# Hyperbolic Functions Calculator

Created by Davide Borchia
Last updated: Jun 23, 2022

Our hyperbolic functions calculator will make these lesser-known functions as easy as they go — and this is not a hyperbola!

Simply input the number of which you want to know the hyperbolic functions' corresponding value: we will give you the result in no time. Here you will learn:

• What are the hyperbolic functions;
• How to calculate the hyperbolic sine and cosine;
• How to calculate the hyperbolic tangent;
• How to calculate the less common hyperbolic functions (hyperbolic secant and hyperbolic cosecant).

If you're interested in other kinds of functions, you can also find the zeros of the polynomial function with our calculator.

## What are hyperbolic functions?

Sine and cosine, the basic trigonometric functions, define a circumference using only an angle, by means of the parametric equation of a circle $(\sin{(t)},\cos{(t)})$. These functions are intimately connected to the exponential function. Introducing a bit of... complexity in their description, we can define the two functions with the expressions starting from the complex circle:

\begin{align*} \sin{(x)}&=\frac{e^{ix}-e^{-ix}}{2i}\\ \\ \cos{(x)}&=\frac{e^{ix}+e^{-ix}}{2} \end{align*}

The imaginary unit $i$ somehow helps create the circle introducing an "independent coordinate".

If we remove it from the formulas, we obtain a map of a hyperbola. The modified functions assume slightly modified names: hyperbolic sine and hyperbolic cosine. Here is how we calculate the hyperbolic cosine and sine using the exponential function:

\begin{align*} \sinh{(x)}&=\frac{e^{x}-e^{-x}}{2}\\ \\ \cosh{(x)}&=\frac{e^{x}+e^{-x}}{2} \end{align*}

The parametric expression $(\sinh{(t)},\cosh{(t))}$ now maps the unit hyperbola.

## Other hyperbolic functions: the hyperbolic tangent and much more

Now that you know the most important hyperbolic functions, we can introduce the remaining ones.

Using the same relations holding for trigonometric functions, we define the hyperbolic tangent and hyperbolic cotangent: we calculate the hyperbolic tangent with the ratio between hyperbolic sine and hyperbolic cosine.

\begin{align*} \tanh{(x)}&=\frac{\sinh{(x)}}{\cosh{(x)}}=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}\\ \\ \coth{(x)}&=\frac{\cosh{(x)}}{\sinh{(x)}}=\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}\\ \end{align*}

Do you see the similarity with the "normal" trigonometric tangent?

And eventually the hyperbolic cosecant and the hyperbolic secant:

\begin{align*} \text{sech}{(x)}&=\frac{1}{\cosh{(x)}}=\frac{2}{e^{x}+e^{-x}}\\ \\ \text{csch}{(x)}&=\frac{1}{\sinh{(x)}}=\frac{2}{e^{x}-e^{-x}}\\ \end{align*}

## Some features of hyperbolic functions

Hyperbolic functions are not periodic. We lost that trait when we moved away from the circle. We now have functions that either follow an exponential growth or asymptotically tend to a given value.

For large positive values, the functions $\text{sinh}$ and $\text{cosh}$ closely approximate each other, thanks to the features of the exponential function: for $x\gg1$, $e^x\gg e^{-x}$. THe same hold for negative values and the pair of funcitons $\text{sinh}$ and $-\text{cosh}$.

Many identities hold for the hyperbolic functions: we remember only the most important one:

$\cosh{(x)}^2 - \sinh{(x)}^2=1$

The rest is for you to explore! After dealing with the hyperbolic functions, we encourage you to see our determinant solver, which is another handy tool.

## Hyperbolic functions and where to find them

Even if they look pretty niche, hyperbolic functions find their fair share of applications inside and outside the academic world.

The most common (and elegant) example of this is the catenary curve: the shape of a rope hanging freely between two supports under the sole influence of the gravitational . The curve is exactly described by a hyperbolic cosine.

Hyperbolic functions also appear when you consider the air resistance in a free-fall problem.

Last but not least, the hyperbolic tangent function often covers the role of the activation function in the field of machine learning. Hyperbolic functions were also proposed in neuron models.

## How to use our hyperbolic functions calculator

Simply insert the desired value of $x$ in the first field of our hyperbolic functions calculator: we will calculate all the six hyperbolic functions. You can also use our calculator in reverse: insert a known value of a hyperbolic function in the correct field, and we will calculate the inverse!

WIth our hyperbolic functions calculator, you can forget about searching for the right button on your calculator: in a single page, you can find:

• Sinh calculator;
• Cosh calculator;
• Tanh calculator;
• Coth calculator;
• Sech calculator; and
• Csch calculator.
Davide Borchia
x
sinh (x)
cosh (x)
tanh (x)
coth (x)
sech (x)
csch (x)
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