Hyperbolic Functions Calculator
Enter any real number x and instantly see all six hyperbolic functions (sinh, cosh, tanh, coth, sech, csch) plus all six inverse hyperbolic functions. Results are shown with full worked steps so you can follow the exponential definitions behind each value. Domain warnings flag when a function is undefined for your input.
What are hyperbolic functions?
Hyperbolic functions are analogs of the trigonometric (circular) functions, but defined using the unit hyperbola x^2 - y^2 = 1 instead of the unit circle x^2 + y^2 = 1. The six standard hyperbolic functions are sinh (hyperbolic sine), cosh (hyperbolic cosine), tanh (hyperbolic tangent), coth (hyperbolic cotangent), sech (hyperbolic secant), and csch (hyperbolic cosecant). They appear constantly in physics (the shape of a hanging chain, the Lorentz factor in special relativity), engineering (heat transfer, transmission lines), and pure mathematics (differential equations, complex analysis).
Exponential definitions and the core identity
Every hyperbolic function is a combination of e^x and e^(-x). The two building blocks are sinh(x) = (e^x - e^(-x)) / 2 and cosh(x) = (e^x + e^(-x)) / 2. The remaining four follow: tanh = sinh/cosh, coth = cosh/sinh, sech = 1/cosh, and csch = 1/sinh. The central identity, analogous to sin^2 + cos^2 = 1 for trigonometry, is cosh^2(x) - sinh^2(x) = 1. This holds for every real x and is easy to verify by substituting the exponential definitions.
Inverse hyperbolic functions and logarithm form
The inverse hyperbolic functions - arsinh, arcosh, artanh, arcoth, arsech, arcsch - each undo the corresponding forward function. Because the hyperbolic functions are built from exponentials, their inverses can be written as natural logarithms. For example, arsinh(x) = ln(x + sqrt(x^2 + 1)), which is defined for all real x. artanh(x) = 0.5 * ln((1+x)/(1-x)) is only defined for |x| < 1. The domain restrictions listed in the reference table above matter: entering a value outside the stated domain returns "undefined" in this calculator.
Hyperbolic functions in science and engineering
A catenary, the curve formed by a flexible chain hanging under gravity, is described by y = a * cosh(x/a). The surface area of a surface of revolution built from a catenary (a catenoid) is the minimum-area surface spanning two circular rings, which is why soap films take this shape. In special relativity, the rapidity parameter is the artanh of the velocity-to-c ratio, making the addition of relativistic velocities equivalent to simple rapidity addition. In signal processing and control theory, the transfer functions of Butterworth and Chebyshev filters involve hyperbolic cosines. In statistics, the Fisher z-transformation for correlation coefficients uses artanh.
Hyperbolic function domains and key identities
| Function | Formula | Domain | Range |
|---|---|---|---|
| sinh(x) | (e^x - e^-x) / 2 | all reals | all reals |
| cosh(x) | (e^x + e^-x) / 2 | all reals | [1, +inf) |
| tanh(x) | sinh(x) / cosh(x) | all reals | (-1, 1) |
| coth(x) | cosh(x) / sinh(x) | x != 0 | (-inf,-1) U (1,+inf) |
| sech(x) | 1 / cosh(x) | all reals | (0, 1] |
| csch(x) | 1 / sinh(x) | x != 0 | all reals except 0 |
| arsinh(x) | ln(x + sqrt(x^2+1)) | all reals | all reals |
| arcosh(x) | ln(x + sqrt(x^2-1)) | x >= 1 | [0, +inf) |
| artanh(x) | 0.5 ln((1+x)/(1-x)) | |x| < 1 | all reals |
| arcoth(x) | 0.5 ln((x+1)/(x-1)) | |x| > 1 | all reals except 0 |
| arsech(x) | ln(1/x + sqrt(1/x^2-1)) | 0 < x <= 1 | [0, +inf) |
| arcsch(x) | ln(1/x + sqrt(1/x^2+1)) | x != 0 | all reals except 0 |
Domain restrictions and principal values for all 12 functions.
Frequently asked questions
What is the difference between sinh and sin?
sin(x) is defined as the y-coordinate on the unit circle (x^2 + y^2 = 1) when the arc length from (1,0) is x. sinh(x) is defined analogously using the unit hyperbola (x^2 - y^2 = 1), but the connection to arc length works differently. Algebraically, sin(x) = (e^(ix) - e^(-ix)) / (2i) whereas sinh(x) = (e^x - e^(-x)) / 2, so sinh(x) = -i * sin(ix). The functions share many structural identities but have different domains and ranges: sin is bounded between -1 and 1, while sinh is unbounded.
Why is cosh(x) always >= 1?
cosh(x) = (e^x + e^(-x)) / 2. By the AM-GM inequality, the arithmetic mean of any two positive numbers is at least their geometric mean, so (e^x + e^(-x)) / 2 >= sqrt(e^x * e^(-x)) = sqrt(1) = 1. Equality holds only when e^x = e^(-x), which happens at x = 0, giving cosh(0) = 1.
Where is coth(x) undefined?
coth(x) = cosh(x) / sinh(x). Since sinh(0) = 0, coth(0) involves division by zero and is undefined. For all other real x, coth is defined and satisfies |coth(x)| > 1. As x approaches 0 from either side, coth(x) tends to +infinity or -infinity.
What is the domain of artanh(x)?
artanh(x) = 0.5 * ln((1+x)/(1-x)) requires both (1+x) > 0 and (1-x) > 0, which means -1 < x < 1. At x = 1 or x = -1 the logarithm diverges. For |x| > 1 you would need arcoth instead.
How do I use this calculator in reverse?
Switch the Mode dropdown to "Inverse". Then enter the value you already know (for example, the tanh output from a previous calculation) into the x field. The calculator shows arsinh, arcosh, artanh, arcoth, arsech, and arcsch for that input. Domain restrictions apply: only the functions whose domain includes your x will return a real number; the others show "undefined".
What is the identity cosh^2(x) - sinh^2(x) = 1?
This is the fundamental hyperbolic identity, analogous to cos^2(x) + sin^2(x) = 1 for circular functions. It follows directly from the exponential definitions: ((e^x + e^(-x))/2)^2 - ((e^x - e^(-x))/2)^2 = (e^(2x) + 2 + e^(-2x))/4 - (e^(2x) - 2 + e^(-2x))/4 = 4/4 = 1. This identity connects cosh and sinh just as the Pythagorean identity connects cos and sin, and it defines the unit hyperbola x^2 - y^2 = 1.