating hyperbolic geometry; no advanced trig or even complex numbers are required. Physicists in the early 20th century were uncomfortable enough with these concepts that they stumbled around for several decades without noticing that the usual trig functions with complex arguments work just fine for special relativity. (Google search for "rapidity"; also Scott Walter; John Rhodes and Mark Semon.) Are there any other interpretations of "angle", other than the "area" interpretation and the "rapidity" interpretations?