Problem 3 - Thomas Calculus in SI Units

Question

Gives a value of \(\sinh x\) or \(\cosh x .\) Use the definitions and the identity \(\cosh ^{2} x-\sinh ^{2} x=1\) to find the values of the remaining five hyperbolic functions. $$\cosh x=\frac{17}{15}, \quad x>0$$

Step-by-Step Solution

Answer

\(\sinh x = \frac{8}{15}, \tanh x = \frac{8}{17}, \coth x = \frac{17}{8}, \text{sech } x = \frac{15}{17}, \text{csch } x = \frac{15}{8}\).

Step 1: Recall Hyperbolic Identities

The hyperbolic identity given in the problem is \( \cosh^2 x - \sinh^2 x = 1 \). We'll use this equation along with the known value of \( \cosh x \).

Step 2: Substitute Known Values

Substitute \( \cosh x = \frac{17}{15} \) into the identity: \( \cosh^2 x = \left(\frac{17}{15}\right)^2 = \frac{289}{225} \). The equation becomes: \( \frac{289}{225} - \sinh^2 x = 1 \).

Step 3: Solve for \( \sinh^2 x \)

Rearrange the identity to find \( \sinh^2 x \): \( \sinh^2 x = \frac{289}{225} - 1 = \frac{289}{225} - \frac{225}{225} = \frac{64}{225} \).

Step 4: Find \( \sinh x \)

Since \( x > 0 \), we have \( \sinh x = \sqrt{\sinh^2 x} = \sqrt{\frac{64}{225}} = \frac{8}{15} \).

Step 5: Calculate \( \tanh x \)

Using \( \tanh x = \frac{\sinh x}{\cosh x} \), calculate: \( \tanh x = \frac{8/15}{17/15} = \frac{8}{17} \).

Step 6: Calculate \( \coth x \)

Using \( \coth x = \frac{1}{\tanh x} \), find \( \coth x = \frac{17}{8} \).

Step 7: Calculate \( \text{sech } x \)

Using \( \text{sech } x = \frac{1}{\cosh x} \), compute: \( \text{sech } x = \frac{15}{17} \).

Step 8: Calculate \( \text{csch } x \)

Using \( \text{csch } x = \frac{1}{\sinh x} \), find \( \text{csch } x = \frac{15}{8} \).

Key Concepts

Hyperbolic Identities

Hyperbolic identities are mathematical relationships similar to the well-known trigonometric identities but involve hyperbolic functions. One central hyperbolic identity is \( \cosh^2 x - \sinh^2 x = 1 \). This identity is analogous to the trigonometric identity \( \cos^2 x + \sin^2 x = 1 \) but has a different structure due to the nature of hyperbolic functions.
To understand this, recall that hyperbolic functions can be expressed using exponential functions by the definitions:

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

This relationship arises because when these definitions are plugged into the identity, the exponential terms cancel appropriately. Understanding that hyperbolic identities are similar to trigonometric ones, with slight differences, helps in applying these concepts to solve problems.

Cosh and Sinh Functions

The hyperbolic cosine and sine functions, \( \cosh x \) and \( \sinh x \), are vital building blocks of hyperbolic function theory. They are defined via exponential functions and retain properties that are reflections of real hyperbolic geometry.
Unlike their trigonometric counterparts, \( \cosh x \) and \( \sinh x \) do not rely on angles or circles but relate instead to areas and hyperbolas. The hyperbolic cosine, \( \cosh x \), grows exponentially as \( x \) increases and is always positive. Meanwhile, the hyperbolic sine, \( \sinh x \), can be positive or negative, which facilitates its use in a broader context.
In applying these functions, remember their core properties:

• \( \cosh(0) = 1 \)
• \( \sinh(0) = 0 \)
• The derivative of \( \cosh x \) is \( \sinh x \)
• The derivative of \( \sinh x \) is \( \cosh x \)

These properties make hyperbolic functions efficient for solving differential equations and certain types of integrals.

Trigonometric Identities

Trigonometric identities relate to the relationships between the basic trigonometric functions such as sine, cosine, and tangent. They provide tools for simplifying expressions and solving trigonometric equations. Some of the most crucial identities include:

• Pythagorean Identity: \( \cos^2 x + \sin^2 x = 1 \)
• Angle Addition Formulas:
  • \( \sin(a + b) = \sin a \cos b + \cos a \sin b \)
  • \( \cos(a + b) = \cos a \cos b - \sin a \sin b \)

When working with trigonometric and hyperbolic identities together, note that the hyperbolic identities (like \( \cosh^2 x - \sinh^2 x = 1 \)) differ in appearance but carry similar roles in calculations.
Both types of identities simplify calculations in calculus and physics. Thus, knowing when and how to apply these identities is crucial to solving many mathematical and applied problem scenarios.