TITLE: Geometric definitions of hyperbolic functions
QUESTION [13 upvotes]: I've learned in school that all the trigonometric functions can be constructed geometrically in terms of a unit circle:

Can the hyperbolic functions be constructed geometrically as well? I know that $\sinh$ and $\cosh$ can be constructed based on the area between a ray through the origin and the unit hyperbola, but what about $\tanh$, $\mathrm{csch}$, $\mathrm{sech}$ and $\coth$?

REPLY [3 votes]: Since you wanted all Pythagorean relationships to be visible, I had to made right-triangles near the origin.  This results in sech being drawn vertically and horizontally, unfortunately.  Otherwise, co-functions are horizontal lines, and the rest are vertical lines.  The triangles on the line y=tanh(A)*x all have horrible hypotenuses, unavoidably.  I did not draw the line y=coth(A)*x, which would've clouded the picture even further.