Hyperbolic functions are similar to trigonometric functions. These functions are denoted by sinh, cosh, tanh, cosech (csch), sech, coth and are pronounced as hyperbolic sin (or sin hyperbolic), hyperbolic cos, hyperbolic tan, hyperbolic cosec, hyperbolic sec, hyperbolic cot. Unlike trigonometric functions which are defined in context with right triangle, hyperbolic functions are defined as exponential expressions.

Sine and cosine are defined as follows:

  • $\sinh x$ = $\frac{e^{x}-e^{-x}}{2}$
  • $\cosh x$ = $\frac{e^{x}+e^{-x}}{2}$

Expressions of all other hyperbolic functions can be derived by using above two relations.

Hyperbolic functions share various properties similar to that of trigonometric functions. Few important properties of hyperbolic functions are as follows:

Hyperbolic functions are defined in terms of exponents as described below:

  1. $\sinh x$ = $\frac{e^{x}-e^{-x}}{2}$
  2. $\cosh x$ = $\frac{e^{x}+e^{-x}}{2}$
  3. $\tanh x$ = $\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}$
  4. $\text{cosech} x$ = $\frac{2}{e^{x}-e^{-x}}$
  5. $\text{sech} x$ = $\frac{2}{e^{x}+e^{-x}}$
  6. $\coth x$ = $\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}$

Hyperbolic functions can be defined in terms of imaginary unit "i", where $i^{2} = -1$:

  1. $\sinh x = -i \sin ix$
  2. $\cosh x = \cos ix$
  3. $\tanh x = - i \tan ix$
  4. $\text{cosech} x = i \csc ix$
  5. $\text{sech} x = \sec ix$
  6. $\coth x = i \cot ix$

T
hese functions are either even or odd.
  1. $\sinh (-x) = -\sinh x$: Odd function.
  2. $\cosh (-x) = \cosh x$: Even function.
  3. $\tanh (-x) = -\tanh x$: Odd function.
  4. $\text{cosech} (-x) = -\text{cosech} x$: Odd function.
  5. $\text{sech} (-x) = \text{sech} x$: Even function.
  6. $\coth (-x) = -\coth x$: Odd function.
Hyperbolic functions follow the following useful identities:

Fro
m the exponential expressions of hyperbolic sine and hyperbolic cosine, we can derive following two relations:
  1. $e^{x} = \cosh x + \sinh x$
  2. $e^{-x} = \cosh x - \sinh x$

Pythagorean trigonometric identities:

  1. $\cosh^{2}x - \sinh ^{2}x = 1$
  2. $\tanh^{2}x + \text{sech} ^{2}x = 1$
  3. $\coth^{2}x - \text{cosech} ^{2}x = 1$ 

Hyperbolic functions of sum and difference:

  1. $\sinh (x \pm y) = \sinh x \cosh y \pm \cosh x \sinh y$
  2. $\cosh (x \pm y) = \cosh x \cosh y \pm \sinh x \sinh y$
  3. $\tan h (x \pm y)$ = $\frac{\tanh x \pm \tanh y}{1 \pm \tanh x \tanh y}$

Double angle formulas:

  1. $\sinh 2x = 2 \sinh x \cosh x$
  2. $\cosh 2x = \cosh ^{2}x + \sinh ^{2}x = 2 \cosh ^{2}x - 1 = 1 + 2 \sinh ^{2}x$
  3. $\tanh 2x$ = $\frac{2 \tanh x}{1 + \tanh ^{2}x}$
Derivatives of hyperbolic functions are listed below:
  • $\frac{\mathrm{d} }{\mathrm{d} x}$$\sinh x = \cosh x$
  • $\frac{\mathrm{d} }{\mathrm{d} x}$$\cosh x = \sinh x$
  • $\frac{\mathrm{d} }{\mathrm{d} x}$$\tanh x = \text{sech} ^{2}x$
  • $\frac{\mathrm{d} }{\mathrm{d} x}$$\coth x = -\text{cosech} ^{2}x$
  • $\frac{\mathrm{d} }{\mathrm{d} x}$$\text{cosech} x = -\text{cosech} x \coth x$
  • $\frac{\mathrm{d} }{\mathrm{d} x}$$\text{sech} x = -\text{sech} x \tanh x$
Inverse hyperbolic functions are the inverse functions of hyperbolic functions. These are denoted by $\sinh ^{-1}$, $\cosh ^{-1}$, $\tanh ^{-1}$, $\text{cosech} ^{-1}$, $\text{sech} ^{-1}$, $\coth ^{-1}$ or by arcsinh, arccosh, arctanh, arccosech, arcsech, arccoth.
Inverse hyperbolic functions are defined in complex plane as follows:
  • $\sinh ^{-1}z = \ln (z + \sqrt{z^{2} + 1})$
  • $\cosh ^{-1}z = \ln (z + \sqrt{(z + 1)(z - 1)})$
  • $\tanh ^{-1}z$ = $\frac{1}{2}$$ \ln $$(\frac{1+z}{1-z})$
All hyperbolic functions share many properties same as trigonometric functions. These functions are denoted in the similar manner as trigonometric functions are represented. Hyperbolic functions also follow almost similar identities as trigonometric functions. Therefore, These hyperbolic functions are also knows as hyperbolic trig functions or hyperbolic trigonometric functions.
Graphs of hyperbolic functions are shown below:

Graph of sinh x:

Graph of sinh x

Graph of cosh x:

Graph of cosh x

Graph of tanh x:

Graph of tanh x

Graph of cosech x:

Graph of cosech x

Graph of sech x:

Graph of sech x

Graph of coth x:

Graph of coth x
Initially, practical applications of hyperbolic functions were not very common. But, in the past recent years, hyperbolic functions have become fairly applicable in various fields.

Important applications of hyperbolic functions are as follows:

Catenary:
Let us consider a string which is supported only at its ends and hangs down making a U-shape curve. This curve is known as catenary.
Hyperbolic Functions Applications
The equation of a catenary is $y = a \cosh $$(\frac{x}{a})$. Catenaries are widely used in various applications related to hanging strings.

Catenoid:
Catenoids are three-dimensional shapes as shown in the following figure:

Hyperbolic Functions Application

The equation of a catenoid is also derived from hyperbolic cosine. Catenoids are frequently used in physics and chemistry.

Hyperbolic functions are commonly used in quantitative applications
in engineering when current, voltage, resistance etc are given in complex form and they are to be transformed into trigonometric functions using hyperbolic trigs. Hyperbolic functions have many different applications in economics and finance also. Hyperbolic functions are widely used in architecture and civil engineering while designing and constructing mega structures.
Given below are few examples based on hyperbolic functions:

Solved Examples

Question 1: Given that $\cosh x$ = $\frac{7}{6}$, determine the value of $\sinh x$ and $\tanh x$.
Solution:
$\cosh ^{2}x - \sinh ^{2}x = 1$

$(\frac{7}{6})$$^{2} - \sinh ^{2}x = 1$

$\sinh ^{2}x$ = $\frac{49}{36}$$ - 1$

$\sinh ^{2}x$ = $\frac{13}{36}$

$\sinh x$ = $\frac{\sqrt{13}}{6}$

$\tanh x$ = $\frac{\sinh x}{\cosh x}$

$\tanh x$ = $\frac{\frac{\sqrt{13}}{6}}{\frac{7}{6}}$

$\tanh x$ = $\frac{\sqrt{13}}{7}$

Question 2: Given that $\sinh x$ = $\frac{6}{5}$. Find the value of $\coth x$ and $\text{sech} x$.
Solution:
$\sinh x$ = $\frac{6}{5}$

Step 1: We know that, $\text{cosech} x$ = $\frac{1}{\sinh x}$

So, $\text{cosech} x$ = $\frac{5}{6}$

Step 2: Using this in the following identity:
$\coth ^{2}x - \text{cosech} ^{2}x = 1$
$\coth ^{2}x = 1 + \text{cosech} ^{2}x$

$\coth ^{2}x = 1 + $$(\frac{5}{6})$$^{2}$

$\coth^{2}x = 1 + $$\frac{25}{36}$

$\coth ^{2}x$ = $\frac{\sqrt{61}}{6}$

$\tanh x$ = $\frac{1}{\coth x}$

$\tanh x$ = $\frac{6}{\sqrt{61}}$

Step 3: Using this in the following identity:
$\tanh ^{2}x + \text{sech} ^{2}x = 1$
$\text{sech} ^{2}x = 1 - \tanh ^{2}x$
$\text{sech} ^{2}x = 1 - $$\frac{36}{61}$

$\text{sech} ^{2}x$ = $\frac{25}{61}$

$\text{sech} x$ = $\frac{5}{\sqrt{61}}$