# Geometric meaning of hyperbolic functions.

Trigonometric functions have strong geometric meanings, which make them useful for dealing with complex number (which can be thought as point on the complex plane). In my point of view, by the e-definition, hyperbolic functions are defined more naturally than trigonometric functions [only involve exp(real number)]. Usually simple things have more applications, so is there geometric meaning of hyperbolic functions (if possible, I'd like to focus on plane geometry)? Thank you.

• Yes, the same as trigonometric functions, but for the hyperbola. A point in the circle is of the form $(\cos(t),\sin(t))$, a point in the hyperbola is of the form $(\cosh(t),\sinh(t))$. – OR. Sep 11 '13 at 13:54
• The first picture in the Wikipedia article explains it. en.wikipedia.org/wiki/Hyperbolic_function – OR. Sep 11 '13 at 13:59
• – Blue Sep 11 '13 at 14:00
• But the hyperbola seems to be only useful in analytic geometry...? – A. Chu Sep 11 '13 at 14:02
• The books Geometry of Surfaces by Stillwell and The Advanced Geometry of Plane Curves and Their Applications by Zwikker might interest you. – Arnie Bebita-Dris Sep 11 '13 at 14:35

Not plane geometry, but spherical, uses the trig functions even more thoroughly than Euclidean geometry does. For instance, the equivalent formula to Pythagoras on the surface of a sphere is $\cos c=\cos a\cos b$. On the hyperbolic plane, you get trigonometry very easily: take any spherical trig formula, and replace the trig functions when applied to sides of triangles with the corresponding hyperbolic formulas, but salt with minus signs occasionally. For instance, the Pythagorean formula is $\cosh c=\cosh a\cosh b$. For another analogous pair, the Law of Cosines on the sphere is $\cos c=\cos a\cos b + \sin a\sin b\cos C$, where as usual, $C$ is the angle opposite the side $c\,$; the corresponding hyperbolic-trig formula is $\cosh c=\cosh a\cosh b-\sinh a\sinh b\cos C$.