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The hyperbolic sine function, denoted as \( \sinh x \), is one of the basic hyperbolic functions. It is defined using exponential functions as \( \sinh x = \frac{e^x - e^{-x}}{2} \). This definition is quite similar to the one for sine in trigonometry, but adapted to hyperbolic geometry.
When you plug in a value of \( x \) into this function, it describes a kind of slope or curve within the hyperbolic plane. You’ll often see \( \sinh x \) taking on both positive and negative values depending on \( x \), unlike \( \cosh x \), which only outputs non-negative values. For instance, in the problem, we have \( \sinh x = -\frac{3}{4} \), showcasing its ability to reach into negative values.

• Defined as \( \sinh x = \frac{e^x - e^{-x}}{2} \)
• Can take on negative values
• Think of it as the hyperbolic equivalent of the sine function

Understanding this function is crucial because it forms a fundamental part of the identity \( \cosh^2 x - \sinh^2 x = 1 \), which is pivotal in solving many problems involving hyperbolic functions.

The hyperbolic cosine function is symbolized as \( \cosh x \) and is defined by the formula \( \cosh x = \frac{e^x + e^{-x}}{2} \). As with \( \sinh x \), \( \cosh x \) uses exponential functions, but it always yields a non-negative result because it's made of the sum of exponentials.
It represents a smooth curve, similar to a parabola, in the hyperbolic space. This is different from the standard cosine curve you may be familiar with in trigonometry, which oscillates between -1 and 1. Since \( \cosh x \) is always positive, it's used to indicate the presence of hyperbolic geometric structures or scenarios where only positive outputs are relevant.

• Defined as \( \cosh x = \frac{e^x + e^{-x}}{2} \)
• Always yields positive values
• Serves as the hyperbolic analog to the cosine function

In the example problem, you initially find \( \cosh x = \pm \frac{5}{4} \) via the hyperbolic identity \( \cosh^2 x - \sinh^2 x = 1 \). However, by definition, you should select \( \cosh x = \frac{5}{4} \) since it's positive.

The hyperbolic tangent function, expressed as \( \tanh x \), is a combination of the hyperbolic sine and cosine functions: \( \tanh x = \frac{\sinh x}{\cosh x} \). This ratio gives \( \tanh x \) properties similar to the trigonometric tangent, but within hyperbolic space.
In practical terms, \( \tanh x \) describes the slope of a line that touches the hyperbola defined by \( \sinh x \) and \( \cosh x \) at a particular point. The function aims to bridge the behaviors of both \( \sinh x \) and \( \cosh x \), revealing an important relationship between them.

• Calculated as \( \tanh x = \frac{\sinh x}{\cosh x} \)
• Used for determining the gradient or slope within hyperbolic curves

For the specific problem, after determining \( \sinh x = -\frac{3}{4} \) and \( \cosh x = \frac{5}{4} \), you find that \( \tanh x = -\frac{3}{5} \), leading to useful outputs for further calculations or analysis.

Hyperbolic identities play a crucial role in simplifying complex calculations and proving hyperbolic equations. A core identity is \( \cosh^2 x - \sinh^2 x = 1 \), closely resembling the Pythagorean identity \( \cos^2 x + \sin^2 x = 1 \) used in trigonometry.
These identities allow you to derive unknowns, such as transforming \( \sinh x \) into \( \cosh x \), and vice versa. For example, when given \( \sinh x = -\frac{3}{4} \), you can solve for \( \cosh x \) using the identity:

• Substituting \( \sinh x \) to solve for \( \cosh x \)
• Finding unknown hyperbolic functions via established relationships

Another application is determining the other hyperbolic functions, including \( \coth x, \sech x, \) and \( \csch x \), all of which are derived from their basic counterparts by using identities to simplify the calculations. The ability to use these identities efficiently is essential in deeper explorations of hyperbolic trigonometry.