Problem 1 - Thomas Calculus

Question

Each of Exercises \(1-4\) 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. $$ \sinh x=-\frac{3}{4} $$

Step-by-Step Solution

Answer

\(\cosh x = \frac{5}{4}, \tanh x = -\frac{3}{5}, \coth x = -\frac{5}{3}, \sech x = \frac{4}{5}, \csch x = -\frac{4}{3}.\)

Step 1: Recognize Given Information

We are given that \( \sinh x = -\frac{3}{4} \). We need to find the values of the other five hyperbolic functions: \( \cosh x, \tanh x, \coth x, \sech x, \text{ and } \csch x \).

Step 2: Calculate \( \cosh x \) using Identity

Use the identity \( \cosh^2 x - \sinh^2 x = 1 \). Substitute \( \sinh x = -\frac{3}{4} \) into the identity:\[ \cosh^2 x - \left(-\frac{3}{4}\right)^2 = 1 \]This simplifies to:\[ \cosh^2 x - \frac{9}{16} = 1 \]Therefore:\[ \cosh^2 x = 1 + \frac{9}{16} = \frac{25}{16} \]So, \( \cosh x = \sqrt{\frac{25}{16}} = \frac{5}{4} \), since \( \cosh x \) must be positive.

Step 3: Calculate \( \tanh x \)

The hyperbolic tangent function \( \tanh x \) is defined as \( \tanh x = \frac{\sinh x}{\cosh x} \).Thus:\[ \tanh x = \frac{-\frac{3}{4}}{\frac{5}{4}} = -\frac{3}{5} \]

Step 4: Calculate \( \coth x \)

The hyperbolic cotangent function \( \coth x \) is the reciprocal of \( \tanh x \), defined as \( \coth x = \frac{1}{\tanh x} \).Thus:\[ \coth x = \frac{5}{-3} = -\frac{5}{3} \]

Step 5: Calculate \( \sech x \)

The hyperbolic secant function \( \sech x \) is the reciprocal of \( \cosh x \), defined as \( \sech x = \frac{1}{\cosh x} \).Thus:\[ \sech x = \frac{1}{\frac{5}{4}} = \frac{4}{5} \]

Step 6: Calculate \( \csch x \)

The hyperbolic cosecant function \( \csch x \) is the reciprocal of \( \sinh x \), defined as \( \csch x = \frac{1}{\sinh x} \).Thus:\[ \csch x = \frac{1}{-\frac{3}{4}} = -\frac{4}{3} \]

Key Concepts

sinh x

The hyperbolic sine function, denoted as \( \sinh x \), is an essential part of the hyperbolic functions family. Just like sine in trigonometry, \( \sinh x \) provides us with a way to relate angle measures to ratios of a right triangle on a hyperbolic plane.
The function is defined by the formula:

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

This formula can be broken down into exponential components, involving the natural constant \( e \), making it very distinct from the trigonometric sine function. One special property of \( \sinh x \) is its symmetry about the origin, indicating it is an odd function, so \( \sinh(-x) = -\sinh(x) \).
For instance, in our exercise, we were given \( \sinh x = -\frac{3}{4} \), which directly reflects its property as an odd function, and helped us calculate other hyperbolic functions.

cosh x

Hyperbolic cosine, or \( \cosh x \), is another crucial hyperbolic function, similar to the trigonometric cosine. However, it deals with the hyperbolic plane and has its unique properties.
Its definition is expressed as:

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

Unlike \( \sinh x \), \( \cosh x \) is always positive and is symmetric about the vertical axis, implying it is an even function: \( \cosh(-x) = \cosh(x) \). This characteristic made it necessary to choose the positive square root in our calculations in the original exercise.
From the identity \( \cosh^2 x - \sinh^2 x = 1 \), we were able to determine that \( \cosh x = \frac{5}{4} \) when \( \sinh x = -\frac{3}{4} \), reinforcing its property of always being non-negative.

hyperbolic identities

Hyperbolic identities are expressions involving hyperbolic functions that hold true for all values of the variables involved, just like their trigonometric counterparts.
One of the fundamental hyperbolic identities is:

• \( \cosh^2 x - \sinh^2 x = 1 \)

This identity is analogous to the Pythagorean identity in trigonometry, \( \cos^2 x + \sin^2 x = 1 \), making it an essential tool in solving hyperbolic function problems. In our problem, this identity was pivotal in solving for \( \cosh x \) when \( \sinh x \) was given.
These identities also include relationships for calculating functions like \( \tanh x = \frac{\sinh x}{\cosh x} \), \( \coth x = \frac{1}{\tanh x} \), \( \sech x = \frac{1}{\cosh x} \), and \( \csch x = \frac{1}{\sinh x} \). Each one of these functions derives from the basic properties of their defining hyperbolic functions, which are essential for more complex calculations in calculus and engineering.