Problem 2

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. $$ \sinh x=\frac{4}{3} $$

Step-by-Step Solution

Verified

Answer

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

1Step 1: Use the Identity

The identity $ \cosh^2 x - \sinh^2 x = 1 $ is a key equation for hyperbolic functions. Here, we know that $ \sinh x = \frac{4}{3} $, so substitute and rearrange the identity to find $ \cosh x $.

2Step 2: Solve for cosh x

Substitute $ \sinh x = \frac{4}{3} $ into the identity to find $ \cosh^2 x $.\[ \cosh^2 x = 1 + \left( \frac{4}{3} \right)^2 = 1 + \frac{16}{9} = \frac{25}{9} \]Therefore, $ \cosh x = \sqrt{\frac{25}{9}} = \frac{5}{3} $.

3Step 3: Calculate the Remaining Hyperbolic Functions

Now that we have $ \sinh x = \frac{4}{3} $ and $ \cosh x = \frac{5}{3} $, use the definitions to find the other hyperbolic functions: $ \tanh x = \frac{\sinh x}{\cosh x} = \frac{\frac{4}{3}}{\frac{5}{3}} = \frac{4}{5} $, $ \coth x = \frac{\cosh x}{\sinh x} = \frac{5}{4} $, $ \sech x = \frac{1}{\cosh x} = \frac{3}{5} $, and $ \csch x = \frac{1}{\sinh x} = \frac{3}{4} $.

Key Concepts

Understanding $\sinh$Exploring $\cosh$Understanding $\tanh$Hyperbolic Function Identities

Understanding $\sinh$

When you first encounter hyperbolic functions, the term $\sinh$ might seem a bit puzzling. However, it's quite straightforward once you break it down. The hyperbolic sine function, or $\sinh(x)$, is analogous to the trigonometric sine function but is defined using exponential functions. In mathematical terms, $\sinh(x) = \frac{e^x - e^{-x}}{2}$. This represents the difference of two exponential functions divided by two, capturing the essence of hyperbolic geometry.

Hyperbolic sine, $\sinh$, is important in many fields like physics and engineering because it describes shapes like hanging cables known as catenaries. Remember, $\sinh(x)$ is an odd function, meaning $\sinh(-x) = -\sinh(x)$. By understanding this fundamental property, you can better grasp how $\sinh$ behaves in different contexts.

Exploring $\cosh$

Just like $\sinh$, the hyperbolic cosine function $\cosh(x)$ mirrors the trigonometric cosine function with its own unique definition. It is mathematically expressed as $\cosh(x) = \frac{e^x + e^{-x}}{2}$. Unlike $\sinh$, $\cosh(x)$ is an even function, meaning $\cosh(-x) = \cosh(x)$, which is significant when calculating hyperbolic identities.

In the problem given, you use $\sinh x = \frac{4}{3}$ and the identity $\cosh^2 x - \sinh^2 x = 1$ to find $\cosh x$. You'll find $\cosh x = \frac{5}{3}$ after solving the identity. These relationships and identities are crucial for solving complex problems in hyperbolic functions and are used to model scenarios in physics and calculus where traditional trigonometric functions fall short.

Understanding $\tanh$

The hyperbolic tangent function $\tanh(x)$ is derived from $\sinh(x)$ and $\cosh(x)$. It's comparable to the trigonometric tangent function and defined as $\tanh(x) = \frac{\sinh(x)}{\cosh(x)}$. Using the values from the solved exercise, where $\sinh x = \frac{4}{3}$ and $\cosh x = \frac{5}{3}$, you can calculate $\tanh x = \frac{4}{5}$.

Just as other functions have their properties, $\tanh(x)$ is an odd function much like $\sinh(x)$. This means that it maintains symmetry about the origin, an important characteristic when graphing and analyzing its behavior. $\tanh$ is especially useful in scenarios such as signal processing and logistic functions, where it serves as a smooth threshold function.

Hyperbolic Function Identities

Hyperbolic function identities are vital tools in understanding the relationships and interactions between different hyperbolic functions. The primary identity for hyperbolic functions is $\cosh^2 x - \sinh^2 x = 1$. This identity is analogous to the trigonometric identity $\cos^2 x + \sin^2 x = 1$ and forms the basis for a range of calculations.

Additional identities help in deriving and manipulating hyperbolic functions further. For instance:

$\cosh^2(x) = 1 + \sinh^2(x)$
$\tanh^2(x) + \sech^2(x) = 1$
$\coth^2(x) - \csch^2(x) = 1$

These identities are not just theoretical; they have practical applications in integration and differential equations. Understanding these identities allows you to switch between functions easily, making them powerful tools in both academic and professional use cases.

Problem 1

Problem 2