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Hyperbolic Functions

Hyperbolic functions are nearly similar to trigonometric functions. They are defined through the algebraic expressions which include
exponential function $e^x$
 and its inverse function $e^{-x}$, where e is the Euler’s constant. The approximate value of e is 2.718281. The basic hyperbolic functions are hyperbolic sine (sinh) and hyperbolic cosine (cosh), from which the hyperbolic tangent (tanh) function is derived. The other functions coth, sech, cosech are reciprocals of the above three functions correspondingly. We shall learn about hyperbolic functions in this article.

Related Calculators
Calculator Functions Calculate Exponential Function
Calculate Inverse Function calculating gamma function
 

Properties of Hyperbolic Functions

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The properties of the hyperbolic functions are analogous to trigonometric functions. Let us discuss in detail,
  1. $sinh(-x)$ = - $sinh\ x$ ($sinh\ x$ is a odd function)
  2. $cosh(-x)$ = $cosh\ x$ ($cosh\ x$ is a even function)
  3. $sinh\ 2x$ = $2sinh\ x\ cosh\ x$
  4. $cosh\ 2x$ = $cosh2\ x$ + $sinh2\ x$
The derivatives of the basic hyperbolic functions are$\frac{d}{dx}$ sinh(x) = cosh x
  1. $\frac{d}{dx}$ $cosh(x)$ = $sinh\ x$
The relation of hyperbolic functions to trigonometric functions is as follows:
  1. $sinh\ x$ = $-\ i\ sin(ix)$
  2. $cosh\ x$ = $cos(ix)$
  3. $tanh\ x$ = $-\ i\ tan(ix)$

Hyperbolic Function Identities

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The hyperbolic identities are also called as hyperbolic functions formulas which are quite similar to that of trigonometric functions as following:
Pythagorean Trigonometric Identities:

$cosh^2(x)$ - $sinh^2(x)$ = $1$

$tanh^2(x)$ + $sech^2(x)$ = $1$

$coth^2(x)$ - $cosech^2(x)$ = $1$

Sum and Difference Identities:

sinh(x $\pm$ y) = sinh x cosh x $\pm$ coshx sinh y

cosh(x $\pm$ y) = cosh x cosh y $\pm$ sinh x sinh y

tanh(x $\pm$ y) = $\frac{\tanh x \pm \tanh y}{1 \pm \tanh x \tanh y}$

coth(x $\pm$ y) = $\frac{\coth x \coth y \pm 1}{\coth y \pm \coth x}$

Sum to Product:

sinh x + sinh y = 2sinh($\frac{x+y}{2}$)cosh($\frac{x-y}{2}$)

sinh x - sinh y = 2cosh($\frac{x+y}{2}$)sinh($\frac{x-y}{2}$)

cosh x + cosh y = 2cosh($\frac{x+y}{2}$)cosh($\frac{x-y}{2}$)

cosh x - cosh y = 2sinh($\frac{x+y}{2}$)sinh($\frac{x-y}{2}$)

Product to Sum:

2sinh x cosh y = sinh(x + y) + sinh(x -y)

2cosh x sinh y = sinh(x + y) - sinh(x - y)

2sinh x sinh y = cosh(x + y) - cosh(x - y)

2cosh x cosh y = cosh(x + y) + cosh(x - y).


Graphs of Hyperbolic Functions

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The basic hyperbolic functions and their graphs are given below:

Hyperbolic Sine Function


The function f : R -> R defined by f(x) = $\frac{e^x - e^{-x}}{2}$ is called hyperbolic sine function and it is denoted by Sinh x.
   $Sinh\ x$ = $\frac{e^x - e^{-x}}{2}$

Graph of y = sinh x


Hyperbolic Sine Function

Hyperbolic Cosine Function

The function f : R -> R defined by f(x) = $\frac{e^x + e^{-x}}{2}$ is called hyperbolic cosine function and it is denoted by cosh x.

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

Graph of y = cosh x


Hyperbolic Cosine Function

Hyperbolic Tangent Function

The function f : R -> R defined by f(x) = $\frac{e^x - e^{-x}}{e^x + e^{-x}}$ is called hyperbolic tangent function and it is denoted by tanh x.

$tanh\ x$ = $\frac{e^x - e^{-x}}{e^x + e^{-x}}$


Graph of y = tanh x


Hyperbolic Tangent Function


Graphs of Other Hyperbolic Functions:

The function f : R -> R defined by f(x) =  $\frac{e^x + e ^{-x}}{e^x - e ^{- x}}$ is called hyperbolic cotangent function and it is denoted by coth x.

$coth\ x$ = $\frac{e^x + e ^{-x}}{e^x - e ^{- x}}$
Graph of y = coth x

graphs of hyperbolic functions

The function f : R -> R defined by f(x) = $\frac{2}{e^x + e ^{-x}}$ is called hyperbolic secant function and it is denoted by sech x.

$sech\ x$ = $\frac{2}{e^x + e ^{-x}}$


Graph of y = sech x

graphs of hyperbolic function

The function f : R -> R defined by f(x) = $\frac{2}{e^x - e^{-x}}$ is called hyperbolic cosecant function and it is denoted by cosech x.

$cosech\ x$ = $\frac{2}{e^x - e^{-x}}$
Graph of y = csch x

graph of hyperbolic functions

Hyperbolic Functions Applications

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Application of the hyperbolic functions play a very important role in the field of science and engineering, it takes place whenever light, velocity, electricity or radioactivity is gradually absorbed or extinguished. Trigonometric functions are intimately related to triangle geometry and hyperbolic functions occur in the theory of triangles in hyperbolic spaces.

Hyperbolic functions are related to hyperbola, in the same way, as the trigonometric functions are related to circle. A unit hyperbola can be defined with the help of the two basic hyperbolic functions as follows:

$x^2 - y^2$ = $1$ where, $x$ = $cosh\ t$ , $y$ = $sinh\ t$

$x^2 - y^2$ = $cosh^2\ t - sinh^2\ t$ = $1$

The main use of these functions is to integrate common and simple functions with less computation and the other use of these functions can be observed in the models of real life problems.

Inverse Hyperbolic Functions

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The inverse hyperbolic functions are the inverse functions of the hyperbolic functions. Inverse hyperbolic functions also called the area hyperbolic functions and are denoted by $\sinh ^{-1}$, $\cosh ^{-1}$, $\tanh ^{-1}$, $\text{csch} ^{-1}$, $\text{sech} ^{-1}$ and $\coth ^{-1}$. The inverse hyperbolic functions provide a hyperbolic angle corresponding to a given value of a hyperbolic function.

The inverse hyperbolic functions are defined in the complex plane by

$\sinh ^{-1} x = \ln(x + \sqrt{1 + x^2})$

$\cosh ^{-1} x = \ln(x + \sqrt{x^2 - 1})$

$\tanh ^{-1} x = \frac{1}{2} (\ln(1 + x) - \ln(1 - x))$

Complex Hyperbolic Functions

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The hyperbolic cosine and hyperbolic sine functions are:

$\sinh z$ = $\frac{e^z - e^{-z}}{2}$

$\cosh z$ = $\frac{e^z + e^{-z}}{2}$

The other complex hyperbolic trigonometric functions are created by using above functions.

$\tanh z$ = $\frac{\sinh z}{\cosh z}$ = $\frac{e^z - e^{-z}}{e^z + e^{-z}}$

$\text{csch} z$ = $\frac{1}{\sinh z}$ = $\frac{2}{e^z - e^{-z}}$

$\text{sech} z$ = $\frac{1}{\cosh z}$ = $\frac{2}{e^z + e^{-z}}$

$\coth z$ = $\frac{\cosh z}{\sinh z}$ = $\frac{e^z + e^{-z}}{e^z - e^{-z}}$.

Hyperbolic Functions Examples

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Given below are some of the examples in solving hyperbolic functions.

Solved Examples

Question 1: Solve $\cosh ^2x - \sinh ^2x$
Solution:
$\cosh x$ = $\frac{e^x + e^{-x}}{2}$

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

Therefore
$\cosh ^2 x - \sinh ^2 x = [\frac{e^x + e^{-x}}{2}]^2 - [\frac{e^x - e^{-x}}{2}]^2$

= $\frac{4 e^{x - x}}{4}$

= $\frac{4 e^0}{4}$

= $\frac{4 \times 1}{4}$

= $1$

Question 2: If $\sinh x$ = $\frac{3}{4}$, calculate $\cosh x$ and $\tanh x$.
Solution:
Step 1: Find the value of cosh x.

We know that, $cosh^2(x)$ - $sinh^2(x)$ = $1$

$cosh^2(x)$ = $1$ + $sinh^2(x)$

= $1 + $$\frac{9}{16}$

= $\frac{16 + 9}{16}$

= $\frac{25}{16}$

$\cosh x$ = $\frac{5}{4}$ ...................(i)

Step 2:

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

= $\frac{\frac{3}{4}}{\frac{5}{4}}$ (Using equation (i))

= $\frac{3}{4}$ $\times$ $\frac{4}{5}$

= $\frac{3}{5}$

$\tanh x$ = $\frac{3}{5}$



More topics in Hyperbolic Functions
Hyperbolic Identities Derivatives of Hyperbolic Functions
Hyperbolic Inverse Functions
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