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.
2. sinh x = 4/3

Verified step by step guidance

1. Recall the fundamental identity for hyperbolic functions: \(\cosh^{2}x - \sinh^{2}x = 1\). Given \(\sinh x = \frac{4}{3}\), substitute this value into the identity to find \(\cosh x\).
2. Calculate \(\cosh x\) by rearranging the identity: \(\cosh^{2}x = 1 + \sinh^{2}x\). Substitute \(\sinh x = \frac{4}{3}\) to get \(\cosh^{2}x = 1 + \left(\frac{4}{3}\right)^{2}\).
3. Take the positive square root of \(\cosh^{2}x\) to find \(\cosh x\), since \(\cosh x\) is always positive for real \(x\).
4. Use the definitions of the other hyperbolic functions in terms of \(\sinh x\) and \(\cosh x\) to find their values: \(\tanh x = \frac{\sinh x}{\cosh x}\), \(\coth x = \frac{\cosh x}{\sinh x}\), \(\sech x = \frac{1}{\cosh x}\), and \(\csch x = \frac{1}{\sinh x}\).
5. Substitute the known values of \(\sinh x\) and \(\cosh x\) into these formulas to express all five remaining hyperbolic functions.

Key Concepts

Hyperbolic Functions Definitions

Hyperbolic functions such as sinh x and cosh x are defined using exponential functions: 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 algebraically.

Fundamental Hyperbolic Identity

The identity cosh²x - sinh²x = 1 is analogous to the Pythagorean identity in trigonometry. It allows you to find one hyperbolic function if the other is known, which is essential for solving problems involving multiple hyperbolic functions.

Other Hyperbolic Functions and Their Relationships

Besides sinh and cosh, the other hyperbolic functions are tanh x, coth x, sech x, and csch x, defined as ratios involving sinh and cosh. Knowing how to express these functions in terms of sinh and cosh is crucial for finding their values once sinh x or cosh x is given.