Question

Find one or more values of each of the following complex expressions in the easiest way you can. \(\tanh (i \pi / 4)\)

Short answer

\(\tanh(i \pi / 4) = i\)

Step-by-step solution

Understand the Hyperbolic Tangent Function

The hyperbolic tangent function, \(\tanh(z)\), is defined as \(\tanh(z) = \frac{\sinh(z)}{\cosh(z)}\), where \(\sinh(z)\) and \(\cosh(z)\) are the hyperbolic sine and cosine functions, respectively.

Substitute the Given Value

Substitute \(z = i \pi / 4\) into the hyperbolic tangent function, yielding \(\tanh(i \pi / 4) = \frac{\sinh(i \pi / 4)}{\cosh(i \pi / 4)}\).

Use Euler's Formulas for Hyperbolic Sine and Cosine

Recall that \(\sinh(ix) = i \sin(x)\) and \(\cosh(ix) = \cos(x)\). Therefore, \(\sinh(i \pi / 4) = i \sin(\pi / 4)\) and \(\cosh(i \pi / 4) = \cos(\pi / 4)\).

Compute the Sine and Cosine Values

We know that \(\sin(\pi / 4) = \cos(\pi / 4) = \frac{\sqrt{2}}{2}\). Therefore, \(\sinh(i \pi / 4) = i \frac{\sqrt{2}}{2}\) and \(\cosh(i \pi / 4) = \frac{\sqrt{2}}{2}\).

Simplify the Expression

Plug these values into the hyperbolic tangent function: \(\tanh(i \pi / 4) = \frac{i \frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = i\).

Key concepts

Hyperbolic Functions

Hyperbolic functions, such as \(\tanh(z)\), \(\text{sinh}(z)\), and \(\text{cosh}(z)\), are analogs of trigonometric functions but for hyperbolic geometry. These functions are defined using exponential functions rather than circular functions.
For example:

• \(\text{sinh}(z) = \frac{e^z - e^{-z}}{2}\)
• \(\text{cosh}(z) = \frac{e^z + e^{-z}}{2}\)
• \(\tanh(z) = \frac{\text{sinh}(z)}{\text{cosh}(z)}\)

These definitions allow for straightforward manipulation when solving complex expressions. In the exercise above, this means using the aforementioned definitions to break down \(\tanh(i \frac{\text{π}}{4})\) step by step.

Euler's Formulas

Euler's formulas are essential when working with complex numbers and functions. Two important formulas are:
\(e^{ix} = \text{cos}(x) + i\text{sin}(x)\) and \(e^{-ix} = \text{cos}(x) - i\text{sin}(x)\).
These can be extended to hyperbolic functions:

• \(\text{sinh}(ix) = i \text{sin}(x)\)
• \(\text{cosh}(ix) = \text{cos}(x)\)

For this exercise, these derivations help transform \(\text{sinh}(i \frac{\text{π}}{4}) \) into a format involving sine and \(\text{cosh}(i \frac{\text{π}}{4}) \) into a cosine term. This simplification is crucial when simplifying the hyperbolic tangent function.

Complex Numbers

Complex numbers extend the concept of one-dimensional real numbers to the two-dimensional complex plane using the imaginary unit \(\text{ i}\) where \(\text{i}^2 = -1\).
They have a form \(a + bi\) where \ a \ and \ b \ are real numbers.
When working with hyperbolic functions and Euler's formulas, we often encounter terms like \(\text{ ix}\).
In the context of this exercise, substituting \(\text{ ix}\) into hyperbolic functions allows us to utilize the sine and cosine equivalents, making it easier to simplify and solve complex expressions, e.g. \(\tanh(i \frac{\text{π}}{4}) = i\).

Hyperbolic Tangent Function

The hyperbolic tangent function is defined as \(\tanh(z) = \frac{\text{sinh}(z)}{\text{cosh}(z)}\).
It shares similar properties with the tangent function in trigonometry but adapted for hyperbolic contexts.
In this exercise, solving \(\tanh(i \frac{\text{π}}{4})\) involves:

• Using the definitions of \(\text{sinh}(ix) = i \text{sin}(x)\)
• and \(\text{cosh}(ix) = \text{cos}(x)\),
• Substituting \(i \frac{\text{π}}{4}\) and simplifying,
• Finally, you get the result \(i\).

Understanding these transformations is especially useful for solving complex expressions efficiently.