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Hyperbolic Functions

Hyperbolic Functions

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 trigonometric 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

Hyperbolic Function Definition

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.

Hyperbolic Functions: Formulas

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".

Deriving the Exponential Forms

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.

Hyperbolic Functions: Graphs

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

Hyperbolic functions graph of hyperbolic sine StudySmarterThe graph of hyperbolic sine using graphical addition - StudySmarter Originals

Hyperbolic functions graph of hyperbolic cosine StudySmarterThe graph of hyperbolic cosine using graphical addition - StudySmarter Originals

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

Hyperbolic functions graph of hyperbolic cosecant StudySmarterThe graph of hyperbolic cosecant - StudySmarter Originals

Hyperbolic functions graph of hyperbolic secant StudySmarterThe graph of hyperbolic secant - StudySmarter Originals

The graph of The graph of

Hyperbolic functions graph of hyperbolic tangent StudySmarterThe graph of hyperbolic tangent - StudySmarter Originals

Hyperbolic functions graph of hyperbolic cotangent StudySmarterThe graph of hyperbolic cotangent - StudySmarter Originals

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 .

Domain and Range of Hyperbolic Functions

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

FunctionDomainRange

Hyperbolic Functions: Properties and Identities

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!

The Relationship Between Trigonometric and Hyperbolic Functions

♦ 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.

Hyperbolic functions graph of unit circle StudySmarterThe graph of a unit circle with angle POQ shaded - StudySmarter Originals

♦ 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.

Hyperbolic functions graph of unit hyperbola StudySmarterThe graph of a unit hyperbola with section P shaded - StudySmarter Originals

Derivatives of Hyperbolic Functions

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,

Integrals of Hyperbolic Functions

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

Inverse 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:

Graphs of Inverse Hyperbolic Functions

The graphs of the inverse hyperbolic functions are shown below.

The graphs of the inverse hyperbolic functions
The graph of The graph of

Hyperbolic functions graph of inverse hyperbolic sine StudySmarterThe graph of inverse hyperbolic sine - StudySmarter Originals

Hyperbolic functions graph of inverse hyperbolic cosecant StudySmarterThe graph of inverse hyperbolic cosecant - StudySmarter Originals

The graph of The graph of

Hyperbolic functions graph of inverse hyperbolic cosine StudySmarterThe graph of inverse hyperbolic cosine - StudySmarter Originals

Hyperbolic functions graph of inverse hyperbolic secant StudySmarterThe graph of inverse hyperbolic secant - StudySmarter Originals

The graph of The graph of

Hyperbolic functions graph of inverse hyperbolic tangent StudySmarterThe graph of inverse hyperbolic tangent - StudySmarter Originals

Hyperbolic functions graph of inverse hyperbolic cotangent StudySmarterThe graph of inverse hyperbolic cotangent - StudySmarter Originals

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 .

Domain and Range of Inverse Hyperbolic Functions

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

FunctionDomainRange

Derivatives of Inverse Hyperbolic Functions

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 .

Hyperbolic Functions: Examples and Applications

Find the value of if .

Solution:

  1. Substitute the values: into the equation.
  2. Simplify:
  3. Since , the only solution is:

Express and as a function of and .

Solution:

  1. Add the two equations for and .
    • Therefore,
  2. Subtract the two equations for and .
    • Therefore,
  3. 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.

Hyperbolic Functions - Key takeaways

  • 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.

Frequently Asked Questions about Hyperbolic Functions

Hyperbolic functions are the trigonometric functions defined using a hyperbola instead of a circle. While the points (cos 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 ex and e-x.

Hyperbolic functions can be evaluated by plugging the value at which you wish to evaluate the function, and simplifying.

Hyperbolic functions are used in engineering and physics applications like the study of waves or vibrations of elastic membranes. They are also used to represent a hanging cable, or catenary, and to design arches to help stabilize structures.

Hyperbolic functions are written similarly to the trigonometric functions for a circle. The six main hyperbolic functions are written as:
sinh = (ex - e-x)/2
cosh = (ex + e-x)/2
tanh = (sinh x)/(cosh x)
csch = 1/(sinh x)
sech = 1/(cosh x)
coth = (cosh x)/(sinh x)

Final Hyperbolic Functions Quiz

Question

Why do we have to restrict the domain of hyperbolic cosine and hyperbolic secant in order to find their inverses?

Show answer

Answer

Because cosine and secant are even functions, and are therefore not one-to-one.

Show question

More about Hyperbolic Functions
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