Hyperbolic Functions Calculator

Hyperbolic functions are analogs of trigonometric functions but defined using the hyperbola rather than the circle. The key functions are sinh, cosh, tanh, coth, sech, and csch. While points (cos x, sin x) trace a unit circle, points (cosh x, sinh x) trace a unit hyperbola.

These are defined using the exponential function: sinh(x) = (eˣ − e⁻ˣ) / 2, cosh(x) = (eˣ + e⁻ˣ) / 2, and tanh(x) = sinh(x) / cosh(x). The remaining functions — coth, sech, and csch — are reciprocals of tanh, cosh, and sinh respectively.

At x = 0, sinh(0) = 0 because (e⁰ − e⁰) / 2 = 0, and cosh(0) = 1 because (e⁰ + e⁰) / 2 = 1. This mirrors the trigonometric identities sin(0) = 0 and cos(0) = 1.

sinh and tanh are odd functions, meaning sinh(−x) = −sinh(x) and tanh(−x) = −tanh(x). cosh is an even function, meaning cosh(−x) = cosh(x). This is analogous to sin being odd and cos being even in trigonometry.

Inverse hyperbolic functions (arcsinh, arccosh, arctanh, arccsch, arcsech, arccoth) return the value of x given a hyperbolic function result. They are expressible as natural logarithms — for example, arcsinh(x) = ln(x + √(x² + 1)) — and are defined over specific domains.

Each inverse function has a restricted domain based on its original function's range. arccosh requires x ≥ 1, arctanh requires |x| < 1, arcsech requires 0 < x ≤ 1, arccoth requires |x| > 1, and arccsch requires x ≠ 0. Inputs outside these domains produce undefined (NaN) results.

Hyperbolic functions appear throughout mathematics, physics, and engineering. cosh describes the shape of a hanging cable (catenary curve), tanh is widely used in neural network activation functions, and sinh/cosh appear in solutions to differential equations in heat transfer and wave mechanics.

Trigonometric functions relate to angles on a unit circle and are periodic. Hyperbolic functions relate to areas on a unit hyperbola and are not periodic. They share similar identities — for instance, cosh²(x) − sinh²(x) = 1, analogous to cos²(x) + sin²(x) = 1.