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Hyperbolic Functions

Q: Why are they called 'hyperbolic'?

They are called hyperbolic because the point $(\cosh(t), \sinh(t))$ will always lie on the unit hyperbola defined by the equation $x^2 - y^2 = 1$. This is parallel to how the point $(\cos(t), \sin(t))$ always lies on the unit circle $x^2 + y^2 = 1$.

Q: Is cosh(x) ever negative?

No, $\cosh(x)$ is never negative. Its definition, $\cosh(x) = (e^x + e^{-x})/2$, involves the sum of two positive terms, so the result is always positive. The minimum value of $\cosh(x)$ is $1$, which occurs at $x=0$.

Q: What is the main difference between the graphs of sin(x) and sinh(x)?

The biggest difference is that $\sin(x)$ is periodic, meaning its graph is a wave that repeats every $2\pi$ units. In contrast, $\sinh(x)$ is not periodic; its graph is an ever-increasing curve that goes to $-\infty$ and $\infty$ without repeating.

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Discover the world of hyperbolic functions, the cousins of our familiar trigonometric functions. This lesson explores their definitions using the exponential function $e^{x}$ , their unique properties, and how they describe shapes like hanging chains and arches, providing a powerful tool for calculus and physics.

Hyperbolic Functions — an original Algebra911 reference diagram defining hyperbolic functions with its key formula and a worked example. — Hyperbolic Functions

What Are Hyperbolic Functions?

Hyperbolic functions are a set of six functions defined using the exponential function, $e^{x}$ . They are named hyperbolic because they have a similar relationship to the unit hyperbola, $x^{2} - y^{2} = 1$ , as the standard trigonometric functions (like sine and cosine) have to the unit circle, $x^{2} + y^{2} = 1$ . While trigonometric functions are often called circular functions, hyperbolic functions provide a powerful alternative toolkit for modeling different kinds of problems, especially in physics, engineering, and calculus.

The three most fundamental hyperbolic functions are the hyperbolic sine ( $\sinh (x)$ ), hyperbolic cosine ( $\cosh (x)$ ), and hyperbolic tangent ( $\tanh (x)$ ). Just as any point on the unit circle can be represented by $(\cos (t), \sin (t))$ , any point on the right branch of the unit hyperbola can be represented by $(\cosh (t), \sinh (t))$ , where $t$ is a parameter related to the area of a specific hyperbolic sector.

The key takeaway is that these functions, while having names similar to trigonometric functions, are fundamentally built from exponentials. This gives them unique properties that make them indispensable for describing phenomena involving exponential growth and decay, such as the shape of a hanging cable or the velocity of a falling object with air resistance.

Exploring the Core Three: Sinh, Cosh, and Tanh

The entire family of hyperbolic functions is built upon the definitions of hyperbolic sine and hyperbolic cosine, which are themselves defined using $e^{x}$ and $e^{- x}$ . Understanding these definitions is the first step to mastering their use.

Hyperbolic Sine (sinh)

The hyperbolic sine, pronounced "sinch," is the odd component of the exponential function $e^{x}$ .

\sinh (x) = \frac{e^{x} - e^{- x}}{2}

Its domain is all real numbers ( $- \infty, \infty$ ), and its range is also all real numbers ( $- \infty, \infty$ ). It is an odd function, meaning $\sinh (- x) = - \sinh (x)$ .

Hyperbolic Cosine (cosh)

The hyperbolic cosine, pronounced "kosh," is the even component of the exponential function $e^{x}$ .

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

Its domain is all real numbers ( $- \infty, \infty$ ), but its range is $[1, \infty)$ . This is because its minimum value occurs at $x = 0$ , where $\cosh (0) = (e^{0} + e^{0}) / 2 = (1 + 1) / 2 = 1$ . It is an even function, meaning $\cosh (- x) = \cosh (x)$ .

Hyperbolic Tangent (tanh)

Just like the standard tangent, the hyperbolic tangent is the ratio of hyperbolic sine to hyperbolic cosine.

\tanh (x) = \frac{\sinh (x)}{\cosh (x)} = \frac{e^{x} - e^{- x}}{e^{x} + e^{- x}}

Its domain is all real numbers ( $- \infty, \infty$ ), and its range is the open interval $(- 1, 1)$ . This means it has two horizontal asymptotes at $y = - 1$ and $y = 1$ .

Example 1

Calculate the exact values of $\sinh (\ln 3)$ and $\cosh (\ln 3)$ .

Solution for $\sinh (\ln 3)$ :

We use the definition $\sinh (x) = (e^{x} - e^{- x}) / 2$ . Substitute $x = \ln 3$ :

\sinh (\ln 3) = \frac{e^{\ln 3} - e^{- \ln 3}}{2}

Recall that $e^{\ln a} = a$ and $e^{- \ln a} = e^{\ln (1 / a)} = 1 / a$ . So:

\sinh (\ln 3) = \frac{3 - 1 / 3}{2} = \frac{8 / 3}{2} = \frac{8}{6} = \frac{4}{3}

Solution for $\cosh (\ln 3)$ :

We use the definition $\cosh (x) = (e^{x} + e^{- x}) / 2$ . Substitute $x = \ln 3$ :

\cosh (\ln 3) = \frac{e^{\ln 3} + e^{- \ln 3}}{2}

\cosh (\ln 3) = \frac{3 + 1 / 3}{2} = \frac{10 / 3}{2} = \frac{10}{6} = \frac{5}{3}

How Do You Graph Hyperbolic Functions?

The graphs of hyperbolic functions have distinct and memorable shapes that arise directly from their exponential definitions.

Graph of $y = \cosh (x)$ : The graph of hyperbolic cosine is a U-shaped curve called a catenary. This is the shape a heavy, flexible chain or cable makes when hanging between two points. The graph is symmetric about the y-axis (since it's an even function) and has its minimum point at $(0, 1)$ . It is always positive and grows exponentially as $x$ moves away from zero.
Graph of $y = \sinh (x)$ : The graph of hyperbolic sine is an S-shaped curve that passes through the origin $(0, 0)$ . It is symmetric about the origin (since it's an odd function). The function is always increasing and, like $\cosh (x)$ , grows exponentially for large positive or negative $x$ .
Graph of $y = \tanh (x)$ : The graph of hyperbolic tangent is also an S-shaped curve passing through the origin. However, it is bounded between two horizontal asymptotes at $y = - 1$ and $y = 1$ . As $x \to \infty$ , $\tanh (x) \to 1$ , and as $x \to - \infty$ , $\tanh (x) \to - 1$ .

Visualizing these graphs helps in understanding the domain, range, and overall behavior of each function.

What Are the Most Important Hyperbolic Identities?

Hyperbolic functions have identities that are strikingly similar to trigonometric identities, but with occasional, crucial sign differences. The most fundamental of these is the hyperbolic equivalent of the Pythagorean identity.

\cosh^{2} (x) - \sinh^{2} (x) = 1

This identity can be proven directly from the exponential definitions.

Example 2

Prove the identity $\cosh^{2} (x) - \sinh^{2} (x) = 1$ .

Solution:

We start with the left-hand side and substitute the definitions of $\cosh (x)$ and $\sinh (x)$ .

\cosh^{2} (x) - \sinh^{2} (x) = {(\frac{e^{x} + e^{- x}}{2})}^{2} - {(\frac{e^{x} - e^{- x}}{2})}^{2}

Now, expand the squares:

= \frac{(e^{x})^{2} + 2 (e^{x}) (e^{- x}) + (e^{- x})^{2}}{4} - \frac{(e^{x})^{2} - 2 (e^{x}) (e^{- x}) + (e^{- x})^{2}}{4}

= \frac{e^{2 x} + 2 e^{0} + e^{- 2 x}}{4} - \frac{e^{2 x} - 2 e^{0} + e^{- 2 x}}{4}

Since $e^{0} = 1$ , we can combine the fractions:

= \frac{(e^{2 x} + 2 + e^{- 2 x}) - (e^{2 x} - 2 + e^{- 2 x})}{4}

= \frac{e^{2 x} + 2 + e^{- 2 x} - e^{2 x} + 2 - e^{- 2 x}}{4}

= \frac{4}{4} = 1

This completes the proof. The left side equals the right side.

Here is a table comparing some common trigonometric and hyperbolic identities. Notice the pattern of sign changes, a rule of thumb known as Osborn's Rule.

Trigonometric Identity	Hyperbolic Identity
$\cos^{2} (x) + \sin^{2} (x) = 1$	$\cosh^{2} (x) - \sinh^{2} (x) = 1$
$1 + \tan^{2} (x) = \sec^{2} (x)$	$1 - \tanh^{2} (x) = {sech}^{2} (x)$
$\sin (x \pm y) = \sin (x) \cos (y) \pm \cos (x) \sin (y)$	\sinh(x \pm y) = \sinh(x)\cosh(y) \pm \cosh(x)\sinh(y)
$\cos (x \pm y) = \cos (x) \cos (y) \mp \sin (x) \sin (y)$	\cosh(x \pm y) = \cosh(x)\cosh(y) \pm \sinh(x)\sinh(y)
$\sin (2 x) = 2 \sin (x) \cos (x)$	$\sinh (2 x) = 2 \sinh (x) \cosh (x)$
$\cos (2 x) = \cos^{2} (x) - \sin^{2} (x)$	$\cosh (2 x) = \cosh^{2} (x) + \sinh^{2} (x)$

How Do You Solve Problems with Hyperbolic Functions?

Solving problems with hyperbolic functions often involves using their definitions or their identities, much like with trigonometric functions. The key is to identify which tool is best for the job.

Example 3

Given that $\sinh (x) = - \frac{3}{4}$ , find the values of $\cosh (x)$ and $\tanh (x)$ .

Solution:

We can use the fundamental identity $\cosh^{2} (x) - \sinh^{2} (x) = 1$ to find $\cosh (x)$ .

\cosh^{2} (x) - {(- \frac{3}{4})}^{2} = 1

\cosh^{2} (x) - \frac{9}{16} = 1

Now, solve for $\cosh^{2} (x)$ :

\cosh^{2} (x) = 1 + \frac{9}{16} = \frac{16}{16} + \frac{9}{16} = \frac{25}{16}

To find $\cosh (x)$ , we take the square root. An important property of $\cosh (x)$ is that it is always positive (its range is $[1, \infty)$ ). Therefore, we only take the positive root.

\cosh (x) = \sqrt{\frac{25}{16}} = \frac{5}{4}

Now that we have both $\sinh (x)$ and $\cosh (x)$ , we can easily find $\tanh (x)$ using its definition:

\tanh (x) = \frac{\sinh (x)}{\cosh (x)} = \frac{- 3 / 4}{5 / 4}

\tanh (x) = - \frac{3}{4} \cdot \frac{4}{5} = - \frac{3}{5}

So, the final answers are $\cosh (x) = 5 / 4$ and $\tanh (x) = - 3 / 5$ .

Key formulas for hyperbolic functions by Algebra911.

What About Csch, Sech, and Coth?

Just as trigonometry has three reciprocal functions (cosecant, secant, cotangent), so do hyperbolic functions. They are defined in a parallel way.

Hyperbolic Cosecant (csch)
$csch (x) = \frac{1}{\sinh (x)} = \frac{2}{e^{x} - e^{- x}}$
The domain of $csch (x)$ is all real numbers except $x = 0$ .
Hyperbolic Secant (sech)
$sech (x) = \frac{1}{\cosh (x)} = \frac{2}{e^{x} + e^{- x}}$
The range of $sech (x)$ is $(0, 1]$ . Its graph is a bell-shaped curve often called the Gudermannian curve.
Hyperbolic Cotangent (coth)
$coth (x) = \frac{1}{\tanh (x)} = \frac{\cosh (x)}{\sinh (x)} = \frac{e^{x} + e^{- x}}{e^{x} - e^{- x}}$
The domain of $coth (x)$ is all real numbers except $x = 0$ . Its range is $(- \infty, - 1) \cup (1, \infty)$ .

These functions appear less frequently than the primary three, but they are important for completing the set of hyperbolic identities and for solving certain types of calculus problems, particularly in integration.

What Are Common Mistakes with Hyperbolic Functions?

When first learning about hyperbolic functions, it's easy to make a few common errors. Being aware of these can help you avoid them.

Confusing Identities: The most frequent mistake is mixing up the signs in identities. For example, writing $\cosh^{2} (x) + \sinh^{2} (x) = 1$ instead of the correct $\cosh^{2} (x) - \sinh^{2} (x) = 1$ . Always double-check the signs when translating a trigonometric identity to its hyperbolic counterpart.
Assuming Periodicity: Trigonometric functions are periodic (e.g., $\sin (x + 2 π) = \sin (x)$ ). Hyperbolic functions are not periodic. Their graphs do not repeat.
Incorrectly Calculating $\cosh (x)$ : When solving $\cosh^{2} (x) = k$ , students sometimes forget that $\cosh (x)$ must be positive and greater than or equal to 1. There is only one valid solution for $\cosh (x)$ , not a $\pm$ solution.
Forgetting the Definitions: When in doubt, always go back to the exponential definitions: $\sinh (x) = (e^{x} - e^{- x}) / 2$ and $\cosh (x) = (e^{x} + e^{- x}) / 2$ . Nearly any identity or value can be derived from these starting points.
Calculator Errors: Make sure you know how to access hyperbolic functions on your calculator. They are usually found in a 'hyperbolic' or 'hyp' menu, separate from the standard 'sin', 'cos', and 'tan' buttons.

Quick Summary and Reference Sheet

This section provides a quick reference for the most important definitions and properties of hyperbolic functions.

Core Definitions

Hyperbolic Sine: $\sinh (x) = \frac{e^{x} - e^{- x}}{2}$
Hyperbolic Cosine: $\cosh (x) = \frac{e^{x} + e^{- x}}{2}$
Hyperbolic Tangent: $\tanh (x) = \frac{\sinh (x)}{\cosh (x)}$

Fundamental Identity

\cosh^{2} (x) - \sinh^{2} (x) = 1

Function Properties

Function	Domain	Range	Symmetry	Key Graph Feature
$\sinh (x)$	$(- \infty, \infty)$	$(- \infty, \infty)$	Odd	Passes through $(0, 0)$
$\cosh (x)$	$(- \infty, \infty)$	$[1, \infty)$	Even	Catenary curve, minimum at $(0, 1)$
$\tanh (x)$	$(- \infty, \infty)$	$(- 1, 1)$	Odd	Asymptotes at $y = - 1$ and $y = 1$

Frequently Asked Questions

Are hyperbolic functions the same as trigonometric functions?

No, they are not the same, but they are analogous. Trigonometric functions relate to the unit circle, while hyperbolic functions relate to the unit hyperbola. They share similar-looking identities but are defined differently, using exponential functions instead of angles in a circle.

Why are they called 'hyperbolic'?

They are called hyperbolic because the point $(\cosh (t), \sinh (t))$ will always lie on the unit hyperbola defined by the equation $x^{2} - y^{2} = 1$ . This is parallel to how the point $(\cos (t), \sin (t))$ always lies on the unit circle $x^{2} + y^{2} = 1$ .

What does the 'h' in sinh and cosh stand for?

The 'h' at the end of sinh, cosh, tanh, etc., stands for 'hyperbolic'. It is used to distinguish these functions from their standard trigonometric counterparts (sin, cos, tan).

Where are hyperbolic functions used in real life?

Hyperbolic functions appear in many areas of science and engineering. The function $\cosh (x)$ describes the shape of a hanging cable or chain (a catenary), which is used in bridge construction. They are also used in relativity theory, electrical engineering, and fluid dynamics.

Is cosh(x) ever negative?

No, $\cosh (x)$ is never negative. Its definition, $\cosh (x) = (e^{x} + e^{- x}) / 2$ , involves the sum of two positive terms, so the result is always positive. The minimum value of $\cosh (x)$ is $1$ , which occurs at $x = 0$ .

How do I type sinh on my calculator?

On most scientific and graphing calculators, hyperbolic functions are not primary buttons. You typically need to press a 'HYP' or 'HYPERBOLIC' key first, and then press the 'sin', 'cos', or 'tan' key to get $\sinh$ , $\cosh$ , or $\tanh$ .

Are there inverse hyperbolic functions?

Yes, there are. Just as trigonometric functions have inverses (like $\arcsin$ or $\sin^{- 1}$ ), hyperbolic functions also have inverses, denoted as $arsinh (x)$ , $arcosh (x)$ , etc. These inverse functions are important because they can be expressed using natural logarithms.

What is the main difference between the graphs of sin(x) and sinh(x)?

The biggest difference is that $\sin (x)$ is periodic, meaning its graph is a wave that repeats every $2 π$ units. In contrast, $\sinh (x)$ is not periodic; its graph is an ever-increasing curve that goes to $- \infty$ and $\infty$ without repeating.