Hyperbolic Function Calculator
Evaluate sinh, cosh, tanh and their reciprocals and inverses, and explore exponential definitions, identities, and domains.
Note
Inverse functions require inputs within their natural domains (e.g., acosh(x) with x ≥ 1).
Identity checks
Notes
How to use
Select a hyperbolic function or its inverse, input a real value for x, and press Calculate to evaluate the function numerically. Clear the inputs to start fresh.
Methodology
The tool computes definitions via exponentials, verifies identities by comparing both sides, and flags domain warnings for inverse functions so you always see reliable output.
Full original guide (expanded)
Definitions & quick reference
Hyperbolic functions arise naturally from exponential expressions. The basic definitions are:
- \(\sinh x = \dfrac{e^x - e^{-x}}{2}\)
- \(\cosh x = \dfrac{e^x + e^{-x}}{2}\)
- \(\tanh x = \dfrac{\sinh x}{\cosh x}\)
- \(\operatorname{sech} x = \dfrac{1}{\cosh x}\), \(\operatorname{csch} x = \dfrac{1}{\sinh x}\), \(\operatorname{coth} x = \dfrac{\cosh x}{\sinh x}\)
Inverse functions include \(\operatorname{arsinh}, \operatorname{arcosh}, \operatorname{artanh}, \operatorname{arcoth}, \operatorname{arsech}, \operatorname{arcsch}\) with their respective domains.
Core identities
\(\cosh^2 x - \sinh^2 x = 1\)
\(\sinh(x + y) = \sinh x \cosh y + \cosh x \sinh y\)
\(\cosh(x + y) = \cosh x \cosh y + \sinh x \sinh y\)
\(\tanh(x + y) = \dfrac{\tanh x + \tanh y}{1 + \tanh x \tanh y}\)
\(\sinh(2x) = 2\sinh x \cosh x\), \(\cosh(2x) = \cosh^2 x + \sinh^2 x\), \(\tanh(2x) = \dfrac{2\tanh x}{1 + \tanh^2 x}\)
Inverse definitions & domain notes
- \(\operatorname{arsinh} x = \ln(x + \sqrt{x^2 + 1})\)
- \(\operatorname{arcosh} x = \ln(x + \sqrt{x^2 - 1})\), valid for \(x \ge 1\)
- \(\operatorname{artanh} x = \tfrac{1}{2}\ln\left(\dfrac{1 + x}{1 - x}\right)\), valid for \(|x| < 1\)
- \(\operatorname{arcoth} x = \tfrac{1}{2}\ln\left(\dfrac{x + 1}{x - 1}\right)\), valid for \(|x| > 1\)
- \(\operatorname{arsech} x = \operatorname{arcosh}(1/x)\), defined for \(0 < x \le 1\)
- \(\operatorname{arcsch} x = \operatorname{arsinh}(1/x)\), defined for \(x \neq 0\)
Why hyperbolic functions matter
They model hanging cables (catenary curves), special relativity, Laplace transforms, and solutions to differential equations. Their complex relationships to trigonometric functions via \(\sin(ix) = i\sinh x\) and \(\cos(ix) = \cosh x\) make them invaluable across mathematics and physics.
FAQ
What is a hyperbolic function? They are analogues of trig functions defined via exponentials and linked to hyperbolas.
How do I use the calculator? Select the function, enter x, and hit Calculate. The tool evaluates the expression, shows definitions, and checks identities.
Are hyperbolic functions periodic? No. They grow exponentially as |x| increases, unlike trig functions that stay bounded.