# Hyperbolic Functions

Hyperbolic Functions
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If you have a long rope, and you hold it from the two ends.

Do you know what is the function that gives this curve?

Hyperbolic function defines this curve and the equation of the curve is: $$\text{f(x) = a cosh}\left(\dfrac{x}{a} \right)$$

In this lesson, you will learn about the hyperbolic functions. We will be covering their formulas, graphs, domain and range, and understand step by step the ways to solve problems on related topics.

You can check out the interactive examples to know more about the lesson and try your hand at solving a few interesting practice questions at the end of the page.

## What Is Meant by a Hyperbolic Function?

As an ordinary trigonometric function is defined for or on a circle, similarly hyperbolic function is defined for hyperbola.

In ordinary trigonometry, we were using sine, cosine, and other functions. Similarly, in hyperbola, we use sinh, cosh, and other hyperbolic functions.

As in ordinary trigonometric function, (cos θ, sin θ) forms a circle of unit radius, similarly in hyperbolic function, (cosh θ, sinh θ) forms right half of equilateral hyperbola.

They also occur in the solutions of many linear differential equations, cubic equations, and Laplace's equation(Laplace's equation is very important in the field of physics, maths, and engineering).

### The basic hyperbolic functions are:

1. Hyperbolic sine or sinh
2. Hyperbolic cosine or cosh

Apart from these two basic functions, the other 4 functions which can be derived from these basic functions are:

• hyperbolic tangent or tanh
• hyperbolic cosecant  or cosech
• hyperbolic secant or sech
• hyperbolic cotangent or coth

## What Are the Formulas of Hyperbolic Function?

The hyperbolic functions are defined through the algebraic expressions that include the exponential function (ex) and its inverse exponential functions (e-x), where e is the Euler’s constant.

Let us see all the hyperbolic functions one by one:

1. Hyperbolic sine function or Sinh $$x$$:

This is the odd part of the exponential functions.

Algebriac expression for hyperbolic sine function is:

$\text {sinh x } = \dfrac{ {(\text e^\text x - \text e^\text {-x})}}2$

2. Hyperbolic cosine function or Cosh $$x$$:

This is the even part of the exponential function.

Algebriac expression for hyperbolic cosine function is:

$\text {cosh x } = \dfrac{ {(\text e^\text x + \text e^\text {-x})}}2$

3. Hyperbolic tangent function or Tanh $$x$$:

$\text {tanh x } = \dfrac {\text { sinh x }}{ \text {cosh x}} = \dfrac{ {(\text e^\text x - \text e^\text {-x})}} {(\text e^\text x + \text e^\text {-x})}$

4. Hyperbolic cotangent $$x$$ or Cot $$x$$:

$\text {coth x } = \dfrac { \text {cosh x}} {\text { sinh x }}= \dfrac{(\text e^\text x + \text e^\text {-x})}{ {(\text e^\text x - \text e^\text {-x})}}$

5. Hyperbolic secant $$x$$ or Sec $$x$$:

$\text {sech x } = \dfrac 2 { {(\text e^\text x + \text e^\text {-x})}}$

6. Hyperbolic cosecant $$x$$ or Cosec $$x$$:
$\text {cosech x } = \dfrac 2{ {(\text e^\text x - \text e^\text {-x})}}$

## Graphical Representation of Hyperbolic Function

Graph of the hyperbolic functions are given below:

1. Hyperbolic sine function or sinh $$x$$:

Domain: R and Range: R

2. Hyperbolic cosine function or cosh $$x$$:

Domain: R and Range: $$[1 , +\infty)$$

3. Hyperbolic tangent function or tanh $$x$$:

Domain: R and Range: $$(-1 , + 1)$$

4. Hyperbolic cotangent $$x$$ or coth $$x$$:

Domain: $$(-\infty , 0)$$ U $$(0 , +\infty)$$ and Range: $$(-\infty , -1)$$ U $$(1 , +\infty)$$

5. Hyperbolic secant $$x$$ or sech $$x$$:

Domain: R and Range: (0 , 1]

6. Hyperbolic cosecant $$x$$ or cosech $$x$$:

Domain: $$(-\infty , 0)$$ U $$(0 , +\infty)$$ and Range: $$(-\infty , 0)$$ U $$(0 , +\infty)$$

## What Are the Properties of the Hyperbolic Function?

The properties of hyperbolic functions are analogous to the properties of trigonometric functions.

Examples

• $$\text {sinh (-x) = – sinh(x)}$$
• $$\text {cosh (-x) = cosh (x)}$$
• $$\text {cosh 2x = 1 + 2 sinh}^2 (\text x)$$
• $$\text {cosh 2x = cosh 2x + sinh 2x}$$
• $$\dfrac d{d \ \text x} \text{ sinh x = cosh x}$$
• $$\dfrac {\text d}{\text d \ \text x} \text{ cosh x = sinh x}$$

Hyperbolic functions can also be deduced from trigonometric functions with complex arguments:

• $$\text {sinh x = i sin(ix)}$$
• $$\text {cosh x = cos(ix)}$$
• $$\text {tanh x = - i tan(ix)}$$
• $$\text {coth x = i cot(ix)}$$
• $$\text {sech x = sec(ix)}$$
• $$\text {cosech x = i cosec(ix)}$$

## Are Hyperbolic Function Identities Similar to Trigonometric Identities?

The hyperbolic function identities are similar to trigonometric identities and can be understood better from below.

### Explanations

Osborn's rule states that, trigonometric identities can be converted into hyperbolic identities when expanded completely in terms of integral powers of sines and cosines, which includes changing sine to sinh, cosine to cosh. Sign of every term that contains a product of two sinh should be replaced.

$$\text {sinh x – sinh y} = 2\text {cosh}\dfrac {\text {(x+y)}}2 \text { sinh}\dfrac {\text {(x+y)}}2$$

$$\text {sinh x + sinh y} = 2\text {sinh}\dfrac {\text {(x+y)}}2 \text { cosh}\dfrac {\text {(x+y)}}2$$

$$\text {cosh x + cosh y} = 2\text {cosh}\dfrac {\text {(x+y)}}2 \text { cosh}\dfrac {\text {(x+y)}}2$$

$$\text {cosh x – cosh y} = 2\text {sinh}\dfrac {\text {(x+y)}}2 \text { sinh}\dfrac {\text {(x+y)}}2$$

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

$$2 \text {cosh x sinh y = sinh(x + y) – sinh(x – y)}$$

$$2 \text {sinh x sinh y = cosh(x + y) – cosh(x – y)}$$

$$2 \text {cosh x cosh y = cosh(x + y) + cosh(x – y)}$$

$$\text {sinh(x ± y) = sinh x cosh x ± coshx sinh y }$$

$$\text {cosh(x ± y) = cosh x cosh y ± sinh x sinh y}$$

$$\text {tanh(x ± y) }= \ \dfrac {\text {(tanh x ± tanh y})} {(1± \text { tanh x tanh y} )}$$

$$\text {coth(x ± y)} = \dfrac {(\text {coth x coth y} ± 1) }{(\text {coth y ± coth x)} }$$

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

$$\text {tanh}^2 (\text x) – \text {sech}^2 (\text x) = 1$$

$$\text {coth}^2 (\text x) – \text {cosech}^2 (\text x) = 1$$

Important Notes
• The hyperbolic function identities are similar to trigonometric identities
• The hyperbolic sine function is the odd part of the exponential functions, and the hyperbolic cosine function is the even part of the exponential functions.

## What Is the Inverse of a Hyperbolic Function?

The inverse of a hyperbolic function is the inverse function of the hyperbolic function.

For example, if $$x$$ = sinh y, then y = sinh-1 $$x$$ is the inverse of the hyperbolic sine function.

The principal values of the inverse hyperbolic functions expressed in terms of logarithmic functions are shown below:

1. $${\displaystyle \operatorname {sinh^{-1}} \text x=\ln \left(\text x+{\sqrt {\text x^{2}+1}}\right)}$$

2. $${\displaystyle \operatorname {cosh^{-1}} \text x=\ln \left( \text x+{\sqrt { \text x^{2}-1}}\right)}$$

3. $${\displaystyle \operatorname {tanh^{-1}} \text x={\frac {1}{2}}\ln \left({\frac {1+\text x}{1-\text x}}\right)}$$

4. $${\displaystyle \operatorname {coth^{-1}} \text x={\frac {1}{2}}\ln \left({\frac {\text x+1}{\text x-1}}\right)}$$

5. $${\displaystyle \operatorname {sech^{-1}} \text x=\ln \left({\frac {1}{\text x}}+{\sqrt {{\frac {1}{\text x^{2}}}-1}}\right)}$$

6. $${\displaystyle \operatorname {cosech^{-1}} \text x=\ln \left({\frac {1}{\text x}}+{\sqrt {{\frac {1}{\text x^{2}}}+1}}\right)}$$

Challenging Questions

• Do you know why the inverse of a hyperbolic function is also called an area hyperbolic function?

## Solved Examples

 Example 1

Daniel wants to simplify: $\dfrac{ \sinh x }{ \cosh y}$ Can you simplify this?

Solution

Since,

$\text {sinh x } = \dfrac{ {(\text e^\text x - \text e^\text {-x})}}2$

and,

$\text {cosh x } = \dfrac{ {(\text e^\text x + \text e^\text {-x})}}2$

Now,\begin{align*} \dfrac {\text {sinh x } }{\text {cosh x }} &= \dfrac{ \dfrac{ {(\text e^\text x - \text e^\text {-x})}}2} { \dfrac{ {(\text e^\text x + \text e^\text {-x})}}2} \\ \dfrac {\text {sinh x } }{\text {cosh x }} &= \dfrac{ {(\text e^\text x - \text e^\text {-x})}} { {(\text e^\text x + \text e^\text {-x})}} \end{align*}

 $$\therefore \dfrac{ {(\text e^\text x - \text e^\text {-x})}} { {(\text e^\text x + \text e^\text {-x})}}$$
 Example 2

Ron is puzzled and is unable to find the value of $$x$$ in $$3\sinh x- 2\cosh x -2 = 0.$$ Can you help him?

Solution

Putting values of cosh $$x$$ and sinh $$x$$, we get

\begin{align*} 3 \dfrac{ {(\text e^\text x - \text e^\text {-x})}}2 - 2 \dfrac{ {(\text e^\text x + \text e^\text {-x})}}2 - 2 &= 0 \\ 3 {(\text e^\text x - \text e^\text {-x})} - 2 {(\text e^\text x + \text e^\text {-x})} - 4 &= 0 \\ \text e^\text x -5 \text e^{\text -x} - 4 &= 0 \\ ( \text e^\text x)^2 -4 \text e^{\text x} - 5 &= 0 \\ (\text e^\text x - 5)(\text e^\text x + 1) &= 0 \end{align*}

Since,

$\text e^\text x ≠ -1$

\begin{align*} \therefore \text e^\text x &= 5 \\ \text x &= \text {ln } 5 \end{align*}

 $$\therefore \text x = \text {ln } 5$$
 Example 3

How will you show that $$\text {cosh x + sinh x} = \text e^\text x$$?

Solution

Since,

$\text {sinh x } = \dfrac{ {(\text e^\text x - \text e^\text {-x})}}2$

and,

$\text {cosh x } = \dfrac{ {(\text e^\text x + \text e^\text {-x})}}2$

LHS

\begin{align*} &= \dfrac{ {(\text e^\text x - \text e^\text {-x})}}{2} + \dfrac{ {(\text e^\text x + \text e^\text {-x})}}2 \\ &= \dfrac{ 2 {(\text e^\text x}) + {(\text e^\text {-x} - \text e^\text {-x})}}2 \\ &= {\text e^\text x} \end{align*}

Hence,

 RHS = LHS
 Example 4

Help Nathan prove that $$\text {cosh (x-y)} = \text {coshx coshy - sinhx sinhy}$$.

Solution

\begin{align*} \text {cosh x } \text {cosh y } &= \dfrac{ {(\text e^\text x + \text e^\text {-x})}}2 \times \dfrac{ {(\text e^\text y + \text e^\text {-y})}}2 \\ \text {sinh x } \text {sinh y } &= \dfrac{ {(\text e^\text x - \text e^\text {-x})}}2 \times \dfrac{ {(\text e^\text y - \text e^\text {-y})}}2 \end{align*}

On subtracting, we get:

$$\text {cosh x } \text {cosh y } - \text {sinh x } \text {sinh y }$$

\begin{align*} &= 2\times \dfrac{ { \text e^\text {x-y} + \text e^\text {-(x-y)}} } 4 \\ &= \dfrac{ { \text e^\text {x-y} + \text e^\text {-(x-y)}} } 2 \\ &= \text{cosh (x-y)} \end{align*}

Hence,

 $$\therefore$$ $$\text {cosh (x-y)} = \text {coshx coshy - sinhx sinhy}$$
 Example 5

Find the logarithmic form of $$\sinh^{-1}\left( \dfrac34\right).$$

Solution

Since,

${\displaystyle \operatorname {sinh^{-1}} \text x=\ln \left(\text x+{\sqrt {\text x^{2}+1}}\right)}$

Putting:

$\text x = \dfrac34$

We get,

\begin{align*} \sinh^{-1} \dfrac34 &=\ln \left(\dfrac 34+{\sqrt { {\dfrac 9{16}+1}}}\right) \\ \sinh^{-1} \dfrac34 &=\ln \left(\dfrac 34+{\sqrt { {\dfrac{ 25}{16}}}}\right) \\ \sinh^{-1} \dfrac34 &=\ln \left(\dfrac 34+{ {\dfrac 5{4}}}\right)\\ \sinh^{-1} \dfrac34 &=\ln \left( \dfrac 8{4} \right) \\ \sinh^{-1} \dfrac34&=\ln \left(\dfrac {8}4\right)\\ \sinh^{-1} \dfrac34 &= \ln 2 \end{align*}

 $$\therefore {\sinh^{-1}} \dfrac34= \ln 2$$

## Interactive Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

## Let's Summarize

The mini-lesson targeted the fascinating concept of hyperbolic function. The math journey around the hyperbolic function started with what a student already knew and went on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever.

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

## 1. What is the difference between trigonometric function and hyperbolic function?

As a trigonometric function is defined for or on a circle, but a hyperbolic function is defined for hyperbola.

In trigonometry, we use sine, cosine, and other functions. Similarly, in hyperbola, we use sinh, cosh, and other hyperbolic functions.

## 2.  How do you derive the inverse of a hyperbolic function?

The standard way to derive the formula for sinh−1x goes like this:

Put y = sinh−1x, so that:

x = sinh y = (ey−e−y)/2.

Rearrange this, to get 2x=ey−e−y.

And hence e2y−2xey−1=0, which is a quadratic equation in ey.

You then solve the quadratic and take logs (and take care of the ± sign you get with the roots of the quadratic).

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