Problem 4

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. $$\cosh x=\frac{13}{5}, \quad x>0$$

Step-by-Step Solution

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Answer

$\sinh x = \frac{12}{5}$; other functions are $\tanh x = \frac{12}{13}$, $\coth x = \frac{13}{12}$, $\text{sech} x = \frac{5}{13}$, $\text{csch} x = \frac{5}{12}$.

1Step 1: Recall Hyperbolic Identities

The hyperbolic identity that we can use here is $ \cosh^2 x - \sinh^2 x = 1 $. This is analogous to the Pythagorean identity in trigonometry but for hyperbolic functions.

2Step 2: Substitute Given Value

Substitute the given value of $ \cosh x = \frac{13}{5} $ into the identity: $ \left(\frac{13}{5}\right)^2 - \sinh^2 x = 1 $.

3Step 3: Simplify Expression

Calculate $ \cosh^2 x $: $ \left(\frac{13}{5}\right)^2 = \frac{169}{25} $. The equation becomes $ \frac{169}{25} - \sinh^2 x = 1 $.

4Step 4: Solve for $\sinh^2 x$

Rearrange to solve for $\sinh^2 x$: $ \sinh^2 x = \frac{169}{25} - 1 $. This simplifies to $ \sinh^2 x = \frac{144}{25} $.

5Step 5: Solve for $\sinh x$

Since $ x>0 $, $ \sinh x = \sqrt{\frac{144}{25}} = \frac{12}{5} $.

6Step 6: Calculate Remaining Hyperbolic Functions

Using the definitions: 1. $ \tanh x = \frac{\sinh x}{\cosh x} = \frac{\frac{12}{5}}{\frac{13}{5}} = \frac{12}{13} $.2. $ \coth x = \frac{1}{\tanh x} = \frac{13}{12} $.3. $ \text{sech} x = \frac{1}{\cosh x} = \frac{5}{13} $.4. $ \text{csch} x = \frac{1}{\sinh x} = \frac{5}{12} $.

Key Concepts

Understanding $ \sinh $The Anatomy of $ \cosh $Hyperbolic Identities Unveiled

Understanding $ \sinh $

The hyperbolic sine function, denoted as $ \sinh x $, is a hyperbolic function closely related to the exponential function. It is defined as $ \sinh x = \frac{e^x - e^{-x}}{2} $. This definition shows that $ \sinh x $ is a measure of the vertical distance from the origin using hyperbolic geometry.
Unlike regular sine and cosine functions, which relate to circles, $ \sinh x $ operates over a hyperbola. Here are some key characteristics:

$ \sinh x $ is an odd function, meaning $ \sinh(-x) = -\sinh(x) $.
The graph of $ \sinh x $ is symmetric over the origin.
As $ x $ becomes very large, $ \sinh x $ approaches $ \frac{e^x}{2} $, indicating rapid growth.

Understanding $ \sinh x $ provides insight into solving problems like the one in the exercise, where it is calculated based on $ \cosh x $ using hyperbolic identities.

The Anatomy of $ \cosh $

Hyperbolic cosine, represented as $ \cosh x $, is another fundamental hyperbolic function. It is defined by the equation $ \cosh x = \frac{e^x + e^{-x}}{2} $. This function describes the distance along the axis of a hyperbola.
Unlike $ \sinh x $, $ \cosh x $ is an even function, with several unique properties:

$ \cosh(-x) = \cosh(x) $, showcasing its even symmetry.
The minimum value of $ \cosh x $ is 1, occurring when $ x = 0 $.
As $ x $ increases, $ \cosh x $ follows an exponential growth, similar to $ \sinh x $, but never goes below 1.

In the exercise, the given value of $ \cosh x = \frac{13}{5} $ sets the stage for finding other hyperbolic functions, including $ \sinh x $, by utilizing the hyperbolic identity.

Hyperbolic Identities Unveiled

Hyperbolic identities are key tools in solving problems involving hyperbolic functions. The main identity used in the exercise is $ \cosh^2 x - \sinh^2 x = 1 $. It's akin to the Pythagorean identity in trigonometry, $ \cos^2 x + \sin^2 x = 1 $, but adjusted for hyperbolic contexts.
These identities help in:

Bridging $ \sinh x $ and $ \cosh x $ through their squared values.
Calculating unknown hyperbolic functions when one is known, like in the solution's steps.

With this identity, determining remaining hyperbolic functions becomes straightforward when one function like $ \cosh x $ is given. Besides the main identity, others include: