Hyperbolic Functions Explorer
Hyperbolic vs circular functions, identity checks, and geometric intuition.
Non-Euclidean Geodesic Projections 🖖
Hyperbolic functions rigorously map analogous trigonometric relationships onto the geometry of the standard hyperbola instead of a closed circle. The defining identity cosh² - sinh² = 1 is the explicit signature of Minkowski spacetime and strictly dictates the catenary curve of a hanging chain.
Just exponentials in disguise 🖖
Strip away the fancy names and sinh and cosh are simply built from the exponential functions e^t and e^-t: cosh(t) = (e^t + e^-t)/2 and sinh(t) = (e^t - e^-t)/2. So cosh is the average of a growing and a shrinking exponential, while sinh is half their difference. That is why cosh never dips below 1 and sinh passes cleanly through the origin.
The bridge that needs no i 🖖
You can convert between hyperbolic and circular functions without any imaginary numbers, using the Gudermannian gd(t): sin(gd t) = tanh(t) and tan(gd t) = sinh(t). Its inverse, ln(tan(π/4 + φ/2)), is exactly the vertical coordinate of the Mercator map projection. So every classic nautical sea chart is quietly drawn using the geometry of sinh and cosh.
Example problems
- identity check - Hyperbolic identity check: cosh^2(1) - sinh^2(1) = 1.
- tanh asymptote - tanh approaches +/-1 as |t| increases, unlike tan which diverges.
- catenary shape - cosh curve is the catenary profile used by hanging chains and cables.
- derivative relation - Derivative relation: d/dx sinh = cosh and d/dx cosh = sinh.