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The hyperbolic sine function, denoted as \( \sinh(x) \), is a fundamental hyperbolic function often used in calculus and engineering. It's defined by the formula:

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

Here, \( e \) represents the base of the natural logarithms, approximately equal to 2.718. The formula essentially combines exponential growth and decay, highlighting its symmetry and relation to the exponential function.
The calculation of \( \sinh(3) \) involves computing \( e^3 \) and \( e^{-3} \), subtracting these results, and dividing by 2. With \( e^3 \approx 20.0855 \) and \( e^{-3} \approx 0.0498 \), the computation gives:

• \( \sinh(3) \approx \frac{20.0855 - 0.0498}{2} \approx 10.0179 \)

Functions like \( \sinh \) have practical applications, such as in modeling the shape of a hanging cable or chain, known as a catenary curve.

The hyperbolic cosine function, \( \cosh(x) \), is defined similarly to \( \sinh \) but instead involves the sum of exponential functions:

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

An interesting property of the \( \cosh \) function is that it's even, meaning that \( \cosh(-x) = \cosh(x) \). This arises from the use of addition rather than subtraction in its formula, resulting in the function's symmetrical nature.
To compute \( \cosh(-2) \), we note that \( \cosh(-2) = \cosh(2) \). Substituting \( x = 2 \) results in:

• \( \cosh(2) \approx \frac{7.3891 + 0.1353}{2} \approx 3.7622 \)

Hyperbolic cosine is used in many scientific fields, including physics and hyperbolic geometry, often to describe certain relations involving time-dependent and spatially periodic phenomena.

The hyperbolic tangent function, \( \tanh(x) \), describes the relationship between the hyperbolic sine and cosine functions:

• \( \tanh(x) = \frac{\sinh(x)}{\cosh(x)} \)

The \( \tanh \) function is inherently related to growth and stabilization processes and finds applications in hyperbolic geometry and signal processing.
For practical purposes, to find \( \tanh(\ln 4) \), you first compute the \( \ln 4 \approx 1.3863 \). Using this, find \( \sinh(1.3863) \) and \( \cosh(1.3863) \) and divide the two:

• \( \tanh(1.3863) \approx \frac{1.875}{2.125} \approx 0.8820 \)

This process highlights the role of hyperbolic tangent in describing situations where there is a need to model rapid changes to saturation, often seen in statistical models and hyperbolic trigonometry.

Inverse hyperbolic functions allow us to retrieve the input used in a hyperbolic function from its output. These functions include \( \sinh^{-1}(x) \), \( \cosh^{-1}(x) \), and \( \tanh^{-1}(x) \).

• \( \sinh^{-1}(x) = \ln(x + \sqrt{x^2 + 1}) \)
• \( \cosh^{-1}(x) = \ln(x + \sqrt{x^2 - 1}) \)
• \( \tanh^{-1}(x) = \frac{1}{2} \ln \left(\frac{1+x}{1-x}\right) \)

These are indispensable in solving equations involving hyperbolic functions where finding the exact parameter \( x \) is required.
For example, calculating \( \sinh^{-1}(-2) \) involves:

• \( \sinh^{-1}(-2) \approx \ln(-2 + \sqrt{5}) \approx -1.4436 \)

Meanwhile, \( \cosh^{-1}(3) \) requires computing:

• \( \cosh^{-1}(3) \approx \ln(3 + \sqrt{8}) \approx 1.7627 \)

The inverse functions are vital in many areas, such as solving differential equations and transforming space variables in mathematical physics.