# Hyperbolic Functions Calculator

Created by Bogna Szyk
Reviewed by Steven Wooding
Last updated: Dec 11, 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. Visit our sin calculator and cosine calculators to explore these topics further.

What's the difference, then? If you plot points with coordinates ($\cos x$, $\sin x$) in a Cartesian coordinate system, they will form a circle. But if you plot points with coordinates ($\cosh x$, $\sinh x$), they will create a hyperbola (as shown below).

## 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 $\sinh x$ – the hyperbolic sine – you need to use the following formula:

$\small \sinh x = \frac{1}{2}(e^x - e^{-x})$

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

$\small \cosh x = \frac{1}{2}(e^x + e^{-x})$

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

\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, $\sinh x$ or $\tanh x$), and this tool will automatically return the value of $x$.

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

\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*}

## FAQ

### What is an hyperbolic function?

A hyperbolic function is a function similar in definition to a trigonometric function but with some major differences:

• Hyperbolic functions corresponds to the parametrization of a hyperbola, and not a circle;
• Hyperbolic functions are not periodic;
• Hyperbolic functions don't require complex numbers in their definition.

### What is the parity of the hyperbolic functions sinh, cosh, and tanh?

The parity of the hyperbolic functions doesn't differ from the one of the normal trigonometric functions:

1. The hyperbolic sine (sinh) has odd parity (reflection on both the y and x-axis);
2. The hyperbolic cosine (cosh) has even parity (reflection only on the y axis);
3. The hyperbolic tangent (tanh) has odd parity as the hyperbolic sine.

### How do I calculate the values of the three most important hyperbolic functions?

To calculate the values of sinh, cosh, and tanh, you only need to know how to calculate the exponential function exp(x). Here is the definition of the three hyperbolic functions:

1. The hyperbolic sine is a subtraction of exponential:
sinh(x) = (exp(x) - exp(-x))/2
2. The hyperbolic cosine is a sum:
cosh(x) = (exp(x) + exp(-x))/2
3. The hyperbolic tangent is the ratio of the previous two hyperbolic functions:
tanh(x) = sinh(x)/cosh(x).

### What are the values of sinh(0) and cosh(0)?

The value of the hyperbolic functions sinh and cosh in the point 0 of the x-axis is, thanks to the properties of the exponential function, the same as the "normal trigonometric functions":

1. The value of the hyperbolic sine is: sinh(0) = 0;
2. The value of the hyperbolic cosine is: cosh(0) = 1.

Since the hyperbolic tangent is defined as the ratio between sinh and cosh, it has the value: tanh(0) = 0`.

Bogna Szyk
x
sinh (x)
cosh (x)
tanh (x)
coth (x)
sech (x)
csch (x)
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