Hyperbolic Functions Calculator

Created by Bogna Szyk
Reviewed by Steven Wooding
Last updated: Jul 12, 2022

Whether a high school student or a proficient mathematician, this hyperbolic functions calculator will surely be useful. It is a tool that computes the values of six basic hyperbolic functions – sinh, cosh, tanh, coth, sech, and csch – all in a blink of an eye. You can also use it to calculate the inverse hyperbolic functions.

What are hyperbolic functions?

Hyperbolic functions are analogical to trigonometric functions that you probably know already, such as sine or cosine. What's the difference, then? If you plot points with coordinates (cos⁑x\cos x, sin⁑x\sin x) in a Cartesian coordinate system, they will form a circle. But if you plot points with coordinates (cosh⁑x\cosh x, sinh⁑x\sinh x), they will create a hyperbola (as shown below).

Plot of cosh and sinh

How to calculate sinh, cosh, and tanh

We can define all of these functions in terms of exponential functions. If you're unsure what these are, head over to our exponent calculator for a more detailed explanation.

If you want to calculate sinh⁑x\sinh x – the hyperbolic sine – you need to use the following formula:

sinh⁑x=12(exβˆ’eβˆ’x)\small \sinh x = \frac{1}{2}(e^x - e^{-x})

The formula for calculating cosh⁑x\cosh x – the hyperbolic cosine – is quite similar:

cosh⁑x=12(ex+eβˆ’x)\small \cosh x = \frac{1}{2}(e^x + e^{-x})

You can calculate tanh⁑x\tanh x, coth⁑x\coth x, sech x\text{sech}\ x and csch x\text{csch}\ x (hyperbolic tangent, cotangent, secant and cosecant) analogically as in trigonometry:

tanh⁑x=sinh⁑xcosh⁑x=(exβˆ’eβˆ’x)(ex+eβˆ’x)coth⁑x=cosh⁑xsinh⁑x=(ex+eβˆ’x)(exβˆ’eβˆ’x),x ⁣≠ ⁣0sech x=1cosh⁑x=2(ex+eβˆ’x)csch x=1sinh⁑x=2(exβˆ’eβˆ’x),x ⁣≠ ⁣0\small \begin{align*} \tanh x &= \frac{\sinh x}{\cosh x} = \frac{(e^x - e^{-x})}{(e^x + e^{-x})}\\[1.5em] \coth x &= \frac{\cosh x}{\sinh x} = \frac{(e^x + e^{-x})}{(e^x - e^{-x})}, x\! \not =\! 0\\[1em] \text{sech}\ x &= \frac{1}{\cosh x} = \frac{2}{(e^x + e^{-x})}\\[1em] \text{csch}\ x &= \frac{1}{\sinh x} = \frac{2}{(e^x - e^{-x})}, x\! \not =\! 0 \end{align*}

Inverse hyperbolic functions

Our hyperbolic functions calculator can also find the values of inverse hyperbolic functions. All you have to do is input the value of one of the functions (for example, sinh⁑x\sinh x or tanh⁑x\tanh x), and this tool will automatically return the value of xx.

The formulas used to compute inverse hyperbolic functions are shown below.

arsinh x=ln⁑(x+x2+1)arcosh x=ln⁑(x+x2βˆ’1)artanh x=12ln⁑(1+x1βˆ’x)arcoth x=12ln⁑(1βˆ’x1+x)arsech x=ln⁑(1+1βˆ’x2x)arcsch x=ln⁑(1x+1x2+1)\small \begin{align*} \text{arsinh}\ x &= \ln \left(x + \sqrt{x^2 + 1}\right)\\[1em] \text{arcosh}\ x &= \ln \left(x + \sqrt{x^2 - 1}\right)\\[1em] \text{artanh}\ x &= \frac{1}{2} \ln\left(\frac{1+x}{1-x}\right)\\[1.5em] \text{arcoth}\ x &= \frac{1}{2} \ln\left(\frac{1-x}{1+x}\right)\\[1.5em] \text{arsech}\ x &= \ln\left(\frac{1 + \sqrt{1 - x^2}}{x}\right)\\[1.5em] \text{arcsch}\ x &= \ln\left(\frac{1}{x} + \sqrt{\frac{1}{x^2} + 1}\right) \end{align*}
Bogna Szyk
sinh (x)
cosh (x)
tanh (x)
coth (x)
sech (x)
csch (x)
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