Problem 1

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

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Answer

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

1Step 1: Recall hyperbolic identities

Recall the identity $ \cosh^2 x - \sinh^2 x = 1 $. This identity will help us find $ \cosh x $, given $ \sinh x = -\frac{3}{4} $.

2Step 2: Find $ \cosh x $

Since $ \sinh x = -\frac{3}{4} $, substitute this into the identity: \[ \cosh^2 x - \left(-\frac{3}{4}\right)^2 = 1 \]. Simplify it to: \[ \cosh^2 x - \frac{9}{16} = 1 \]. Then, solve for $ \cosh^2 x $: \[ \cosh^2 x = 1 + \frac{9}{16} = \frac{25}{16} \]. Thus, $ \cosh x = \pm \frac{5}{4} $.

3Step 3: Determine the correct sign for $ \cosh x $

$ \cosh x $ is always positive, so we choose $ \cosh x = \frac{5}{4} $.

4Step 4: Find $ \tanh x $

$ \tanh x = \frac{\sinh x}{\cosh x} = \frac{-\frac{3}{4}}{\frac{5}{4}} = -\frac{3}{5} $.

5Step 5: Find $ \coth x $

$ \coth x = \frac{1}{\tanh x} = \frac{1}{-\frac{3}{5}} = -\frac{5}{3} $.

6Step 6: Find $ \sech x $

$ \sech x = \frac{1}{\cosh x} = \frac{1}{\frac{5}{4}} = \frac{4}{5} $.

7Step 7: Find $ \csch x $

$ \csch x = \frac{1}{\sinh x} = \frac{1}{-\frac{3}{4}} = -\frac{4}{3} $.

Key Concepts

Hyperbolic IdentityHyperbolic SineHyperbolic CosineHyperbolic TangentInverse Hyperbolic Functions

Hyperbolic Identity

Hyperbolic identities are essential building blocks in understanding hyperbolic functions. One of the fundamental hyperbolic identities is $ \cosh^2 x - \sinh^2 x = 1 $. This identity is akin to the Pythagorean identity in trigonometry $ \cos^2 x + \sin^2 x = 1 $. It shows the relationship between hyperbolic sine and hyperbolic cosine.
To use this identity, first substitute the given value into it. For example, if $ \sinh x = -\frac{3}{4} $, you substitute it into the identity to find $ \cosh x $.
Knowing this identity allows you to calculate one function if you know the other, keeping calculations straightforward and interconnected. This way, it's simpler to extend knowledge from one function to others.

Hyperbolic Sine

The hyperbolic sine function, denoted as $ \sinh x $, is defined as $ \sinh x = \frac{e^x - e^{-x}}{2} $. This formula helps understand how the function behaves, expressing growth and decay.
In the given problem, $ \sinh x = -\frac{3}{4} $. This value is used in computing other hyperbolic functions through identities.

Recognizing that $ \sinh x $ can take positive and negative values helps to build an intuitive sense of its graph, where it smoothly transitions through the origin.
Its unique property allows it to aid in finding other functions when plugged into the identity $ \cosh^2 x - \sinh^2 x = 1 $.

Hyperbolic Cosine

The hyperbolic cosine function, denoted as $ \cosh x $, is defined as $ \cosh x = \frac{e^x + e^{-x}}{2} $. It contrasts with $ \sinh x $ by taking only non-negative values.
In this context, using the identity $ \cosh^2 x - \sinh^2 x = 1 $ with $ \sinh x = -\frac{3}{4} $, leads to $ \cosh x = \frac{5}{4} $.

Note that $ \cosh x $ is always positive, making it reliable for determining values without guessing.
This property gives the hyperbolic cosine a parabolic nature, reflected in its graph, moving upwards from its vertex, showing exponential growth and symmetry.

Hyperbolic Tangent

Hyperbolic tangent, $ \tanh x $, is a ratio between hyperbolic sine and cosine: $ \tanh x = \frac{\sinh x}{\cosh x} $. It provides a measure of the steepness of the resultant line from these functions.
Using the previously found values, we calculate $ \tanh x = \frac{-\frac{3}{4}}{\frac{5}{4}} = -\frac{3}{5} $.

Range-bound by -1 and 1, $ \tanh x $ takes values gently transitioning as $ x $ increases, portraying a sigmoid shape.
This function is particularly useful in calculations involving hyperbolic angles where one deals with exponential cases.

Inverse Hyperbolic Functions

Inverse hyperbolic functions allow for finding the original value of the hyperbolic function. While not directly explored in the exercise, they play a crucial role.
For each hyperbolic function like $ \sinh x $, there's an inverse, denoted as $ \text{arsinh} x $, retrieving the initial input.

The inverse functions expand the utility of hyperbolic functions in solving exponential equations.
They give solutions that logarithmic functions influence, revealing interesting properties such as $ \text{arsinh} x = \ln(x + \sqrt{x^2 + 1}) $.

Understanding these inverse functions supports detailed problem-solving skills, making them vital for learners delving into higher-level mathematics.

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