Problem 14 - Essential Calculus

Question

Use the value of the given hyperbolic function to find the values of the other hyperbolic functions at \(x\) \(\tanh x=\frac{1}{2}\)

Step-by-Step Solution

Answer

The values of the other hyperbolic functions at \(x\) are \(\sinh x= \frac{1}{\sqrt{3}}\) and \(\cosh x = \frac{2}{\sqrt{3}}\).

Step 1: Identify the hyperbolic function identity

Set up the identity: \(\tanh x = \frac{\sinh x}{\cosh x}\), which is the form of the given equation. By plugging in the given \(\tanh x = \frac{1}{2}\), the equation becomes: \(\frac{1}{2} = \frac{\sinh x}{\cosh x}\).

Step 2: Solve for \(sinh x\)

Solving for \(\sinh x\), we can see that \(\sinh x\) equates to half of \(\cosh x\): \(\sinh x = \frac{1}{2}\cosh x\)

Step 3: Apply the identity for \(\cosh x\) and \(\sinh x\)

Using the identity \(\cosh^2x - \sinh^2x = 1\), we can substitute \(\sinh x = \frac{1}{2}\cosh x\) from Step 2 into the identity equation. This gives us \(\cosh^2x - (\frac{1}{2}\cosh x )^2 = 1\). Simplifying, we get a quadratic equation, \(\frac{3}{4}\cosh^2x - 1 = 0\).

Step 4: Solve the quadratic equation to find value of \(\cosh x\)

From the quadratic equation, solve for \(\cosh x\). The solutions are \(\cosh x = \frac{-b}{2a} ± \sqrt{\frac{b^2 - 4ac}{4a^2}}\), substituting \(a = \frac{3}{4}\), \(b=0\), and \(c=-1\), the roots are \(±\frac{2}{\sqrt{3}}\). Since \(\cosh x≥1\), negative root is not possible, hence, \(\cosh x = \frac{2}{\sqrt{3}}\).

Step 5: Substitute the value of \(\cosh x\) in equation for \(\sinh x\)

Substitute \(\cosh x = \frac{2}{\sqrt{3}}\) from Step 4 into the equation for \(\sinh x\) found in Step 2: \(\sinh x = \frac{1}{2}\times \frac{2}{\sqrt{3}}= \frac{1}{\sqrt{3}}\).

Key Concepts

Hyperbolic Identities

Hyperbolic identities are similar to trigonometric identities, but instead they involve hyperbolic functions. These functions include sinh (hyperbolic sine), cosh (hyperbolic cosine), and tanh (hyperbolic tangent). Much like their trigonometric counterparts, they help us solve hyperbolic equations and find useful relationships between these functions.
For instance, the identity for tanh is given by:

• \( \tanh x = \frac{\sinh x}{\cosh x} \)

This means that the hyperbolic tangent of \( x \) is the ratio of the hyperbolic sine to the hyperbolic cosine of \( x \).
Another important hyperbolic identity is:

• \( \cosh^2x - \sinh^2x = 1 \)

This identity is crucial when solving equations that involve hyperbolic functions since it relates \( \sinh x\) and \( \cosh x \) directly. Understanding and using these identities is fundamental when working with hyperbolic functions.

Solving Quadratic Equations

Quadratic equations are equations of the form \( ax^2 + bx + c = 0 \). Solving them often requires using the quadratic formula, which is:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This formula calculates the roots of the quadratic equation, essentially solving for the value(s) of \( x \).
In our exercise, using the identity \( \cosh^2x - \sinh^2x = 1 \), we arranged it into a quadratic equation form:

• \( \frac{3}{4} \cosh^2x - 1 = 0 \)

Then, by solving this equation using the quadratic formula where \( a = \frac{3}{4} \), \( b = 0 \), and \( c = -1 \), we find the possible values of \( \cosh x \).
Solving quadratic equations is a powerful tool for finding solutions within algebra and calculus.

Tanh Function

The tanh function is one of the primary hyperbolic functions and is noted for its utility in modeling real-world scenarios, such as growth patterns and various scientific calculations. The hyperbolic tangent function \( \tanh x \) has properties and graphs similar to the trigonometric tangent function, but it diverges in its application.
It is given by the ratio:

• \( \tanh x = \frac{\sinh x}{\cosh x} \)

This shows tanh as essentially a balance between \( \sinh x \) and \( \cosh x \), just like tangent is a ratio of sine and cosine.
When solving for other hyperbolic functions, knowing \( \tanh x \) helps in finding \( \sinh x \) and \( \cosh x \), serving as a foundational step in such exercises.

Cosh and Sinh Relationship

The relationship between \( \cosh x\) and \( \sinh x \) is a defining feature of hyperbolic functions and is captured in their main identity. This identity \( \cosh^2x - \sinh^2x = 1 \) shows how the functions are interlinked.
This relation provides a way to calculate one if the other is known, which is invaluable when certain values are given, such as in our exercise with \( \tanh x = \frac{1}{2} \).

• If \( \tanh x = \frac{\sinh x}{\cosh x} \), then \( \sinh x = \frac{1}{2} \cosh x \)

By substituting back into the identity \( \cosh^2x - \sinh^2x = 1 \), it becomes possible to solve for \( \cosh x \) and, subsequently, \( \sinh x \).
Understanding and using the cosh and sinh relationship simplifies working with hyperbolic functions and allows for solving complex problems with ease.