Ch. 7 - Transcendental Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 7 - Transcendental Functions

Problem 7.7.1

Chapter 7, Problem 7.7.1

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.
1. sinh x = -3/4

Verified step by step guidance

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

Substitute the given value \(\sinh x = -\frac{3}{4}\) into the identity: \(\cosh^{2}x - \left(-\frac{3}{4}\right)^{2} = 1\).

Simplify the equation to solve for \(\cosh^{2}x\): \(\cosh^{2}x - \frac{9}{16} = 1\) which leads to \(\cosh^{2}x = 1 + \frac{9}{16}\).

Take the positive square root to find \(\cosh x\) because \(\cosh x\) is always positive: \(\cosh x = \sqrt{1 + \frac{9}{16}}\).

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

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definitions of Hyperbolic Functions

Hyperbolic functions sinh x and cosh x are defined as sinh x = (e^x - e^(-x))/2 and cosh x = (e^x + e^(-x))/2. Understanding these definitions helps relate the functions to exponential expressions and is fundamental for calculating other hyperbolic functions.

Hyperbolic Identity: cosh²x - sinh²x = 1

This identity is analogous to the Pythagorean identity in trigonometry and allows finding cosh x when sinh x is known, or vice versa. It ensures the relationship between cosh and sinh values is consistent and is essential for solving for unknown hyperbolic functions.

Definitions of Other Hyperbolic Functions

The remaining hyperbolic functions—tanh x, coth x, sech x, and csch x—are defined in terms of sinh x and cosh x, such as tanh x = sinh x / cosh x. Knowing these definitions allows computation of all hyperbolic functions once sinh x and cosh x are determined.

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