Problem 64 - Thomas Calculus

Question

When hyperbolic function keys are not available on a calculator, it is still possible to evaluate the inverse hyperbolic functions by expressing them as logarithms, as shown here. $$ \begin{array}{rlrl}{\sinh ^{-1} x} & {=\ln \left(x+\sqrt{x^{2}+1}\right),} & {-\infty < x < \infty} \\ {\cosh ^{-1} x} & {=\ln \left(x+\sqrt{x^{2}+1}\right),} & {x \geq 1} \\ {\tanh ^{-1} x} & {=\frac{1}{2} \ln \frac{1+x}{1-x},} & {|x| < 1} \\ {\operatorname{sech}^{-1} x} & {=\ln \left(\frac{1+\sqrt{1-x^{2}}}{x}\right),} & {0 < x \leq 1} \\\ {\operatorname{csch}^{-1} x} & {=\ln \left(\frac{1}{x}+\frac{\sqrt{1+x^{2}}}{|x|}\right),} & {x \neq 0} \\\ {\operatorname{coth}^{-1} x} & {=\frac{1}{2} \ln \frac{x+1}{x-1},} & {|x| > 1}\end{array} $$ Use the formulas given above to express the numbers in Exercises \(61-66\) in terms of natural logarithms. $$ \operatorname{coth}^{-1}(5 / 4) $$

Step-by-Step Solution

Answer

\( \operatorname{coth}^{-1} \left( \frac{5}{4} \right) = \frac{1}{2} \ln(9) \)

Step 1: Identify the Formula

From the provided formulas, identify that the appropriate formula to use for \( \operatorname{coth}^{-1}(x) \) is: \[ \operatorname{coth}^{-1} x = \frac{1}{2} \ln \frac{x+1}{x-1} \] which is valid for \( |x| > 1 \).

Step 2: Substitute the Value into the Formula

Substitute \( x = \frac{5}{4} \) into the formula: \[ \operatorname{coth}^{-1} \left( \frac{5}{4} \right) = \frac{1}{2} \ln \left( \frac{\frac{5}{4} + 1}{\frac{5}{4} - 1} \right) \]

Step 3: Simplify the Expression Inside the Logarithm

Simplify the expression \( \frac{\frac{5}{4} + 1}{\frac{5}{4} - 1} \): \[ \frac{5}{4} + 1 = \frac{5}{4} + \frac{4}{4} = \frac{9}{4} \] \[ \frac{5}{4} - 1 = \frac{5}{4} - \frac{4}{4} = \frac{1}{4} \] Thus, the expression becomes \( \frac{9/4}{1/4} = 9 \).

Step 4: Final Calculation of the Expression

Substitute the simplified expression into the logarithm: \[ \operatorname{coth}^{-1} \left( \frac{5}{4} \right) = \frac{1}{2} \ln(9) \] This represents the inverse hyperbolic cotangent in terms of natural logarithms.

Key Concepts

Natural Logarithms

Natural logarithms are a specific type of logarithm that have the base of the mathematical constant \( e \), which is approximately 2.718. The natural logarithm is denoted as \( \ln(x) \). Using natural logarithms is particularly powerful in calculus and other areas of mathematics due to the properties of \( e \).

Here's why they are handy:

• Integration and Differentiation: Natural logarithms make the process of integration and differentiation straightforward as the derivative of \( \ln(x) \) is \( \frac{1}{x} \).
• Inverse Functions: Inverse hyperbolic functions often use natural logarithms because they provide a convenient way to transition between the two types.

In the exercise above, this property is used for expressing \( \operatorname{coth}^{-1}(x) \) as a natural logarithm function, allowing it to be evaluated even when the calculator does not have hyperbolic functionalities.

Hyperbolic Functions

Hyperbolic functions are mathematical functions similar to trigonometric functions but are based on hyperbolas instead of circles. They are defined using exponential functions. The basic hyperbolic functions include \( \sinh(x) \), \( \cosh(x) \), and \( \tanh(x) \), along with their inverse counterparts.

One interesting aspect of these functions is how they relate to exponential expressions:

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

In the solution, inverse hyperbolic functions like \( \operatorname{coth}^{-1} \) are transformed into natural logarithm forms to compute them easily. Understanding hyperbolic functions is essential because they arise in many real-world contexts, such as physics and engineering.

Trigonometric Identities

Trigonometric identities are formulas involving trigonometric functions that are always true for defined variable values. Although not as directly related to hyperbolic functions, these identities provide a foundation that is built upon in hyperbolic functions.

Some well-known trigonometric identities include:

• \( \sin^2(x) + \cos^2(x) = 1 \)
• \( 1 + \tan^2(x) = \sec^2(x) \)

Hyperbolic functions have their analogs, known as hyperbolic identities:

• \( \cosh^2(x) - \sinh^2(x) = 1 \)
• \( \tanh^2(x) + \sech^2(x) = 1 \)

These identities help in simplifying and transforming hyperbolic functions, which is crucial when using logarithms to express inverse hyperbolic functions during calculations. Understanding both trigonometric and hyperbolic identities enables a smoother transition when working with more complex functions.