Problem 62 - 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. $$ \cosh ^{-1}(5 / 3) $$

Step-by-Step Solution

Answer

\( \cosh^{-1}(5/3) = \ln(3) \)

Step 1: Identify the appropriate formula

To express \( \cosh^{-1}(\frac{5}{3}) \) using natural logarithms, we need to use the formula for the inverse hyperbolic cosine: \( \cosh^{-1} x = \ln \left(x + \sqrt{x^2 - 1}\right) \). In this case, our \( x \) value is \( \frac{5}{3} \).

Step 2: Substitute the value into the formula

Substitute \( x = \frac{5}{3} \) into the formula: \( \cosh^{-1} \left( \frac{5}{3} \right) = \ln \left( \frac{5}{3} + \sqrt{\left(\frac{5}{3}\right)^2 - 1} \right) \).

Step 3: Calculate the expression inside the square root

Calculate \( \left( \frac{5}{3} \right)^2 - 1 \). This becomes \( \frac{25}{9} - 1 = \frac{25}{9} - \frac{9}{9} = \frac{16}{9} \).

Step 4: Simplify the square root

Take the square root of \( \frac{16}{9} \), which is \( \sqrt{\frac{16}{9}} = \frac{4}{3} \).

Step 5: Complete the logarithmic expression

Substitute the square root back into the logarithmic expression: \( \ln \left( \frac{5}{3} + \frac{4}{3} \right) = \ln \left( \frac{9}{3} \right) = \ln(3) \).

Key Concepts

Natural Logarithms

Natural logarithms are a specific type of logarithm. They are based on the constant \( e \), which is approximately 2.71828. The natural logarithm is usually denoted by \( \ln(x) \), where \( x \) is a positive number.
This base \( e \) is unique because it arises naturally in various mathematical contexts, including calculus and complex analysis.
Natural logarithms simplify many mathematical operations due to their unique properties:

• They convert multiplication into addition: \( \ln(a \cdot b) = \ln(a) + \ln(b) \).
• They convert division into subtraction: \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \).
• They handle powers with ease: \( \ln(a^b) = b \cdot \ln(a) \).

These properties make natural logarithms especially useful in simplifying complex expressions and solving equations, as seen in the step-by-step solution of the problem involving inverse hyperbolic functions.

Hyperbolic Functions

Hyperbolic functions are analogs of trigonometric functions but instead are related to hyperbolas. They include \( \sinh(x) \), \( \cosh(x) \), and \( \tanh(x) \), among others.
In comparison to trigonometric functions, which are based on circles, hyperbolic functions provide solutions to different sets of problems that often appear in calculus and physics.
For hyperbolic functions:

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

While the trigonometric functions relate to the unit circle, hyperbolic functions describe a hyperbola, which is why their properties and graphs are different. This distinction is crucial for understanding how they transform into their inverse forms using logarithms.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, operations, and functions that collectively represent a certain value or relation. Understanding the structure of these expressions is essential for developing effective strategies to solve problems.
In our exercise, mathematical expressions are used to convert inverse hyperbolic functions into a simpler logarithmic form. This involves substituting values into complex formulae and simplifying them using principles of algebra and arithmetic.

• Always substitute known values for variables carefully.
• Simplify inside out, beginning with powers and roots before moving on to addition or subtraction.
• Use properties like the identity \( \ln(x + y) \) to simplify complex expressions into more manageable forms.

Proper manipulation of mathematical expressions allows us to evaluate unfamiliar or complicated functions, such as inverse hyperbolic functions, more easily and accurately.