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Hyperbolic Functions

- Calculus
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In math, there exist certain **even and odd** combinations of the **natural exponential functions** and which appear so frequently that they earned themselves their own special names. They are, in many ways, analogous to the trigonometric functions. For instance, they share the **same relationship to the hyperbola** that the t**rigonometric functions have to the circle**. Because of this, these special functions are called **hyperbolic functions**.

This article discusses the basic hyperbolic functions and their properties, identities, derivatives, integrals, inverses, and examples in detail.

- Hyperbolic function definition
- Hyperbolic functions: formulas
- Hyperbolic functions: graphs
- Hyperbolic functions: properties and identities
- Derivatives of hyperbolic functions
- Integrals of hyperbolic functions
- Inverse hyperbolic functions
- Hyperbolic functions: examples and applications

What are the hyperbolic functions?

The **hyperbolic functions** are essentially the trigonometric functions of the hyperbola. They extend the notion of the parametric equations for the unit circle, where , to the parametric equations for the unit hyperbola, and are defined in terms of the natural exponential function (where is Euler's number), giving us the following two fundamental hyperbolic formulas:

Based on these two definitions: **hyperbolic cosine** and **hyperbolic sine**, the rest of the six main hyperbolic functions can be derived as shown in the table below.

The formulas for the hyperbolic functions are listed below:

The 6 main hyperbolic formulas | |||

Hyperbolic sine: | Hyperbolic cosecant: | ||

Hyperbolic cosine: | Hyperbolic secant: | ||

Hyperbolic tangent: | Hyperbolic cotangent: |

Whereis pronounced "cinch", is pronounced "cosh", is pronounced "tanch", is pronounced "coseech", is pronounced "seech", andis pronounced "cotanch".

A key characteristic of the hyperbolic trigonometric functions is their similarity to the trigonometric functions, which can be seen from **Euler's formula**:

Solving this for both cosine and sine gives us:

Which is strikingly similar to the hyperbolic cosine and sine functions:

But notice that the hyperbolic functions do not have the imaginary part that Euler's formula does.

Why are the imaginary numbers missing from the hyperbolic functions?

Because when we solve Euler's formula for the hyperbolic functions, the imaginary component does not exist within the solution to the hyperbolic functions.

The graphs of the two fundamental hyperbolic functions: hyperbolic sine and hyperbolic cosine, can be sketched using **graphical addition** as shown below.

The graph of | The graph of |

The graphs of the rest of the six main hyperbolic functions are shown below.

The graphs of hyperbolic cosecant, secant, tangent, and cotangent | |

The graph of | The graph of |

The graph of | The graph of |

Notice that these hyperbolic functions all have horizontal (green) and/or vertical (pink) asymptotes. The graph of hyperbolic secant has a global maximum at the point .

While we are looking at the graphs of the hyperbolic functions, let's take note of their domains and ranges!

Function | Domain | Range |

The hyperbolic function properties and identities are also quite similar to those of their trigonometric counterparts:

Properties:

They can be deduced from trigonometric functions that have

**complex arguments**:

Identities:

Let's test our understanding of these identities!

Prove that (a) and (b) .

Solution:

(a) We start with the definitions of hyperbolic cosine and hyperbolic sine, then simplify:

(b) We start with the proof from part (a):

Now, if we divide both sides by , we get:

This simplifies to:

This example gave us some insight into **why **these are called hyperbolic functions. Let's dive a bit deeper into it!

♦ Say we have a real number, . Then the point is on the unit circle:

because

In fact, the real number can also be interpreted as the radian measure of in the image below. This is the reason the trigonometric functions are sometimes called **circular **functions.

More importantly, the real number **represents twice the area of the shaded section** of the circle.

♦ Likewise, if is any real number, then the point is on the **right half of the unit hyperbola**:

because

where .

In this case, however, **does not represent the measure of an angle**, but instead, **it ****represents twice the area of the shaded hyperbolic section** in the image below.

The derivatives of hyperbolic functions are also analogous to the trigonometric functions. We list these derivatives in the table below.

Derivatives of the 6 main hyperbolic functions | |

Beware! While the **values of the derivatives are the same** as with the trigonometric functions, the **signs** for the derivatives of **hyperbolic cosine** and **hyperbolic secant** are **opposite **to their trigonometric counterparts.

It is also important to note that any of these differentiation rules can be combined using The Chain Rule. For instance,

The derivatives of hyperbolic functions are simpler to calculate because of their use of and the simplicity of its derivation.

For instance,

Just as the derivatives of hyperbolic functions are analogous to their trigonometric counterparts, so are the integrals of hyperbolic functions. We list these integrals in the table below.

Integrals of the 6 main hyperbolic functions | |

Other useful integrals of hyperbolic functions are listed below.

More integrals of hyperbolic functions | |

Based on the graphs of the hyperbolic functions, we can see that (and its reciprocal, ) and (and its reciprocal, ) are **one-to-one functions**, but (and its reciprocal, ) are not.

This is because **cosine **and **secant **are **even ****functions**, while **sine**, **cosecant**, **tangent**, and **cotangent **are **odd ****functions**.

Since cosine and secant are even functions, and are therefore not one-to-one, we have to **restrict their domain** to find their inverses.

So, with cosine's and secant's domains restricted to the interval , all the hyperbolic functions are one-to-one, and we can define the inverse hyperbolic functions as:

Their formulas are:

The 6 main inverse hyperbolic functions | |||

Inverse hyperbolic sine: | Inverse hyperbolic cosecant: | ||

Inverse hyperbolic cosine: | Inverse hyperbolic secant: | ||

Inverse hyperbolic tangent: | Inverse hyperbolic cotangent: |

Notice that the inverse hyperbolic functions all involve logarithmic functions. This is because the hyperbolic functions involve exponential functions, and exponential and logarithmic functions are inverses of each other!

Let's look at how the inverse of sinh (also called arc sinh) is derived. The derivation for the others follows a similar pattern.

Suppose that:

This means that:

By the definition of hyperbolic sine:

Rearranging this, we get:

Then, multiplying both sides by , we have:

Now we solve it like we would a quadratic function, thinking of as , and get the solution:

Since , the only possible solution is the positive one:

Finally, taking the natural logarithm of both sides, we get:

The graphs of the inverse hyperbolic functions are shown below.

The graphs of the inverse hyperbolic functions | |

The graph of | The graph of |

The graph of | The graph of |

The graph of | The graph of |

Notice that inverse hyperbolic cosecant, secant, tangent, and cotangent have horizontal (green) and/or vertical (pink) asymptotes. The graphs of inverse hyperbolic cosine and inverse hyperbolic secant have a definite beginning point at .

While we are looking at the graphs of the inverse hyperbolic functions, let's take note of their domains and ranges!

Function | Domain | Range |

All the inverse hyperbolic functions are differentiable because all the hyperbolic functions are differentiable. The derivatives of the inverse hyperbolic functions are listed below.

The derivatives of the inverse hyperbolic functions | |

Let's prove that .

Find the value of if .

Solution:

- Substitute the values: into the equation.
- Simplify:
- Since , the only solution is:

Express and as a function of and .

Solution:

- Add the two equations for and .
- Therefore,

- Subtract the two equations for and .
- Therefore,

- If we combine equations (1) and (2), we get:
- This is Euler's formula for the hyperbolic function.

There are several real-world applications for hyperbolic functions, such as:

describing the decay of light, velocity, electricity, or radioactivity

modeling the velocity of a wave as it moves across a body of water

the use of hyperbolic cosine to describe the shape of a hanging wire (called a

**catenary)**.

Perhaps the most famous of these is the description of the hanging wire.

- The
**hyperbolic functions**are essentially the trigonometric functions of the hyperbola.- This is why they are sometimes called hyperbolic trig functions.

- There are 6 hyperbolic functions:
- hyperbolic sine -
- hyperbolic cosine -
- hyperbolic tangent -
- hyperbolic cosecant -
- hyperbolic secant -
- hyperbolic cotangent -

- Their properties and identities are analogous to those of trigonometric functions.

*x*, sin *x*) form a circle with a unit radius, the points (cosh *x*, sinh *x*) form the right half of a unit hyperbola. These functions are defined in terms of the exponential functions e^{x} and e^{-x}.

sinh = (e^{x} - e^{-x})/2

cosh = (e^{x} + e^{-x})/2

tanh = (sinh *x*)/(cosh *x*)

csch = 1/(sinh *x*)

sech = 1/(cosh *x*)

coth = (cosh *x*)/(sinh *x*)

More about Hyperbolic Functions

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