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Hyperbolic Trigonometry - mklabgr - 09-11-2026
Author: Ploy Wattanawanichkul
Date: August 2, 2021
This report gives an accessible introduction to hyperbolic trigonometry, explaining how the hyperbolic functions $\sinh x$, $\cosh x$, and $\tanh x$ arise and how they relate to ordinary circular trigonometry. While $\sin\theta$ and $\cos\theta$ parametrize the unit circle $x^2+y^2=1$, the functions $\cosh\theta$ and $\sinh\theta$ parametrize the unit hyperbola through the fundamental identity
$\cosh^2\theta-\sinh^2\theta=1$.
An especially elegant geometric analogy is that, just as the angle $\theta$ on the unit circle corresponds to twice the area of a circular sector, the parameter $\theta$ for the unit hyperbola corresponds to twice an associated hyperbolic area. The paper also develops the connection between Euclidean and hyperbolic geometry using the Poincaré disk model and the Bolyai–Lobachevsky formula for the angle of parallelism,
$\Pi(d)=2\arctan(e^{-d})$,
which leads to relations such as
$\sin(\Pi(x))=\operatorname{sech}(x)$
and
$\cos(\Pi(x))=\tanh(x)$.
The report then presents hyperbolic counterparts of familiar trigonometric laws. For a right hyperbolic triangle,
$\cosh c=\cosh a\cosh b$.
For an arbitrary hyperbolic triangle, the law of sines becomes
$\frac{\sin A}{\sinh a}=\frac{\sin B}{\sinh b}=\frac{\sin C}{\sinh c}$,
while the hyperbolic law of cosines is
$\cosh c=\cosh a\cosh b-\sinh a\sinh b\cos C$.
A particularly important observation is that Euclidean geometry appears as the small-scale approximation of hyperbolic geometry. Using Taylor expansions,
$\sinh x\approx x$
and
$\cosh x\approx1+\frac{x^2}{2}$
for small $x$. Consequently, the hyperbolic formulas reduce approximately to familiar Euclidean relations such as
$c^2=a^2+b^2$
and
$c^2=a^2+b^2-2ab\cos C$.
Thus, for sufficiently small triangles, hyperbolic and Euclidean geometry behave almost identically. As the dimensions of the triangle increase, however, the differences between the two geometries become increasingly significant.
The final section demonstrates that hyperbolic functions are not merely theoretical. The classic catenary, the curve formed by a freely hanging chain or cable, is described by
$y=\frac{\cosh(ax)}{a}$.
The paper derives this equation from the balance of forces and an associated differential equation. Catenary shapes occur naturally in architecture and engineering, including arches and suspended cables. The related catenoid describes the minimal surface formed by a soap film stretched between two circular rings. Hyperbolic functions also appear in subjects such as the Mercator projection and special relativity.
Key takeaways
- Hyperbolic trigonometry is built around the unit hyperbola $x^2-y^2=1$, just as ordinary trigonometry is associated with the unit circle $x^2+y^2=1$.
- The central identity is $\cosh^2x-\sinh^2x=1$.
- Hyperbolic geometry possesses its own versions of the Pythagorean theorem, law of sines, and law of cosines.
- For sufficiently small distances, hyperbolic geometry becomes approximately Euclidean.
- The Bolyai–Lobachevsky formula provides an elegant bridge between ordinary and hyperbolic trigonometric functions.
- The catenary $y=\frac{1}{a}\cosh(ax)$ is one of the clearest physical applications of hyperbolic functions.
- The report illustrates an important mathematical principle: familiar Euclidean formulas can often be understood as local approximations of more general geometric relationships.
ARTICLE [PDF]