Hyperbolic Functions Calculator
Calculate all six hyperbolic functions for any x.
Includes inverse formulas, the cosh²-sinh²=1 identity, and applications from catenaries to neural networks.
Hyperbolic functions are the exponential cousins of sine and cosine. Where the trig functions trace a unit circle, these trace a unit hyperbola.
The definitions come directly from the exponential function:
sinh(x) = (eˣ − e⁻ˣ) / 2 cosh(x) = (eˣ + e⁻ˣ) / 2 tanh(x) = sinh(x) / cosh(x)
The three reciprocal functions follow: csch(x) = 1/sinh(x), sech(x) = 1/cosh(x), coth(x) = 1/tanh(x). Both csch and coth are undefined at x = 0.
The fundamental identity is cosh²(x) − sinh²(x) = 1. This mirrors sin²+cos²=1 and falls out directly from the exponential definitions.
tanh(x) is bounded between −1 and +1. sinh and cosh grow without bound as x increases.
Where these show up in practice
The catenary is the most famous application. A hanging chain or cable takes the shape y = a·cosh(x/a) — not a parabola, though the two look similar at small deflections. Suspension bridge cables, power lines, and arch bridges all involve catenaries.
Neural networks use tanh as an activation function because it is smooth everywhere, differentiable, and maps any real input to (−1, 1). Before ReLU became dominant, tanh was the standard choice for hidden layers.
sinh and cosh appear in solutions to the heat equation, the Laplace equation, and in special relativity where hyperbolic functions connect velocity to rapidity.
Inverse hyperbolic functions
These have closed-form logarithm expressions:
arcsinh(x) = ln(x + √(x²+1)) arccosh(x) = ln(x + √(x²−1)) for x ≥ 1 arctanh(x) = ½·ln((1+x)/(1−x)) for |x| < 1
JavaScript provides Math.sinh, Math.cosh, and Math.tanh natively since ES6.
One thing beginners get wrong: cosh(x) is always ≥ 1. It cannot equal zero, so sech(x) is always defined and always in (0, 1].