Each of Exercises 1–4 gives a value of sinh x or cosh x. Use the definitions and the identity cosh²x - sinh²x = 1 to find the values of the remaining five hyperbolic functions.
4. cosh x = 13/5, x>0

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Recall the fundamental identity for hyperbolic functions: \(\cosh^{2}x - \sinh^{2}x = 1\).

Given \(\cosh x = \frac{13}{5}\) and \(x > 0\), substitute into the identity to find \(\sinh x\): \(\left(\frac{13}{5}\right)^{2} - \sinh^{2}x = 1\).

Solve for \(\sinh^{2}x\): \(\sinh^{2}x = \left(\frac{13}{5}\right)^{2} - 1\).

Since \(x > 0\), take the positive square root to find \(\sinh x\): \(\sinh x = \sqrt{\left(\frac{13}{5}\right)^{2} - 1}\).

Use the definitions of the other hyperbolic functions in terms of \(\sinh x\) and \(\cosh x\) to find their values: - \(\tanh x = \frac{\sinh x}{\cosh x}\) - \(\coth x = \frac{\cosh x}{\sinh x}\) - \(\sech x = \frac{1}{\cosh x}\) - \(\csch x = \frac{1}{\sinh x}\).

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Hyperbolic Functions and Their Definitions

Hyperbolic functions such as sinh x and cosh x are analogs of trigonometric functions but based on exponential functions. Specifically, sinh x = (e^x - e^{-x})/2 and cosh x = (e^x + e^{-x})/2. Understanding these definitions helps in expressing and manipulating hyperbolic functions.

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Fundamental Hyperbolic Identity

The identity cosh²x - sinh²x = 1 is a key relationship between hyperbolic sine and cosine, similar to the Pythagorean identity in trigonometry. It allows solving for one function when the other is known, which is essential for finding the remaining hyperbolic functions.

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Definition of Remaining Hyperbolic Functions

Besides sinh and cosh, the other hyperbolic functions include tanh x = sinh x / cosh x, coth x = cosh x / sinh x, sech x = 1 / cosh x, and csch x = 1 / sinh x. Knowing these definitions enables calculation of all hyperbolic functions once sinh x and cosh x are determined.

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