Problem 18 - Single Variable Calculus

Question

If \(\cosh x=\frac{5}{4}\), find the values of the other hyperbolic functions at \(x\).

Step-by-Step Solution

Answer

Positive case: \(\sinh x = \frac{3}{4}\), \(\tanh x = \frac{3}{5}\), \(\coth x = \frac{5}{3}\), \(\sech x = \frac{4}{5}\), and \(\csch x = \frac{4}{3}\) Negative case: \(\sinh x = -\frac{3}{4}\), \(\tanh x = -\frac{3}{5}\), \(\coth x = -\frac{5}{3}\), \(\sech x = \frac{4}{5}\), and \(\csch x = -\frac{4}{3}\)

Step 1: Find \(\sinh x\) using the identity \(\cosh^2 x - \sinh^2 x = 1\)

We know that \(\cosh^2 x - \sinh^2 x = 1\). Given that \(\cosh x = \frac{5}{4}\), we can substitute this value into the identity: \(\left(\frac{5}{4}\right)^2 - \sinh^2 x = 1\) Now, solve for \(\sinh^2 x\): \(\sinh^2 x = \left(\frac{5}{4}\right)^2 - 1\) \(\sinh^2 x = \frac{9}{16}\) Now, find the square root of both sides: \(\sinh x = \pm\frac{3}{4}\) Without extra information, we cannot determine the sign of \(\sinh x\). Therefore, we'll consider both cases.

Step 2: Calculate \(\tanh x\), \(\coth x\), \(\sech x\), and \(\csch x\) for each case

First, let's consider the positive case, where \(\sinh x = \frac{3}{4}\). 1. \(\tanh x = \frac{\sinh x}{\cosh x} = \frac{\frac{3}{4}}{\frac{5}{4}} = \frac{3}{5}\) 2. \(\coth x = \frac{\cosh x}{\sinh x} = \frac{\frac{5}{4}}{\frac{3}{4}} = \frac{5}{3}\) 3. \(\sech x = \frac{1}{\cosh x} = \frac{1}{\frac{5}{4}} = \frac{4}{5}\) 4. \(\csch x = \frac{1}{\sinh x} = \frac{1}{\frac{3}{4}} = \frac{4}{3}\) Now, let's consider the negative case, where \(\sinh x = -\frac{3}{4}\). 1. \(\tanh x = \frac{\sinh x}{\cosh x} = \frac{-\frac{3}{4}}{\frac{5}{4}} = -\frac{3}{5}\) 2. \(\coth x = \frac{\cosh x}{\sinh x} = \frac{\frac{5}{4}}{-\frac{3}{4}} = -\frac{5}{3}\) 3. \(\sech x = \frac{1}{\cosh x} = \frac{1}{\frac{5}{4}} = \frac{4}{5}\) 4. \(\csch x = \frac{1}{\sinh x} = \frac{1}{-\frac{3}{4}} = -\frac{4}{3}\) Therefore, depending on the sign of \(\sinh x\), the values of the other hyperbolic functions are: Positive case: 1. \(\sinh x = \frac{3}{4}\) 2. \(\tanh x = \frac{3}{5}\) 3. \(\coth x = \frac{5}{3}\) 4. \(\sech x = \frac{4}{5}\) 5. \(\csch x = \frac{4}{3}\) Negative case: 1. \(\sinh x = -\frac{3}{4}\) 2. \(\tanh x = -\frac{3}{5}\) 3. \(\coth x = -\frac{5}{3}\) 4. \(\sech x = \frac{4}{5}\) 5. \(\csch x = -\frac{4}{3}\)

Key Concepts

Hyperbolic Identities

Hyperbolic identities play a critical role in understanding how hyperbolic functions relate to one another. Just as with circular trigonometric functions, hyperbolic functions have a set of fundamental identities that often aid in solving complex mathematical problems.

One such identity is \(\cosh^2 x - \sinh^2 x = 1\), which is analogous to the Pythagorean identity in trigonometry \(\cos^2 x + \sin^2 x = 1\). This basic identity helped us determine the function value of \(\sinh x\) when given \(\cosh x = \frac{5}{4}\). By substitution and simplification, we obtained \(\sinh x\) as \(\frac{3}{4}\), albeit the sign ambiguity.

Another useful set of identities, the reciprocal identities, ensures we can find \(\sech x\), \(\csch x\), and \(\coth x\) from their counterparts. Similarly, the quotient identity \(\tanh x = \frac{\sinh x}{\cosh x}\) is used to connect the hyperbolic sine and cosine functions to hyperbolic tangent function values. Understanding and applying these identities properly is critical in simplifying hyperbolic expressions and solving equations involving hyperbolic functions.

Hyperbolic Function Values

To thoroughly appreciate the solutions of hyperbolic function problems, it is beneficial to grasp the concept of hyperbolic function values. Hyperbolic function values are outcomes of the functions \(\sinh x\), \(\cosh x\), \(\tanh x\), \(\sech x\), \(\csch x\), and \(\coth x\) for a particular value of \(x\).

The values of these functions, as shown in the provided example, can be derived from each other using hyperbolic identities. It's essential to recognize that the value of \(\sinh x\) can be either positive or negative. This changes the corresponding function values for \(\tanh x\), \(\coth x\), and \(\csch x\). Such duality is typical in hyperbolic functions and emphasizes the need to consider the context of a problem to determine the appropriate solution.

For instance, knowing that \(\cosh x = \frac{5}{4}\) and using the derived value of \(\sinh x\), the calculation of the remaining function values became straightforward. Each hyperbolic function has a unique behavior that distinguishes it from the others, aiding us in solving a broad spectrum of mathematical problems where they are applicable.

Inverse Hyperbolic Functions

Inverse hyperbolic functions allow us to solve for the angle \(x\) when the value of a hyperbolic function is known. They are the counterparts to the hyperbolic functions \(\sinh x\), \(\cosh x\), and \(\tanh x\), with their respective inverses \(\text{arsinh} x\), \(\text{arcosh} x\), and \(\text{artanh} x\). These inverses can be understood as the 'angles' (or more appropriately, the hyperbolic angles) that produce the given hyperbolic function value.

For example, if we needed to find \(x\) given \(\tanh x = \frac{3}{5}\), we would use \(\text{artanh} x\) to obtain the original angle. It is crucial to keep in mind that the domains of these functions are restricted so that they can be one-to-one and their inverses can be properly defined. Just like trigonometric functions have inverse functions used for solving equations, finding angles, and various applications, the inverse hyperbolic functions provide us with similar capabilities in the hyperbolic context. These tools are invaluable for fields such as calculus, physics, and engineering, where hyperbolic functions frequently arise.