Question

Identities Prove each identity using the definitions of the hyperbolic functions. \(\tanh (-x)=-\tanh x\)

Short answer

**Question:** Prove that \(\tanh (-x) = -\tanh x\). **Answer:** To prove the identity, we used the definition of the hyperbolic tangent function as \(\tanh x = \frac{\sinh x}{\cosh x}\). We then replaced x by -x and simplified the expression, which resulted in \(\tanh (-x) = -\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} = -\tanh x\). Therefore, we have shown that \(\tanh (-x) = -\tanh x\).

Step-by-step solution

Write the definition of the hyperbolic tangent function

Recall that the hyperbolic tangent function can be defined as: \(\tanh x = \frac{\sinh x}{\cosh x}.\)

Apply the definition of the hyperbolic sine and cosine functions #

To rewrite the expression, consider the definitions of the hyperbolic sine and cosine functions: \(\sinh x = \frac{e^{x} - e^{-x}}{2}, \quad \cosh x = \frac{e^{x} + e^{-x}}{2}.\) Now substitute these expressions into the definition we derived in Step 1: \(\tanh x = \frac{\frac{e^{x} - e^{-x}}{2}}{\frac{e^{x} + e^{-x}}{2}}.\)

Replace x by -x #

To find the expression for the hyperbolic tangent of -x, replace x by -x in the expression: \(\tanh (-x) = \frac{\frac{e^{-x} - e^{x}}{2}}{\frac{e^{-x} + e^{x}}{2}}.\)

Simplify the expression #

Next, simplify the expression: \(\tanh (-x) = \frac{e^{-x} - e^{x}}{e^{-x} + e^{x}}.\) Now compare both expressions for \(\tanh x\) and \(\tanh (-x)\). Observe that \(\tanh(-x)\) is the negation of \(\tanh(x)\): \(\tanh (-x) = -\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} = -\tanh x.\)

Conclusion #

We have shown that \(\tanh (-x) = -\tanh x.\) Thus, the identity is proven.

Key concepts

Understanding the Hyperbolic Tangent

The hyperbolic tangent function, often abbreviated to \(\tanh\), is one of the several hyperbolic functions that are analogues to the familiar trigonometric functions. It behaves similarly to the geometric tangent function, but it actually relates to hyperbolas instead of circles.

When we define \(\tanh x\) using hyperbolic sine \(\sinh x\) and hyperbolic cosine \(\cosh x\), as \(\tanh x = \frac{\sinh x}{\cosh x}\), we can gain an understanding of its properties through the behavior of \(\sinh x\) and \(\cosh x\). For instance, the hyperbolic tangent exhibits odd symmetry, meaning it satisfies the property \(\tanh (-x)=-\tanh x\). This is reflected in the steps of our proof where we replaced \(x\) with \(-x\) and noticed the sign reversal, proving its oddness.

One notable feature of the \(\tanh\) function is how it approaches asymptotes at \(-1\) and \(1\) as the value of \(x\) goes to negative and positive infinity, respectively. This characteristic is essential in various fields including signal processing, and computational neuroscience.

The Relationship Between Hyperbolic Sine and Cosine

Hyperbolic sine and cosine, denoted as \(\sinh x\) and \(\cosh x\), are intimately connected, much like their circular counterparts in trigonometry.

By definition, \(\sinh x = \frac{e^{x} - e^{-x}}{2}\) and \(\cosh x = \frac{e^{x} + e^{-x}}{2}\). These functions are derived from the exponential functions \(e^{x}\) and \(e^{-x}\), and they depict the shape of a hyperbola when plotted. In contrast to the periodic sine and cosine functions, \(\sinh x\) and \(\cosh x\) are not periodic and continue to grow exponentially.

A remarkable identity that links these functions is the hyperbolic Pythagorean identity: \(\cosh^2 x - \sinh^2 x = 1\). This identity is a reflection of the relationship between the sides of a right-angled hyperbola, much like the Pythagorean theorem is a statement about the sides of a right-angled triangle. Understanding these relationships is crucial for grasping the more complex behaviors of hyperbolic functions.

Exponential Functions and Their Core Properties

Exponential functions form the backbone of hyperbolic functions and have a ubiquitous presence in mathematics and related disciplines. An exponential function is of the form \(f(x) = e^{x}\), where \(e\) is a mathematical constant referred to as Euler's number, approximately equal to 2.71828.

The function \(e^{x}\) exhibits unique characteristics that make it a fundamental concept in various areas like calculus, complex analysis, and even finance. One of its distinctive properties is that the rate of growth of \(e^{x}\) is equal to its value at any point on its curve, making it the only function to have this property.

Exponential functions also have clear rules for manipulation: scaling the input by a constant factor results in a corresponding scaling of the output, and the function's inverse \(e^{-x}\) is simply its reflection with respect to the y-axis. These qualities, combined with their infinite domain and range, open the door to vast applications such as compound interest computation, population growth modeling, and even in the definitions of the hyperbolic functions we discussed earlier.