exponential function $e^x$

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The properties of the hyperbolic functions are analogous to trigonometric functions. Let us discuss in detail,

The derivatives of the basic hyperbolic functions are$\frac{d}{dx}$ sinh(x) = cosh x

**Pythagorean Trigonometric Identities****:**

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

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

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

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.

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

### Hyperbolic Cosine Function

### Hyperbolic Tangent Function

### Graphs of Other Hyperbolic Functions:

**The inverse hyperbolic functions are defined in the complex plane by **

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

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

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

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

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

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

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

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

**Question 1: **Solve $\cosh ^2x - \sinh ^2x$

** Solution: **
**Question 2: **If $\sinh x$ = $\frac{3}{4}$, calculate $\cosh x$ and $\tanh x$.

** Solution: **
**Step 1:** Find the value of cosh 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))

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

= $\frac{3}{5}$

$\tanh x$ = $\frac{3}{5}$

- $sinh(-x)$ = - $sinh\ x$ ($sinh\ x$ is a odd function)
- $cosh(-x)$ = $cosh\ x$ ($cosh\ x$ is a even function)
- $sinh\ 2x$ = $2sinh\ x\ cosh\ x$
- $cosh\ 2x$ = $cosh2\ x$ + $sinh2\ x$

- $\frac{d}{dx}$ $cosh(x)$ = $sinh\ x$

- $sinh\ x$ = $-\ i\ sin(ix)$
- $cosh\ x$ = $cos(ix)$
- $tanh\ x$ = $-\ i\ tan(ix)$

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

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

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

**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).

The basic hyperbolic functions and their graphs are given below:

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**

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**

** **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

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

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**

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**

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.

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

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

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

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

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

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

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

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

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

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

Therefore

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

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

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

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

= $1$

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

Therefore

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

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

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

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

= $1$

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)

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

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