Ch. 7 - Transcendental Functions

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

Hass 15th Edition

Ch. 7 - Transcendental Functions

Problem 7.7.4

Chapter 7, Problem 7.7.4

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

Verified step by step guidance

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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Key Concepts

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

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.

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.

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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Related Practice

Textbook Question

Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.)

17. lim(x→∞)arcsec(x)

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Textbook Question

17. Show that √(10x+1) and √(x+1) grow at the same rate as x→∞ by showing that they both grow at the same rate as √x as x→∞.

Textbook Question

Since the hyperbolic functions can be expressed in terms of exponential functions, it is possible to express the inverse hyperbolic functions in terms of logarithms, as shown in the following table.

sinh⁻¹x = ln(x + √(x² + 1)), -∞ < x < ∞

cosh⁻¹x = ln(x + √(x² - 1)), x ≥ 1

tanh⁻¹x = (1/2)ln((1+x)/(1-x)), |x| < 1

sech⁻¹x = ln((1+√(1-x²))/x), 0 < x ≤ 1

csch⁻¹x = ln(1/x + √(1+x²)/|x|), x ≠ 1

coth⁻¹x = (1/2)ln((x+1)/(x-1)), |x| > 1

Use these formulas to express the numbers in Exercises 61–66 in terms of natural logarithms.

63. tanh⁻¹(-1/2)

Textbook Question

Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.)

13. lim(x → 1⁻)arcsin(x)

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