Problem 11 - Advanced Engineering Mathematics

Question

Find all values of the given quantity. \(\sinh ^{-1} \frac{4}{3}\)

Step-by-Step Solution

Answer

The value of \( \sinh^{-1} \frac{4}{3} \) is approximately 1.0986.

Step 1: Understand the Definition of Inverse Hyperbolic Sine

The inverse hyperbolic sine, denoted as \( \sinh^{-1}(x) \), is the function that gives the value of \( y \) such that \( \sinh(y) = x \). It can be expressed in terms of a natural logarithm as follows: \( \sinh^{-1}(x) = \ln(x + \sqrt{x^2 + 1}) \).

Step 2: Apply the Formula to the Given Quantity

We need to apply the formula for the inverse hyperbolic sine to \( \sinh^{-1} \frac{4}{3} \). Substitute \( x = \frac{4}{3} \) into the formula: \[ \sinh^{-1}\left(\frac{4}{3}\right) = \ln\left(\frac{4}{3} + \sqrt{\left(\frac{4}{3}\right)^2 + 1}\right) \].

Step 3: Simplify Inside the Logarithm

Calculate \( \left(\frac{4}{3}\right)^2 + 1 \). This simplifies to \( \frac{16}{9} + 1 = \frac{25}{9} \). Now find the square root: \( \sqrt{\frac{25}{9}} = \frac{5}{3} \).

Step 4: Complete the Logarithm Calculation

Plug the value from Step 3 back into the equation: \[ \ln\left(\frac{4}{3} + \frac{5}{3}\right) = \ln\left(\frac{9}{3}\right) = \ln(3) \].

Step 5: Evaluate the Logarithm

The natural logarithm of 3 is approximately \( 1.0986 \). Therefore, \( \sinh^{-1} \frac{4}{3} \approx 1.0986 \).

Key Concepts

Inverse Hyperbolic Sine

The inverse hyperbolic sine, denoted as \( \sinh^{-1}(x) \), is a crucial mathematical concept. It finds the value \( y \) for which the hyperbolic sine \( \sinh(y) = x \). In simpler terms, it's like asking, "What number, when used in the hyperbolic sine function, gives me \( x \)?" For example, when you are given \( \sinh^{-1} \frac{4}{3} \), you are essentially looking for a number that, when used in the hyperbolic sine function, equals \( \frac{4}{3} \). This function helps in expanding the understanding of real variables in the context of hyperbolic functions, making complex exponential growth patterns easier to comprehend.

Natural Logarithm

A natural logarithm is a special kind of logarithm. It is represented by \( \ln(x) \) and is based on the number \( e \), approximately 2.71828. The natural logarithm is the opposite operation of taking the exponential. It answers the question: "To what power must \( e \) be raised to get \( x \)?" In the context of inverse hyperbolic functions, the formula \( \sinh^{-1}(x) = \ln(x + \sqrt{x^2 + 1}) \) is used because the natural logarithm can effectively express certain exponential relationships. This connection allows for elegant transformations and solutions in calculus and other higher-level mathematics.

Hyperbolic Functions

Hyperbolic functions include \( \sinh(x), \cosh(x), \tanh(x) \), and their inverses. They are analogs of trigonometric functions, but for a hyperbola rather than a circle. Just like \( \sin \) and \( \cos \) are related to circular functions, \( \sinh \) and \( \cosh \) relate to hyperbolic ones. These functions have applications in various fields like physics, engineering, and hyperbolic geometry. They describe rapidly growing processes, analogous to how trigonometric functions describe wave-like periodic processes. Understanding these functions is essential for solving problems that involve exponential growth or decay, such as calculating the shape of a hanging cable or the speed of a vehicle undergoing constant acceleration.

Mathematical Simplification

Mathematical simplification is an important skill that involves reducing a math expression to its simplest form. This can include consolidating terms, reducing fractions, or applying specific algebraic identities. In solving \( \sinh^{-1} \frac{4}{3} \), simplification reveals the expression \( \ln\left(\frac{4}{3} + \sqrt{\left(\frac{4}{3}\right)^2 + 1}\right) \). Computing the square and sum inside the square root, then simplifying the \ln\ expression, showcases how simplification helps in clearly understanding and solving complex problems by breaking them down to easier sub-problems. This skill not only makes calculations easier, but also helps uncover underlying relationships and insights within mathematical problems.