91Ó°ÊÓ

The inverse hyperbolic tangent, denoted as \(\tanh^{-1}(x)\), is the function that 'reverses' the hyperbolic tangent function. It answers the question: for a given value \(x\), what hyperbolic angle \(y\) satisfies \(x = \tanh(y)\)?
To find \(\tanh^{-1}(0.75)\), we use the formula: \(\tanh^{-1}(x) = \frac{1}{2} \times \text{ln}\frac{1+x}{1-x}\).
Let's substitute \(x = 0.75\):
\(\tanh^{-1}(0.75) = \frac{1}{2} \times \text{ln}\frac{1+0.75}{1-0.75}\).
Simplifying the fraction inside the logarithm gives us: \(\tanh^{-1}(0.75) = \frac{1}{2} \times \text{ln}\frac{1.75}{0.25} = \frac{1}{2} \times \text{ln}(7)\). By using a calculator, you get \(\tanh^{-1}(0.75) \approx 0.972\). This tells us the value of \(y\) such that \(\tanh(y) = 0.75\).
In summary:

• Inverse hyperbolic tangent solves for the hyperbolic angle \(y\) given \(x\) in \(\tanh(y)\).
• The formula \(\frac{1}{2} \times \text{ln}\frac{1+x}{1-x}\) helps calculate it.

The inverse hyperbolic cosine, denoted as \(\text{cosh}^{-1}(x)\), is a function that 'reverses' the hyperbolic cosine. Similar to before, it answers: for a given value \(x\), what hyperbolic angle \(y\) satisfies \(x = \text{cosh}(y)\)?
To find \(\text{cosh}^{-1}(2)\), we use the formula: \(\text{cosh}^{-1}(x) = \text{ln}(x + \text{sqrt}(x^2 - 1))\).
Let's substitute \(x = 2\):
\(\text{cosh}^{-1}(2) = \text{ln}(2 + \text{sqrt}(2^2 - 1))\).
Simplifying inside the logarithm gives us: \(\text{cosh}^{-1}(2) = \text{ln}(2 + \text{sqrt}(4 - 1)) = \text{ln}(2 + \text{sqrt}(3))\).
Using a calculator, you get \(\text{cosh}^{-1}(2) \approx 1.317\). This value of \(y\) such that \(\text{cosh}(y) = 2\) is obtained.
In summary:

• Inverse hyperbolic cosine solves for the hyperbolic angle \(y\) given \(x\) in \(\text{cosh}(y)\).
• The formula \(\text{ln}(x + \text{sqrt}(x^2 - 1))\) helps calculate it.

Logarithms, often abbreviated as 'ln' for natural logarithm, is a fundamental mathematical concept.
Definition: A logarithm answers the question: 'To what power must we raise a specific base to get a given number?' For natural logarithms, the base is 'e' (approximately 2.718).
For example, if \(e^y = x\), then \(y = \text{ln}(x)\).

Hyperbolic functions often utilize logarithms for their inverse forms. For instance:

• The formula for \( \tanh^{-1}(0.75) = \frac{1}{2} \text{ln} \frac{1+0.75}{1-0.75} \).
• The formula for \( \text{cosh}^{-1}(2) = \text{ln}(2 + \text{sqrt}(4-1)) \).

Logarithms simplify calculations and are essential tools for solving equations involving exponential growth, compounding interests, and various scientific calculations.