Question

Prove the expression for \(\cosh ^{-1}(x)\). Multiply \(x=\cosh (y)=(1 / 2)\left(e^{y}-e^{-y}\right)\) by \(2 e^{y}\) and solve for \(y .\) Does your expression match the textbook?

Short answer

\( \cosh^{-1}(x) = \ln(x + \sqrt{x^2 - 1}) \) matches the solution.

Step-by-step solution

Understanding the Given Function

The expression for the hyperbolic cosine function is given as \( x = \cosh(y) = \frac{1}{2}(e^y + e^{-y}) \). This is the function we need to manipulate to find an expression for \( y \), which is \( \cosh^{-1}(x) \).

Multiply by \(2e^y\)

To eliminate the fraction in the hyperbolic cosine function, we multiply the entire equation by \( 2e^y \), resulting in \( 2xe^y = e^{2y} + 1 \). This manipulation is key to isolating \( y \).

Rearrange and Form a Quadratic Equation

Rearrange the terms to form the equation \( e^{2y} - 2xe^y + 1 = 0 \). This is a quadratic equation in terms of \( e^y \).

Solve the Quadratic Equation

Using the quadratic formula \( e^y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1 \), \( b = -2x \), and \( c = 1 \), we have \(e^y = \frac{2x \pm \sqrt{(2x)^2 - 4}}{2} \).

Simplify the Expression

Simplify the expression to find \( e^y = x \pm \sqrt{x^2 - 1} \). Since \( e^y > 0 \) for all real numbers, we select the positive root, \( e^y = x + \sqrt{x^2 - 1} \).

Solve for \(y\)

Take the natural logarithm of both sides to solve for \( y \): \( y = \ln(x + \sqrt{x^2 - 1}) \). Thus, \( \cosh^{-1}(x) = \ln(x + \sqrt{x^2 - 1}) \).

Key concepts

Hyperbolic Cosine

The hyperbolic cosine function, denoted as \( \cosh(y) \), is a mathematical function that acts similarly to the standard cosine function, but it is defined using exponential functions. It is expressed as:

• \( \cosh(y) = \frac{1}{2}(e^y + e^{-y}) \)

This formula combines two exponential terms, \( e^y \) and \( e^{-y} \), and averages them. The hyperbolic cosine function has applications in various fields such as engineering and physics, especially when dealing with wave equations and electrical circuits.
It has a unique property that its output is always greater than or equal to 1. This is because the terms \( e^y \) and \( e^{-y} \) are always positive, ensuring the sum is positive too.
To determine the inverse function, \( \cosh^{-1}(x) \), we manipulate the regular \( \cosh(y) \) expression to isolate \( y \). This involves mathematical techniques like solving a quadratic equation, as detailed in the given exercise.

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in a single variable that can be expressed in the standard form:

• \( ax^2 + bx + c = 0 \)

Here, \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). Quadratic equations can be solved using various methods, including factoring, completing the square, and the quadratic formula.
In the solution process for \( \cosh^{-1}(x) \), we arrive at a quadratic equation in terms of \( e^y \) by transforming the equation to \( e^{2y} - 2xe^y + 1 = 0 \).
Using the quadratic formula, we solve for \( e^y \):

• \( e^y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

where \( a = 1 \), \( b = -2x \), and \( c = 1 \). The positive root is chosen since exponential functions cannot be negative.

Natural Logarithm

The natural logarithm is the inverse operation of exponentiation with base \( e \), a fundamental constant approximately equal to 2.718. It's represented as \( \ln(x) \) and provides the power to which \( e \) must be raised to yield \( x \).
When solving for the inverse hyperbolic cosine function \( \cosh^{-1}(x) \), the natural logarithm is a critical step. After isolating \( e^y \), we end up with:

• \( e^y = x + \sqrt{x^2 - 1} \)

To solve for \( y \), we take the natural logarithm:

• \( y = \ln(x + \sqrt{x^2 - 1}) \)

This is how we find the inverse hyperbolic cosine of \( x \), expressed as \( \cosh^{-1}(x) = \ln(x + \sqrt{x^2 - 1}) \). The natural logarithm simplifies the task of isolating \( y \) in our original equation.

Hyperbolic Identities

Hyperbolic identities are mathematical equations expressing relationships between hyperbolic functions, similar to trigonometric identities. They are useful in simplifying complex expressions and proving new equations.
Some fundamental hyperbolic identities include:

• \( \cosh^2(y) - \sinh^2(y) = 1 \)
• \( 1 - \tanh^2(y) = \text{sech}^2(y) \)

In our exploration of the inverse hyperbolic cosine, these identities can aid in checking consistency or simplifying expressions. For instance, the identity \( \cosh^2(y) - \sinh^2(y) = 1 \) verifies the intrinsic relationship between \( \cosh \) and \( \sinh \).
Understanding these identities deepens comprehension of how hyperbolic functions interact and assures learners of the correctness of derived expressions. They are pivotal when dealing with hyperbolic equations or transformations.