Problem 61 - 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. $$\sinh ^{-1}(-5 / 12)$$

Step-by-Step Solution

Answer

\(\sinh^{-1}(-\frac{5}{12}) = \ln(\frac{2}{3})\)

Step 1: Identify the appropriate formula

We're asked to evaluate \(\sinh^{-1}(-5/12)\). The given formula is \(\sinh^{-1}x = \ln(x+\sqrt{x^2+1})\). Here, \(x = -\frac{5}{12}\).

Step 2: Substitute the value of x

Substitute \(x = -\frac{5}{12}\) into the formula:\[\sinh^{-1}(-\frac{5}{12}) = \ln\left(-\frac{5}{12} + \sqrt{(-\frac{5}{12})^2 + 1}\right)\]

Step 3: Calculate inside the square root

First, compute \((-\frac{5}{12})^2 + 1\):\[(-\frac{5}{12})^2 = \frac{25}{144}\]\[\frac{25}{144} + 1 = \frac{25}{144} + \frac{144}{144} = \frac{169}{144}\]

Step 4: Evaluate the square root

Find \(\sqrt{\frac{169}{144}}\):\[\sqrt{\frac{169}{144}} = \frac{\sqrt{169}}{\sqrt{144}} = \frac{13}{12}\]

Step 5: Simplify the expression

Substitute the result of the square root back into the logarithm:\[\sinh^{-1}(-\frac{5}{12}) = \ln\left(-\frac{5}{12} + \frac{13}{12}\right)\]Simplify inside the parentheses:\[-\frac{5}{12} + \frac{13}{12} = \frac{8}{12} = \frac{2}{3}\]

Step 6: Evaluate the logarithm

Calculate the natural logarithm of the simplified term:\[\sinh^{-1}(-\frac{5}{12}) = \ln(\frac{2}{3})\]

Key Concepts

Natural Logarithms

Natural logarithms are a type of logarithm that use the base \( e \), where \( e \approx 2.71828 \). The natural logarithm is commonly denoted as \( \ln(x) \). It represents the power to which \( e \) must be raised to produce the number \( x \). This mathematical function is extensively used in calculus, sciences, and engineering because it naturally exemplifies growth processes.
In the context of inverse hyperbolic functions, natural logarithms act as tools to transition from complex functions to simpler expressions. This is demonstrated in the formulas used to convert inverse hyperbolic expressions to logarithmic equivalents, making calculations possible even without a calculator's hyperbolic function keys. This adaptability springs from the properties of logarithms, which handle exponential relationships effectively.

Hyperbolic Functions

Hyperbolic functions like \( \sinh(x) \), \( \cosh(x) \), and \( \tanh(x) \) resemble the standard trigonometric functions but are based on hyperbolas rather than circles.
They find applications in various fields including physics, engineering, and hyperbolic geometry. These functions are defined using exponential functions, which helps present graceful and useful formula transformations.
For instance, the inverse hyperbolic sine function \( \sinh^{-1}(x) \) can be expressed in terms of a natural logarithm, assisting in our given problem:

• \( \sinh^{-1}(x) = \ln(x + \sqrt{x^2 + 1}) \)

This relation offers a method to compute inverse hyperbolic functions without direct calculator support.

Trigonometry

Trigonometry traditionally deals with relationships involving angles and lengths in right-angled triangles through trigonometric functions such as sine, cosine, and tangent.
Hyperbolic functions extend these classical ideas, relating more to the geometry of hyperbolas. Instead of grids and circles, they form another layer, making it essential to understand underlying principles like the Taylor Series and exponential functions.
Understanding how these translate into hyperbolic expressions is crucial when dealing with real-world applications that need hyperbolic computations. It allows conversion of geometric problems into solvable equations using known mathematical constants and forms.

Mathematics Problem Solving

Effective mathematics problem-solving often involves breaking down complex problems into simpler, more manageable parts. In our exercise, we applied this method to evaluate \( \sinh^{-1}(-5/12) \) step-by-step using its logarithmic equivalent.
Key components of successful problem solving include:

• Identifying the correct formula based on the problem constraints.
• Substituting known values systematically.
• Simplifying expressions by calculating step-by-step, such as managing inner square roots before outer expressions.
• Verifying results through iterative checking for precision.

Different strategies might be needed depending on whether the mathematical issue involves algebra, calculus, or other realms, but each requires a clear understanding of core principles and logical progression.