Hyperbolic Function Calculator – sinh, cosh, tanh and Their Inverses
Hyperbolic functions are the exponential-based cousins of the familiar trigonometric functions. Instead of coordinates on a unit circle, they come from points on a hyperbola, and instead of an angle, they take a plain real number as input. This calculator evaluates all six hyperbolic functions — sinh, cosh, tanh, csch, sech, and coth — plus their inverses (asinh, acosh, atanh, acsch, asech, acoth), showing the exponential-form derivation behind each result.
The Exponential Definitions
Every hyperbolic function reduces to a combination of eˣ and e⁻ˣ. The two core building blocks are sinh(x) = (eˣ − e⁻ˣ) / 2 and cosh(x) = (eˣ + e⁻ˣ) / 2. All the others follow directly: tanh(x) = sinh(x) / cosh(x), and the reciprocal functions csch, sech, and coth are simply 1 / sinh(x), 1 / cosh(x), and 1 / tanh(x) respectively. This calculator computes each result with JavaScript's native Math.sinh, Math.cosh, and Math.tanh functions for full double-precision accuracy.
The Hyperbolic Pythagorean Identity
Just as sin²(θ) + cos²(θ) = 1 holds for every angle, the hyperbolic analog cosh²(x) − sinh²(x) = 1 holds for every real number x. Enabling the identity panel substitutes your input into both terms and shows that the difference always resolves to exactly 1, which is a useful sanity check when learning how these functions relate to each other.
Inverse Hyperbolic Functions and Their Domains
The inverse functions recover x from a hyperbolic value and are defined using natural logarithms: asinh(x) = ln(x + √(x² + 1)) accepts any real number, but acosh(x) = ln(x + √(x² − 1)) requires x ≥ 1 because cosh itself never dips below 1. Likewise, atanh(x) = ½ · ln((1 + x) / (1 − x)) only accepts −1 < x < 1, since tanhapproaches but never reaches ±1. The calculator checks each function's domain before computing and reports a clear message instead of a misleading NaN when an input falls outside it.
cosh(x) has a range of y ≥ 1 for all real x, there is no real number whose hyperbolic cosine equals 0.5 — the inverse simply has nothing to invert.Handling Overflow for Large Inputs
Because sinh and coshgrow exponentially, JavaScript's double-precision floating-point numbers overflow to Infinity once x exceeds roughly 710. Rather than displaying a raw Infinity, the calculator detects this overflow condition and reports that the result is too large to represent precisely.
Real-World Applications
Hyperbolic cosine describes the shape of a hanging cable or chain through the catenary curve, y = a · cosh(x/a), which engineers use to design suspension cables, power-line sag, and arches like the Gateway Arch. In special relativity, hyperbolic tangent relates velocity to rapidity, a quantity that adds linearly under Lorentz transformations where ordinary velocities do not. In machine learning, tanh is a widely used neural-network activation function because it squashes any real input into the range (−1, 1). Electrical engineers also rely on hyperbolic functions to model voltage and current along transmission lines.
Reading the Curve Plot
The curve plot renders the selected function across a window centered on your input value, with the evaluated point marked directly on the curve. This makes it easy to see at a glance how steeply sinh and cosh rise compared to the gentler, bounded S-shape of tanh, which flattens out toward ±1 as x grows large in either direction.