91Ó°ÊÓ

The hyperbolic sine function, denoted as \(\text{sinh}\), is analogous to the sine function in trigonometry, but it applies to hyperbolas instead of circles. The definition of hyperbolic sine is given by:
\( \text{sinh} \, x = \frac{e^x - e^{-x}}{2} \). Here, \(e\) represents the base of the natural logarithm.
The hyperbolic sine function exhibits a number of crucial properties:

• \(\text{sinh}(0) = 0\)
• It is an odd function, meaning \(\text{sinh}(-x) = -\text{sinh}(x)\)
• Its derivative is \(\text{cosh}(x)\), the hyperbolic cosine function

Consider the exercise where our goal is to prove \(\text{sinh}(x+y)\). By substituting the definitions, we get:
\[ \text{sinh}(x+y) = \frac{e^{x+y} - e^{-(x+y)}}{2} = \frac{e^x e^y - e^{-x} e^{-y}}{2} \]. Now, reorganizing these terms using \( \text{sinh} \, x \) and \( \text{cosh} \, x \), we finally derive the sum formula:
\[ \text{sinh}(x+y) = \text{sinh} \, x \, \text{cosh} \, y + \text{cosh} \, x \, \text{sinh} \, y \]This establishes a parallel with the sine sum formula in trigonometry.

The hyperbolic cosine function, denoted as \(\text{cosh}\), is used in describing the shape of a hanging cable or chain. Defined similarly to hyperbolic sine, its formula is:
\[ \text{cosh} \, x = \frac{e^x + e^{-x}}{2} \] Like hyperbolic sine, \( e \) here stands for the base of the natural logarithm.
Some key properties of \( \text{cosh} \, x \) are:

• \(\text{cosh}(0) = 1\)
• It is an even function, meaning \(\text{cosh}(-x) = \text{cosh}(x)\)
• Its derivative is \(\text{sinh}(x)\)

In our exercise for proving \(\text{cosh}(x+y)\), we expand using the definition:
\[ \text{cosh}(x+y) = \frac{e^{x+y} + e^{-(x+y)}}{2} = \frac{e^x e^y + e^{-x} e^{-y}}{2} \]
Substituting back the expressions for \( \text{sinh} \, x \) and \( \text{cosh} \, x \), we get:
\[ \text{cosh}(x+y) = \text{cosh} \, x \, \text{cosh} \, y + \text{sinh} \, x \, \text{sinh} \, y \] This similarly mirrors the cosine sum formula in trigonometry.

The hyperbolic tangent function, denoted as \(\text{tanh}\), is defined as the ratio of the hyperbolic sine to the hyperbolic cosine:
\[ \text{tanh} \, x = \frac{ \text{sinh} \, x }{ \text{cosh} \, x } \] This function is useful for describing smooth transitions and is found in many areas including engineering and financial models.
Key properties include:

• \(\text{tanh}(0) = 0\)
• \(-1 < \text{tanh}(x) < 1\) holds for all real numbers x
• Its derivative is \(1 - \text{tanh}^2(x)\)

In solving our exercise equation, we use the previously proven formulas for \(\text{sinh}(x+y)\) and \(\text{cosh}(x+y)\) to derive \(\text{tanh}(x+y)\):
\[ \text{tanh}(x+y) = \frac{\text{sinh}(x+y)}{\text{cosh}(x+y)} = \frac{\text{sinh} \, x \, \text{cosh} \, y + \text{cosh} \, x \, \text{sinh} \, y}{\text{cosh} \, x \, \text{cosh} \, y + \text{sinh} \, x \, \text{sinh} \, y} = \frac{\frac{\text{sinh} \, x}{\text{cosh} \, x} + \frac{\text{sinh} \, y}{\text{cosh} \, y}}{1 + \frac{\text{sinh} \, x \, \text{sinh} \, y}{\text{cosh} \, x \, \text{cosh} \, y} } = \frac{\text{tanh} \, x + \text{tanh} \, y}{1 + \text{tanh} \, x \, \text{tanh} \, y} \] This formula for \(\text{tanh}(x+y)\) is similar to the addition formula for the tangent function in trigonometry.