Understanding Hyperbolic Functions: Definition, Formulas, and Examples

In mathematics, there are special functions closely related to trigonometric functions that often appear in equations. These functions are known as hyperbolic functions and are analogous to trigonometric functions. Let's dive deeper into the concept of hyperbolic functions, including their definition, formulas, properties, and examples.

Defining Hyperbolic Functions

Hyperbolic functions are essentially the trigonometric functions for the hyperbola. They are defined in terms of the natural exponential function, using Euler's number, and extend the concept of parametric equations from the unit circle to the unit hyperbola. The two fundamental hyperbolic functions are the hyperbolic sine and hyperbolic cosine, which can be expressed as follows:

• Hyperbolic sine: sinh(x) = (e^x - e^-x) / 2
• Hyperbolic cosine: cosh(x) = (e^x + e^-x) / 2

From these two functions, we can derive the remaining six hyperbolic functions, which are listed in the table below.

Formulas for Hyperbolic Functions

• Hyperbolic sine: sinh(x)
• Hyperbolic cosine: cosh(x)
• Hyperbolic tangent: tanh(x)
• Hyperbolic cosecant: csch(x)
• Hyperbolic secant: sech(x)
• Hyperbolic cotangent: coth(x)

These functions are often abbreviated as "sinh", "cosh", "tanh", "csch", "sech", and "coth" respectively.

The Similarity to Trigonometric Functions

An interesting characteristic of hyperbolic functions is their resemblance to trigonometric functions. This can be seen in Euler's formula, which connects the trigonometric functions with the natural exponential function. Solving this formula for both cosine and sine results in equations that closely resemble the hyperbolic cosine and sine functions. However, unlike Euler's formula, the hyperbolic functions do not include an imaginary component.

This is because when we solve for the hyperbolic functions, the imaginary component is absent. The connection between these two types of functions is further demonstrated by their graphs, which have a striking resemblance to the graphs of trigonometric functions.

Properties and Identities

The properties and identities of hyperbolic functions are also similar to those of trigonometric functions. For instance, the Pythagorean identity for hyperbolic functions is comparable to that of trigonometric functions:

sinh²(x) + cosh²(x) = 1

Similarly, there are other identities, such as the double-angle identity and half-angle identity, that are analogous to those of trigonometric functions. These identities can be proven using the definitions of hyperbolic sine and cosine, along with some algebraic manipulation.

Domain and Range

Just like trigonometric functions, hyperbolic functions have similar domains and ranges. The domain of both types of functions is all real numbers, while the range is limited. Additionally, the graphs of hyperbolic functions have horizontal and/or vertical asymptotes, resembling those of trigonometric functions.

Derivatives of Hyperbolic Functions

The derivatives of hyperbolic functions follow the same rules as those of trigonometric functions. For instance, the derivative of sinh(x) is cosh(x), and the derivative of cosh(x) is sinh(x). This makes it easier to differentiate equations involving hyperbolic functions.

In summary, understanding the basics of hyperbolic functions can greatly aid in solving various mathematical problems and equations. These functions have a strong connection to trigonometric functions and share similar properties and identities. Familiarizing yourself with the fundamentals of hyperbolic functions can make it easier to utilize them in a variety of applications.

The points (cos x, sin x) create a circle with a unit radius, while the points (cosh x, sinh x) form the right half of a unit hyperbola. These functions are defined in terms of ex and e-x, exponential functions. They are known as the 6 hyperbolic functions, including hyperbolic sine, cosine, tangent, cosecant, secant, and cotangent.

The 6 Hyperbolic Functions

• Hyperbolic sine (sinh x)
• Hyperbolic cosine (cosh x)
• Hyperbolic tangent (tanh x)
• Hyperbolic cosecant (csch x)
• Hyperbolic secant (sech x)
• Hyperbolic cotangent (coth x)

Similar to trigonometric functions, the properties and identities of hyperbolic functions are analogous. This includes the derivatives and integrals, which are simpler to calculate compared to their trigonometric counterparts. The table below lists the derivatives of hyperbolic functions, which can also be combined using the Chain Rule.

Derivatives of Hyperbolic Functions

The derivatives of hyperbolic functions involve the use of exponential functions and have a simple derivation process. This is why they are simpler to calculate compared to their trigonometric counterparts. The table below lists the derivatives of hyperbolic functions, and they can also be combined using the Chain Rule.

Integrals of Hyperbolic Functions

Similarly, the integrals of hyperbolic functions are also analogous to those of trigonometric functions. They are listed in the table below.

In addition to these integrals, there are other useful ones such as:

• ∫sech² x dx = tanh x + C
• ∫csch² x dx = -coth x + C

Inverse Hyperbolic Functions

The inverse hyperbolic functions are defined in terms of logarithmic functions, as the hyperbolic functions involve exponential functions and logarithmic functions are their inverses. The inverse hyperbolic functions are:

• arcsinh x = ln(x + √1 + x²)
• arccosh x = ln(x + √x² - 1)
• arctanh x = ½ln((1 + x)/(1 - x))
• arccsch x = ln(x + √1 + x²)
• arcsech x = ln((1 + √1 - x²)/x)
• arccoth x = ½ln((x + 1)/(x - 1))

Note: The domains of cosh x and sech x must be restricted to the interval [-1, 1] to define their inverses.

Derivatives of Inverse Hyperbolic Functions

All the inverse hyperbolic functions are differentiable because their hyperbolic counterparts are differentiable. The table below shows their derivatives.

Graphs of Inverse Hyperbolic Functions

The graphs of the inverse hyperbolic functions have some interesting properties, such as:

• The graphs of inverse hyperbolic secant and inverse hyperbolic cosecant have horizontal and/or vertical asymptotes.
• The graphs of inverse hyperbolic cosine and inverse hyperbolic secant have a definite starting point at x=1.

The domains and ranges of the inverse hyperbolic functions are as follows:

• arcsinh x: domain and range ∈ ℕ
• arccosh x: domain ∈ [1, ∞), range ∈ ℕ
• arctanh x: domain ∈ (-1, 1), range ∈ ℕ
• arccsch x: domain and range ∈ ℝ ∞
• arcsech x: domain ∈ (0, 1], range ∈ ℕ
• arccoth x: domain ∈ (-∞, -1) ∪ (1, ∞), range ∈ ℕ

Example and Applications of Hyperbolic Functions

Let's look at an application of hyperbolic functions in solving a real-world problem. We are given the equation cosh x = 2. We can solve for x using the inverse hyperbolic cosine function:

arccosh(2) = ln(2 + √2² - 1) = ln(2 + √3)

Therefore, x = arccosh(2) ≈ 1.317.

Hyperbolic functions have various real-world applications, 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

The most famous application of hyperbolic functions is in describing the shape of a hanging wire, which follows a catenary curve.

Key Takeaways

In summary, hyperbolic functions are defined using a hyperbola and are analogous to trigonometric functions. They have real-world applications, and their derivatives and integrals can be calculated using simple rules.

The inverse hyperbolic functions are a set of mathematical functions that are defined using logarithmic functions and have unique properties when plotted on a graph. Understanding these functions can offer valuable insights into a wide range of mathematical and physical phenomena.

How to Calculate Hyperbolic Functions

Evaluating hyperbolic functions is a simple process – you just plug in the desired value and simplify the equation to get the result.

Applications of Hyperbolic Functions

Hyperbolic functions have many practical uses in engineering and physics. They are commonly used to analyze wave patterns and vibrations in elastic materials, as well as to model hanging cables and design structural arches for stability.

Notation for Hyperbolic Functions

The notation for hyperbolic functions is similar to that of trigonometric functions used for circles. There are six important hyperbolic functions, which are expressed in the following forms:

• 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)