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Magazine Chapter 16: Problem 5
Evaluate: (a) \(\tanh ^{-1} 0.75\) and (b) \(\cosh ^{-1} 2\).
Short Answer
Expert verified
(a) \(\tanh^{-1} 0.75 \approx 0.97295\), (b) \(\cosh^{-1} 2 \approx 1.31696\).
Step by step solution
01
Understanding Hyperbolic Functions
The hyperbolic tangent function, \(\tanh(x)\), is defined as \(\frac{\sinh(x)}{\cosh(x)}\). Similarly, hyperbolic cosine, \(\cosh(x)\), is defined as \(\frac{e^x + e^{-x}}{2}\).
02
Formula for Inverse Hyperbolic Tangent
To solve for \(\tanh^{-1} y\), we use the formula: \(\tanh^{-1} y = \frac{1}{2} \ln\left(\frac{1+y}{1-y}\right)\).
03
Calculate \(\tanh^{-1} 0.75\)
Substitute \(y = 0.75\) in the formula: \(\tanh^{-1} 0.75 = \frac{1}{2} \ln\left(\frac{1+0.75}{1-0.75}\right)\). This simplifies to \(\frac{1}{2} \ln\left(\frac{1.75}{0.25}\right)\). Calculate inside the logarithm: \(\frac{1.75}{0.25} = 7\). Therefore, \(\tanh^{-1} 0.75 = \frac{1}{2} \ln(7)\).
04
Solve \(\ln(7)\)
Calculate \(\ln(7)\), which is approximately 1.9459. Thus, \(\tanh^{-1} 0.75 \approx \frac{1}{2} \times 1.9459 = 0.97295\).
05
Formula for Inverse Hyperbolic Cosine
The inverse hyperbolic cosine function is given by the formula \(\cosh^{-1} x = \ln(x + \sqrt{x^2-1})\).
06
Calculate \(\cosh^{-1} 2\)
Substitute \(x = 2\) into the formula: \(\cosh^{-1} 2 = \ln(2 + \sqrt{2^2 - 1}) = \ln(2 + \sqrt{3})\).
07
Solve \(\ln(2 + \sqrt{3})\)
Find \(\sqrt{3}\), which is approximately 1.732. Compute \(2 + 1.732 = 3.732\) and then \(\ln(3.732)\), which is approximately 1.31696. Therefore, \(\cosh^{-1} 2 \approx 1.31696\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Hyperbolic Tangent
The hyperbolic tangent function, denoted as \( \tanh(x) \), is a unique function that has applications in algebra and calculus. It is formed using the hyperbolic sine and cosine functions as follows:
\[ \tanh(x) = \frac{\sinh(x)}{\cosh(x)} \]
This means it describes a ratio between the hyperbolic sine and hyperbolic cosine. Unlike regular trigonometric tangent functions, which are based on sine and cosine, hyperbolic functions feature exponential functions providing a smooth curve between -1 and 1.
\[ \tanh(x) = \frac{\sinh(x)}{\cosh(x)} \]
This means it describes a ratio between the hyperbolic sine and hyperbolic cosine. Unlike regular trigonometric tangent functions, which are based on sine and cosine, hyperbolic functions feature exponential functions providing a smooth curve between -1 and 1.
- Domain: All real numbers
- Range: Values strictly between -1 and 1
Hyperbolic Cosine
Hyperbolic cosine, \( \cosh(x) \), is one of the foundational hyperbolic functions, defined through exponential functions:
\[ \cosh(x) = \frac{e^x + e^{-x}}{2} \]
This differs from the trigonometric cosine function despite the similarity in their names, as it represents the average of exponential functions rather than relying on angle measure.
\[ \cosh(x) = \frac{e^x + e^{-x}}{2} \]
This differs from the trigonometric cosine function despite the similarity in their names, as it represents the average of exponential functions rather than relying on angle measure.
- Domain: All real numbers
- Range: All real numbers \( y \geq 1 \)
Natural Logarithm
The natural logarithm, written as \( \ln(x) \), is a logarithmic function based on the constant \( e \), where \( e \approx 2.71828 \). It is crucial for calculating growth processes and inverse exponentials.
The natural logarithm answers the question: "To what power must \( e \) be raised to produce \( x \)?"
The natural logarithm answers the question: "To what power must \( e \) be raised to produce \( x \)?"
- Domain: \( x > 0 \)
- Range: All real numbers
Inverse Functions
Inverse functions reverse the effects of original functions, answering what input they would need when given an output. For hyperbolic functions:
- \( \tanh^{-1}(y) \): Finds input \( x \) such that \( \tanh(x) = y \) using the formula:
\[ \tanh^{-1}(y) = \frac{1}{2} \ln\left(\frac{1+y}{1-y}\right) \] - \( \cosh^{-1}(x) \): Solves for \( x \) such that \( \cosh(x) = x \) with:
\[ \cosh^{-1}(x) = \ln(x + \sqrt{x^2-1}) \]
