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
x
sinh (x)
cosh (x)
tanh (x)
coth (x)
sech (x)
csch (x)
Check out 36 similar algebra calculators πŸ”‘
Absolute value equationAbsolute value inequalitiesAdding and subtracting polynomials… 33 more
People also viewed…

30 60 90 triangle

How to solve the 30 60 90 triangle? What are the 30 60 90 triangle rules? The 30 60 90 triangle calculator is a safe bet for your geometry problems!

Podcasts

Do you feel like you could be doing something more productive or educational while on a bus? Or while cleaning the house? Well, why don't you dive into the rich world of podcasts! With this podcast calculator, we'll work out just how many great interviews or fascinating stories you can go through by reclaiming your 'dead time'!

Scatter plot

This scatter plot calculator will allow you to visualize any set of 2D data points.

Significant figures

The significant figures calculator performs operations on sig figs and shows you a step-by-step solution!
Copyright by Omni Calculator sp. z o.o.
Privacy policy & cookies
main background