Hyperbolic angle

Martin McBride
2022-04-09

If we plot a parametric curve such that $x=\cos{a}$ and $y=\sin{a}$, the result will be a circle. The parameter a represents the angle of the line from the origin to the point (x, y).

If we plot a parametric curve such that $x=\cosh{a}$ and $y=\sinh{a}$, the result will be a hyperbola. The parameter a is called the hyperbolic angle by analogy with the circle case.

Inverse hyperbolic functions

The inverse sin function $\arcsin{x}$ gives the value of the angle whose sine is x. On a unit circle, the value of the angle is equal to the length of the arc of the circle formed by that angle (when measured in radians). Historically this was an important measure in architecture, so the inverse sine is called the arcsin, along with arccos and arctan.

This is not true of the hyperbolic functions, however the area enclosed by a unit hyperbola is related to the hyperbolic angle (the area is a/2). So the inverse hyperbolic functions are called arsinh, arcos, and artanh. The "ar" is short for area. This is why the inverse sinh is arsinh rather than arcsinh.