91Ó°ÊÓ

The hyperbolic sine function, denoted as \( \sinh x \), is one of the fundamental hyperbolic functions, much like the sine function in trigonometry. It can be defined using the exponential function as follows:

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

This formula illustrates that \( \sinh x \) is derived from the difference of two exponential functions, divided by 2. In terms of personal intuition, \( \sinh x \) can be considered a measure of an imaginary angle in hyperbolic geometry.

In our problem, we found sinh \( x \) using an identity involving \( \cosh x \). By solving \( \cosh^2 x - \sinh^2 x = 1 \), we calculated that \( \sinh x = \frac{12}{5} \). This shows how \( \sinh x \) and \( \cosh x \) are intricately linked, similar to sin and cos in circular trigonometry.

The hyperbolic cosine function, denoted as \( \cosh x \), is similar to the cosine function in trigonometry, but defined using hyperbolic geometry. Its definition through exponential functions is:

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

This describes \( \cosh x \) as the average of two exponential functions. Unlike \( \sinh x \), the \( \cosh x \) function is always positive and does not oscillate, akin to the trigonometric cosine.

In the given exercise, \( \cosh x = \frac{13}{5} \). This value was utilized to find other hyperbolic functions using hyperbolic identities, showcasing the importance of \( \cosh x \) in computations involving hyperbolic functions.

Hyperbolic identities are relationships involving hyperbolic functions that mirror trigonometric identities. One of the most important hyperbolic identities is:

• \( \cosh^2 x - \sinh^2 x = 1 \)

This equation is the hyperbolic equivalent of the Pythagorean identity \( \cos^2 x + \sin^2 x = 1 \) in trigonometry.

In our exercise, this identity was crucial to solve for \( \sinh x \). It demonstrates the deep connection between \( \cosh x \) and \( \sinh x \), allowing us to solve for remaining functions when one is known. Understanding how to apply this identity is essential in many areas of mathematics, including calculus and hyperbolic geometry.

The hyperbolic tangent function, denoted as \( \tanh x \), is analogous to the tangent function in trigonometry. It is defined as:

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

This function measures the steepness or slope at a particular point on the hyperbola.

For the problem at hand, we found \( \tanh x \) to be \( \frac{12}{13} \) using the calculated values of \( \sinh x \) and \( \cosh x \). This shows how hyperbolic tangent can be derived easily with a grasp on its component functions, \( \sinh x \) and \( \cosh x \). It is a useful function, particularly in the areas of hyperbolic geometry and complex analysis.

The hyperbolic secant function, \( \sech x \), is the hyperbolic counterpart to the secant function in trigonometry. It is defined as:

• \( \sech x = \frac{1}{\cosh x} \)

It represents the reciprocal of the hyperbolic cosine function. A unique property of \( \sech x \) is that it approaches zero as \( x \) increases, reflecting the non-periodic nature of hyperbolic functions.

In this exercise, \( \sech x \) was found to be \( \frac{5}{13} \) by taking the reciprocal of \( \cosh x = \frac{13}{5} \). Understanding \( \sech x \) and its properties is key in contexts where \( \cosh x \) is used, such as engineering disciplines dealing with structures and cables.