Hyperbolic Functions Calculator

Hyperbolic functions are a family of mathematical functions built from exponential growth and decay. They resemble ordinary trigonometric functions in notation and many identities, but they describe a hyperbola rather than a circle. The hyperbolic sine, written sinh(x), equals the difference between e raised to x and e raised to negative x, divided by two. The hyperbolic cosine, cosh(x), equals their sum divided by two. Their ratio is the hyperbolic tangent, tanh(x). Three reciprocal functions complete the usual set of six. Hyperbolic cosecant csch(x) is one divided by sinh(x), hyperbolic secant sech(x) is one divided by cosh(x), and hyperbolic cotangent coth(x) is cosh(x) divided by sinh(x). Since sinh(0) is zero, csch(0) and coth(0) are undefined. Cosh(x) is always positive for real x, so sech(x) is defined for every real input. The calculator identifies the undefined cases instead of displaying an infinite or misleading numeric result. Hyperbolic functions appear naturally when differential equations combine a quantity with its second derivative. The hanging shape of a flexible cable is a catenary described by cosh, not a parabola. These functions also model special relativity, electrical transmission lines, heat transfer, fluid flow, minimal surfaces, and certain probability distributions. Engineers and scientists often need several related values at once, which is why this calculator reports the entire family from a single input. Several identities help verify the results. The difference cosh(x) squared minus sinh(x) squared always equals one. Tanh(x) approaches one as x grows positively and negative one as x grows negatively, but it never reaches either limit for a finite real input. Sinh is an odd function, so changing the sign of x changes the sign of its output. Cosh and sech are even functions, so their values remain unchanged when x changes sign. The input x is dimensionless. If a physical formula begins with a measured quantity, that quantity is normally multiplied or divided by constants so the function receives a dimensionless argument. Results are calculated with the browser's standard double-precision exponential functions and rounded to eight decimal places for readable display. Very large magnitudes may exceed the finite numeric range, while ordinary mathematical and engineering inputs produce stable results.