Problem 11 - Advanced Engineering Mathematics

Question

In Problems 1-14, find all values of the given quantity. $$ \sinh ^{-1} \frac{4}{3} $$

Step-by-Step Solution

Answer

\( \ln(3) \)

Step 1: Understand the Definition of Inverse Hyperbolic Sine

The inverse hyperbolic sine function, denoted as \( \sinh^{-1}(x) \), is the value whose hyperbolic sine is \( x \). This means if \( y = \sinh^{-1}(x) \), then \( \sinh(y) = x \).

Step 2: Use the Inverse Hyperbolic Sine Formula

The inverse hyperbolic sine is given by the formula: \( \sinh^{-1}(x) = \ln(x + \sqrt{x^2 + 1}) \). We will apply this to find \( \sinh^{-1}\left(\frac{4}{3}\right) \).

Step 3: Substitute and Simplify

Substitute \( x = \frac{4}{3} \) into the formula:\[ \sinh^{-1}\left(\frac{4}{3}\right) = \ln \left( \frac{4}{3} + \sqrt{\left(\frac{4}{3}\right)^2 + 1} \right) \]Calculate inside the square root first:\[ \left(\frac{4}{3}\right)^2 = \frac{16}{9} \]\[ \sqrt{\frac{16}{9} + 1} = \sqrt{\frac{16}{9} + \frac{9}{9}} = \sqrt{\frac{25}{9}} = \frac{5}{3} \]

Step 4: Compute the Expression

Now substitute back into the exponential expression:\[ \sinh^{-1}\left(\frac{4}{3}\right) = \ln \left( \frac{4}{3} + \frac{5}{3} \right) \]\[ \sinh^{-1}\left(\frac{4}{3}\right) = \ln \left( \frac{9}{3} \right) = \ln(3) \]

Step 5: Conclusion

We have found that \( \sinh^{-1}\left( \frac{4}{3} \right) = \ln(3) \). This is the value whose hyperbolic sine is \( \frac{4}{3} \).

Key Concepts

Understanding the Hyperbolic Sine

The hyperbolic sine function, often denoted as \( \sinh(x) \), is a crucial concept in advanced mathematics, especially when dealing with hyperbolic functions. It is defined as: \[\sinh(x) = \frac{e^x - e^{-x}}{2}\] where \( e \) is the base of the natural logarithm. This expression shows how hyperbolic sine is closely related to exponential functions.Some key points about \( \sinh(x) \) include:

• The domain is all real numbers, which means you can input any real value and get a valid result.
• The range is also all real numbers; hyperbolic sine can output any real number.
• It is an odd function, so \( \sinh(-x) = -\sinh(x) \).

Hyperbolic functions like \( \sinh(x) \) often appear in the study of calculus and differential equations as they relate to complex exponential functions. They are also very useful in physics, especially in models involving hyperbolic geometry and relativity.

The Role of the Natural Logarithm

The natural logarithm, denoted \( \ln(x) \), is the inverse of the exponential function with the base \( e \), where \( e \approx 2.71828 \). When it comes to inverse hyperbolic functions like \( \sinh^{-1}(x) \), the natural logarithm plays a pivotal role due to its elegant properties and relationship with exponential functions.Here's how the natural logarithm functions:

• \( \ln(e^x) = x \); it effectively "un-does" the exponential function.
• It is defined only for positive real numbers. You can't calculate \( \ln(x) \) for \( x \leq 0 \).
• The natural logarithm is a continuous and increasing function.
• Its derivative is \( \frac{1}{x} \), which is a common component in integration processes.

When solving for the inverse hyperbolic sine, \( \sinh^{-1}(x) \), natural logarithms help us take the result of \( x + \sqrt{x^2 + 1} \) and convert it into a value through logarithmic transformations, making such calculations much more manageable in advanced mathematics.

Applications in Advanced Mathematics

Inverse hyperbolic functions, such as \( \sinh^{-1}(x) \), play a significant role in advanced mathematics. These functions are inverses of normal hyperbolic functions and help resolve more complex equations by simplifying calculations involving geometry and multiple variables.Their applications include:

• Solving Equations: Typical in dealing with differential equations, offering solutions where simple algebraic methods falter.
• Geometry and Hyperbolas: Useful for calculations involving hyperbolic structures or forms in both theoretical and applied mathematics.
• Physics: Key in special relativity, where space-time is expressed in hyperbolic terms to model the geodesics (shortest paths between two points) in the universe.

Each usage leverages the unique properties of inverse hyperbolic functions, merging them with logarithmic and exponential functions to uncover solutions conducive to more advanced studies. Understanding these layers in detail equips students with the analytical skills to tackle complex mathematical challenges effectively.