Problem 62 - Thomas Calculus in SI Units

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. (We give one derivation in Section 8.4.) $$\begin{aligned}&\sinh ^{-1} x=\ln (x+\sqrt{x^{2}+1}), \quad-\infty < x<\infty\\\&\cosh ^{-1} x=\ln (x+\sqrt{x^{2}-1}), \quad x \geq 1\\\ &\tanh ^{-1} x=\frac{1}{2} \ln \frac{1+x}{1-x}, \quad|x| < 1\\\&\operatorname{sech}^{-1} x=\ln \left(\frac{1+\sqrt{1-x^{2}}}{x}\right), \quad0 < x \leq 1\\\&\operatorname{csch}^{-1} x=\ln \left(\frac{1}{x}+\frac{\sqrt{1+x^{2}}}{|x|}\right), \quad x \neq 0\\\&\operatorname{coth}^{-1} x=\frac{1}{2} \ln \frac{x+1}{x-1}, \quad |x|>1\end{aligned}$$ Use the formulas in the box here to express the numbers in terms of natural logarithms. $$\cosh ^{-1}(5 / 3)$$

Step-by-Step Solution

Answer

\(\cosh^{-1}\left(\frac{5}{3}\right) = \ln(3)\)

Step 1: Identify the appropriate formula

To express \(\cosh^{-1}\left(\frac{5}{3}\right)\) in terms of a natural logarithm, identify the formula for the inverse hyperbolic cosine function: \(\cosh^{-1}x = \ln(x + \sqrt{x^2 - 1})\), with \(x \geq 1\).

Step 2: Plug 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: Simplify inside the square root

Calculate \(\left(\frac{5}{3}\right)^2\) and subtract 1:\[\left(\frac{5}{3}\right)^2 = \frac{25}{9}, \quad \frac{25}{9} - 1 = \frac{16}{9}\]. Hence, \(\sqrt{\frac{16}{9}} = \frac{4}{3}\).

Step 4: Add the terms

Add \(\frac{5}{3}\) and \(\frac{4}{3}\) together: \(\frac{5}{3} + \frac{4}{3} = \frac{9}{3} = 3\).

Step 5: Take the natural logarithm

Substitute back into the formula: \(\ln\left(3\right)\). This is the expression of \(\cosh^{-1}\left(\frac{5}{3}\right)\) in terms of a natural logarithm.

Key Concepts

Logarithmic Expressions

Logarithms are a fundamental concept in mathematics used to represent exponents in a unique way. In simple words, a logarithm tells us the power to which a certain number, called the base, must be raised to obtain another number. An exponential relation like \(b^y = x\) can be expressed in logarithmic form as \(\log_b(x) = y\), where \(b\) is the base, \(y\) is the exponent, and \(x\) is the result.

• Base 10 is called the common logarithm while base \(e\) (approximately equal to 2.718) is known as the natural logarithm, denoted as \(\ln(x)\).
• Logarithmic expressions are incredibly useful in situations involving growth and decay, such as compound interest, population growth, or radioactive decay.

In mathematics, logarithms can also be used to express inverse hyperbolic functions. By representing a function as a logarithm, we can solve equations more easily, especially when a calculator does not have specific function keys. For example, the inverse hyperbolic cosine \(\cosh^{-1}(x)\) can be expressed as \(\ln(x + \sqrt{x^2 - 1})\). This allows us to find the hyperbolic cosine's inverse directly using logarithmic properties.

Hyperbolic Functions

Hyperbolic functions are analogs of trigonometric functions but are based on hyperbolas rather than circles. They are essential in calculus, especially when dealing with exponential growth and decay. The basic hyperbolic functions are the hyperbolic sine \(\sinh \), cosine \(\cosh \), and tangent \(\tanh \). Their reciprocal functions are \(\csch \), \(\sech \), and \(\coth \).

• One peculiar property of hyperbolic functions is their relationship to the exponential function. For instance, \(\sinh(x) = \frac{e^x - e^{-x}}{2}\) and \(\cosh(x) = \frac{e^x + e^{-x}}{2}\).
• These functions resemble trigonometric identities but have unique properties such as \(\cosh^2(x) - \sinh^2(x) = 1\).

Inverse hyperbolic functions, like \(\sinh^{-1}(x)\), \(\cosh^{-1}(x)\), and \(\tanh^{-1}(x)\), work similarly to their trigonometric counterparts in resolving exponential models and complex equations. They are especially useful in calculus, where they can be represented using natural logarithms to simplify integration and differentiation processes.

Calculus

Calculus is a branch of mathematics that deals with continuous change. It enables us to calculate rates of change (derivatives) and quantities (integrals). These calculations are essential in a wide range of fields including physics, engineering, economics, and medicine. Calculus provides methods to model continuously changing phenomena, making it invaluable for problem-solving and decision making.

• Derivatives measure how a function changes as its input changes. They are the core concept behind differential calculus.
• Integrals, on the other hand, measure the accumulation of quantities, like areas under curves. They are a fundamental aspect of integral calculus.

Hyperbolic functions are closely related to calculus. Since they are linked to exponential functions, they can be easier to integrate and differentiate compared to their trigonometric counterparts. The representation of these functions using logarithmic expressions simplifies handling them within calculus. For example, when finding the derivative of \(\cosh^{-1}(x)\), using the expression \(\ln(x + \sqrt{x^2 - 1})\) simplifies the differentiation process. Thus, linking hyperbolic functions and logarithmic expressions into calculus provides powerful tools for analysis.