Problem 66 - Thomas Calculus

Question

Use these formulas to express the numbers in Exercises \(61-66\) in terms of natural logarithms. $$\operatorname{csch}^{-1}(-1 / \sqrt{3})$$

Step-by-Step Solution

Answer

\( \operatorname{csch}^{-1}(-1/\sqrt{3}) = \ln(2 - \sqrt{3}) \).

Step 1: Recall Key Formula

Recall the identity for the hyperbolic cosecant: \( \operatorname{csch}^{-1}(x) = \ln\left(\frac{1}{x} + \sqrt{\frac{1}{x^2} + 1}\right) \). This general formula will help us express the given exercise in terms of natural logarithms.

Step 2: Substitute the Given Value

Substitute \(-\frac{1}{\sqrt{3}}\) into the formula: \( \operatorname{csch}^{-1}(-\frac{1}{\sqrt{3}}) = \ln\left(-\sqrt{3} + \sqrt{\left(-\sqrt{3}\right)^2 + 1}\right) \). This substitution helps translate the inverse hyperbolic function into a logarithm expression directly.

Step 3: Simplify Under the Square Root

Calculate under the square root: \( \left(-\sqrt{3}\right)^2 + 1 = 3 + 1 = 4 \). Simplifying the square root part reduces the complexity of the expression.

Step 4: Evaluate the Expression

Substitute back the values: \( \operatorname{csch}^{-1}(-\frac{1}{\sqrt{3}}) = \ln\left(-\sqrt{3} + \sqrt{4}\right) \). Now, evaluate the expression: \( \sqrt{4} = 2 \).

Step 5: Complete the Calculation

Plug back into the equation: \( \ln\left(-\sqrt{3} + 2\right) \). Therefore, the expression becomes \( \ln(2 - \sqrt{3}) \), which is the simplified logarithmic form of the given hyperbolic cosecant inverse.

Key Concepts

Natural Logarithms

Natural logarithms are a fundamental part of calculus and mathematical analysis. They are logarithms to the base of 'e', where 'e' is approximately 2.71828. Unlike regular logarithms, which can have any positive number as a base, the natural logarithm specifically uses this transcendental number.

Natural logarithms have some unique properties:

• They are denoted by 'ln' instead of 'log'.
• The natural logarithm of 1 is zero, \( ext{ln}(1) = 0 \).
• The derivative of \( ext{ln}(x) \) is \( rac{1}{x} \), which is highly useful in calculus.
• Any exponential growth or decay in nature often follows patterns involving 'e', such as population growth or radioactive decay.

In the given problem, we used a natural logarithm to express the result of an inverse hyperbolic function calculation. This conversion is crucial as it offers a standard form that is easier to evaluate and manipulate in further mathematical operations.

Csch Function

The hyperbolic cosecant, denoted as \( ext{csch}(x) \), is one of the hyperbolic functions closely related to the trigonometric cosecant function. It is defined as the reciprocal of the hyperbolic sine function, \( ext{sinh}(x) \).

Mathematically, this relationship is expressed as:\[ ext{csch}(x) = \frac{1}{ ext{sinh}(x)} \]Like other hyperbolic functions, \( ext{csch}(x) \) is often used in engineering, physics, and hyperbolic geometry as it models certain kinds of curves and waves.

There is also the inverse function, \( ext{csch}^{-1}(x) \), which is essential when working backward from a value to find the angle or length in hyperbolic analysis. This inverse is what we worked with in the exercise to find its expression in terms of natural logarithms. It holds a specific formula that relates the angle directly to a natural logarithmic expression, helping us simplify complex calculations.

Hyperbolic Functions

Hyperbolic functions are analogs of trigonometric functions but for hyperbolas rather than circles. They include functions such as \( ext{sinh}(x) \), \( ext{cosh}(x) \), and \( ext{tanh}(x) \), among others.

These functions have similar properties to trigonometric functions and often arise in calculus, serving vital roles in solving differential equations and representing the shapes and surfaces, such as in physics and geometry.

• \( ext{sinh}(x) = \frac{e^x - e^{-x}}{2} \)
• \( ext{cosh}(x) = \frac{e^x + e^{-x}}{2} \)
• Other hyperbolic functions are derived from these core definitions.

Due to their properties, these functions provide insight into problems involving hyperbolic curves and offer unique solutions that aren't possible with regular trigonometric functions. In the problem, we transformed an inverse hyperbolic function solution into a natural logarithm form, illustrating the versatile nature of these functions.

Logarithmic Transformation

Logarithmic transformation is a mathematical process that allows converting multiplication operations into addition operations using logarithms. This property makes it a powerful tool for simplifying expressions and solving equations.

In mathematical terms, a logarithmic transformation takes a function \( y = f(x) \) and transforms it into \( ext{ln}(y) = ext{ln}(f(x)) \). By doing this, it converts exponential trends into linear trends, which are easier to analyze and visualize.

• Logarithmic transformations reduce skewness in data for statistical interpretations.
• They help deal with exponential growth by simplifying the calculations.

In the exercise, we performed a logarithmic transformation by converting the \( ext{csch}^{-1}(x) \) formula into a natural logarithm expression. This approach simplifies the complex inverse function, making it more manageable for further calculations and interpretations. This transformation is particularly useful in many mathematical, scientific, and engineering applications.