Problem 66 - 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. $$ \operatorname{csch}^{-1}(-1 / \sqrt{3}) $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Formula

The problem requires us to evaluate \( \operatorname{csch}^{-1}(-1/\sqrt{3}) \). According to the given formula, the inverse hyperbolic cosecant, \( \operatorname{csch}^{-1} x \), is expressed as \( \ln \left(\frac{1}{x} + \frac{\sqrt{1 + x^2}}{|x|}\right) \) for \( x eq 0 \).

Step 2: Substitute the Value of x

Substitute \( x = -\frac{1}{\sqrt{3}} \) into the formula. This gives us:\[ \operatorname{csch}^{-1}(-1/\sqrt{3}) = \ln \left(\frac{1}{-1/\sqrt{3}} + \frac{\sqrt{1 + (-1/\sqrt{3})^2}}{| -1/\sqrt{3} |}\right) \]

Step 3: Simplify the Expression

Simplify \( \frac{1}{-1/\sqrt{3}} \) to \( -\sqrt{3} \). Also, compute \( (-1/\sqrt{3})^2 = 1/3 \), leading to \( 1 + 1/3 = 4/3 \). The expression becomes:\[ \ln \left( -\sqrt{3} + \frac{\sqrt{4/3}}{1/\sqrt{3}} \right) \]

Step 4: Calculate Square Roots and Fractions

Further simplify by calculating \( \sqrt{4/3} = \frac{2}{\sqrt{3}} \) and then \( \frac{\sqrt{4/3}}{1/\sqrt{3}} = 2 \). The expression simplifies to:\[ \ln ( -\sqrt{3} + 2 ) \]

Step 5: Complete the Calculation

Finally, the expression is simplified to \( \ln ( 2 - \sqrt{3} ) \). This is the natural logarithm form for \( \operatorname{csch}^{-1}(-1/\sqrt{3}) \).

Key Concepts

Logarithms

Logarithms are mathematical operations that help us work with large numbers more easily by expressing them as powers of a base. The most common type is the logarithm base 10, but for advanced mathematics, we often use the natural logarithm. Logarithms help solve equations in which the unknown appears as the exponent of some other quantity.

They have properties like:

• Product rule: \(\log_b(xy) = \log_b(x) + \log_b(y)\)
• Quotient rule: \(\log_b(x/y) = \log_b(x) - \log_b(y)\)
• Power rule: \(\log_b(x^y) = y\cdot\log_b(x)\)

These rules make manipulating expressions with logarithms manageable. Logarithms are used in solving exponential and logarithmic equations, in fields like engineering, science, and finance.
In the context of inverse hyperbolic functions, logarithms allow us to express these functions in terms of known mathematical forms, making them easier to evaluate without special calculator keys.

Hyperbolic Functions

Hyperbolic functions are analogs of ordinary trigonometric functions but for a hyperbola rather than a circle. They include functions such as hyperbolic sine (\(\sinh\)), hyperbolic cosine (\(\cosh\)), and hyperbolic tangent (\(\tanh\)). These functions arise in many areas of mathematics, including calculus, complex analysis, and geometry.

The formulas:

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

Just like trigonometric functions, hyperbolic functions have inverses. These inverse functions are expressed using logarithms, which help evaluate inverse hyperbolic values without a dedicated calculator key.

Natural Logarithm

The natural logarithm, denoted as \(\ln(x)\), is a logarithm to the base \(e\), where \(e\) is an irrational constant approximately equal to 2.71828. It is widely used in mathematics when dealing with continuous growth processes or decay processes, such as in finance, physics, and calculus.

A few key points about the natural logarithm are:

• The natural logarithm of 1 is 0: \(\ln(1) = 0\)
• The natural logarithm is undefined for non-positive numbers.
• It is the inverse function of the exponential function \(e^x\).

The transformation of hyperbolic functions through logarithmic expressions often results in natural logarithms. For example, in the inverse hyperbolic function \(\operatorname{csch}^{-1}(x)\), simplifying using the given formula leads to a solution expressed as a natural logarithm.

Cosecant Function

The cosecant function, \(\operatorname{csc}(x)\), is the reciprocal of the sine function. In trigonometry, it is defined as \(\operatorname{csc}(x) = \frac{1}{\sin(x)}\). It is particularly important when considering angles whose sine values are very small, as it becomes easier to work with their reciprocals.

In hyperbolic functions, the hyperbolic cosecant, \(\operatorname{csch}(x)\), is similarly defined but within the context of hyperbolic sine:

• \(\operatorname{csch}(x) = \frac{1}{\sinh(x)}\)

Inverses of the cosecant functions, both ordinary and hyperbolic, provide a way to derive angles or hyperbolic measures that produce a given sine or hyperbolic sine. This is particularly useful when the hyperbolic sine values are yet unknown or cumbersome to compute directly, prompting the use of logarithmic expressions to simplify solutions.