Calculate sinh, cosh, tanh, coth, sech, csch and their inverses for any real number. Includes value comparison bars, reference table, and identity verification.
Hyperbolic functions are analogs of the ordinary trigonometric functions but are defined using the exponential function rather than the unit circle. The six hyperbolic functions — sinh, cosh, tanh, coth, sech, and csch — arise naturally in many areas of mathematics and physics, including the description of catenary curves, the solutions of certain differential equations, special relativity, and the geometry of hyperbolas.
The hyperbolic sine and cosine are defined as sinh(x) = (eˣ − e⁻ˣ)/2 and cosh(x) = (eˣ + e⁻ˣ)/2. From these, the other four functions are derived just as tangent, cotangent, secant, and cosecant are derived from sine and cosine. These functions satisfy identities that closely parallel trigonometric identities, such as cosh²(x) − sinh²(x) = 1, the hyperbolic analog of cos² + sin² = 1.
Each hyperbolic function has an inverse, which can be expressed in terms of logarithms. For instance, arcsinh(x) = ln(x + √(x² + 1)). Inverse hyperbolic functions are used to solve hyperbolic equations and appear in integration formulas.
This calculator computes all six hyperbolic function values and their inverses for any input. It displays comparison bars so you can visually gauge relative magnitudes, a reference table over a customizable range, and an identity verification panel that confirms key hyperbolic identities at your chosen x value. Use the presets to explore standard values quickly and adjust the decimal precision to suit your needs.
Hyperbolic Functions Calculator helps you solve hyperbolic functions problems quickly while keeping each step transparent. Instead of redoing long algebra by hand, you can enter Input value x, Decimal precision, Table range start once and immediately inspect sinh(x), cosh(x), tanh(x) to validate your work.
This is useful for homework checks, classroom examples, and practical what-if analysis. You keep the conceptual understanding while reducing arithmetic mistakes in multi-step calculations.
sinh(x) = (eˣ − e⁻ˣ)/2; cosh(x) = (eˣ + e⁻ˣ)/2; tanh(x) = sinh(x)/cosh(x); coth(x) = cosh(x)/sinh(x); sech(x) = 1/cosh(x); csch(x) = 1/sinh(x). Fundamental identity: cosh²(x) − sinh²(x) = 1.
Result: sinh(x) shown by the calculator
Using the preset "x = 0", the calculator evaluates the hyperbolic functions setup, applies the selected algebra rules, and reports sinh(x) with supporting checks so you can verify each transformation.
This calculator takes Input value x, Decimal precision, Table range start, Table range end and applies the relevant hyperbolic functions relationships from your chosen method. It returns both final and intermediate values so you can audit the process instead of treating it as a black box.
Start with the primary output, then use sinh(x), cosh(x), tanh(x), coth(x) to confirm signs, magnitude, and internal consistency. If anything looks off, change one input and compare the updated outputs to isolate the issue quickly.
A strong workflow is manual solve first, calculator verify second. Repeating that loop improves speed and accuracy because you learn to spot common setup errors before they cost points on multi-step algebra problems.
They appear in engineering (catenary cables, transmission lines), physics (special relativity, wave equations), mathematics (complex analysis, integration), and machine learning (activation functions like tanh). Use this as a practical reminder before finalizing the result.
They are defined directly from the exponential function: sinh(x) = (eˣ − e⁻ˣ)/2 and cosh(x) = (eˣ + e⁻ˣ)/2. All other hyperbolic functions are ratios of these two.
cosh²(x) − sinh²(x) = 1, analogous to cos²(x) + sin²(x) = 1 for circular trigonometric functions. Keep this note short and outcome-focused for reuse.
Yes. arcsinh, arccosh, etc., are inverse hyperbolic functions. They are sometimes written as sinh⁻¹, cosh⁻¹ or with the "ar" prefix (arsinh, arcosh) in European notation.
arccosh(x) is defined only for x ≥ 1, because cosh(x) always returns values ≥ 1. Apply this check where your workflow is most sensitive.
Yes. For complex z, sinh(z) and cosh(z) are entire functions. In fact, sin(ix) = i·sinh(x) and cos(ix) = cosh(x), connecting circular and hyperbolic functions via Euler's formula.