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

Step-by-Step Solution

Answer

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

Step 1: Identify the Given Function

The exercise requires expressing \( \sinh^{-1}(-\frac{5}{12}) \) in terms of natural logarithms. We will use the formula \( \sinh^{-1} x = \ln \left(x+\sqrt{x^{2}+1}\right) \).

Step 2: Insert the Value into the Formula

Substitute \( x = -\frac{5}{12} \) into the inverse hyperbolic sine formula. This gives us:\[ \sinh^{-1} \left(-\frac{5}{12}\right) = \ln \left( -\frac{5}{12} + \sqrt{ \left( -\frac{5}{12} \right)^2 + 1 } \right) \]

Step 3: Simplify the Expression Inside the Square Root

Calculate \( \left( -\frac{5}{12} \right)^2 + 1 \):\[ \left( -\frac{5}{12} \right)^2 = \frac{25}{144} \]\[ \frac{25}{144} + 1 = \frac{25}{144} + \frac{144}{144} = \frac{169}{144} \]

Step 4: Find the Square Root

Calculate the square root of \( \frac{169}{144} \):\[ \sqrt{ \frac{169}{144} } = \frac{13}{12} \]

Step 5: Complete the Expression

Substitute the result from Step 4 back into the equation:\[ \sinh^{-1} \left(-\frac{5}{12}\right) = \ln \left(-\frac{5}{12} + \frac{13}{12}\right) \]Simplify the addition:\[ -\frac{5}{12} + \frac{13}{12} = \frac{8}{12} = \frac{2}{3} \]

Step 6: Finalize the Expression

The expression simplifies to:\[ \sinh^{-1} \left(-\frac{5}{12}\right) = \ln \left( \frac{2}{3} \right) \]

Key Concepts

Natural Logarithms

Natural logarithms are mathematical functions that calculate the power to which the base of the natural logarithm, which is Euler's number (approximately 2.71828), must be raised to produce a given number. They are denoted by \( \ln(x) \). One unique characteristic of natural logarithms is their application in solving equations involving exponential growth or decay, such as population growth models and radioactive decay.
Natural logarithms simplify complex problems, especially those involving growth rates over time. Importantly, in mathematics, especially calculus, natural logs are preferred due to their intrinsic connection with derivatives and antiderivatives.

• They transform multiplicative relationships into additive ones.
• They are the inverse functions of natural exponential functions.
• They regularly appear in formulas for inverse hyperbolic functions.

Understanding how to convert between exponential and logarithmic forms can greatly benefit students when solving related equations.

Hyperbolic Sine

The hyperbolic sine, denoted as \( \sinh(x) \), is part of a group of hyperbolic functions similar to the common trigonometric functions like sine and cosine. These functions arise naturally in solutions to certain types of differential equations and are connected to the geometry of hyperbolas, whereas trigonometric functions are related to circles.
The formula for the inverse hyperbolic sine, \( \sinh^{-1}(x) \), is expressed in terms of logarithms: \( \sinh^{-1}(x) = \ln(x + \sqrt{x^2 + 1}) \). This formulation is particularly useful because it allows computations even when hyperbolic function keys are unavailable, using only basic logarithm functions.

• Hyperbolic functions stretch and transform in ways that resemble hyperbolic geometry.
• They frequently appear in physics, particularly in calculations of rapidity (special relativity) and catenary curves.
• Inverse hyperbolic sine returns a real number that can be interpreted as a hyperbolic angle.

The hyperbolic sine function itself relates to the average increase or decrease in intensity and is hence important in color processing and radiowave descriptions.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, operations, and functions that define a value. They are essential for communicating mathematical ideas clearly and solving problems effectively. The expression \( \sinh^{-1}(-\frac{5}{12}) = \ln \left( \frac{2}{3} \right) \) illustrates the transformation from hyperbolic to logarithmic form.
This specific transformation involves several steps including squaring, fraction operations, square roots, and simplifying fractions. By performing these operations systematically:

• Start by squaring fractions to obtain accurate results, \(\left(- \frac{5}{12}\right)^2 = \frac{25}{144}\).
• Use the properties of square roots to simplify expressions \(\sqrt{\frac{169}{144}}=\frac{13}{12}\).
• Simplify expressions by combining fractions \(-\frac{5}{12} + \frac{13}{12} = \frac{2}{3}\).

By understanding how mathematical expressions convert into meaningful results, students can tackle complex mathematical problems with confidence.