Solve for tanh(x) in the Equation 6sinh(x) + 2cosh...

It seems like your query got cut off. From what I can infer, you're asking for the solution of an equation that involves hyperbolic functions like \( \sinh(x) \) and \( \cosh(x) \), and you'd like to solve for \( \tanh(x) \). Since the full equation wasn't provided, I'll assume it might look something like this:

\[
6\sinh(x) + 2\cosh(x) = 0
\]

### Step-by-Step Solution:

1. **Rewrite the equation**:
   \[
   6\sinh(x) + 2\cosh(x) = 0
   \]
   Divide the entire equation by 2:
   \[
   3\sinh(x) + \cosh(x) = 0
   \]

2. **Express \( \sinh(x) \) and \( \cosh(x) \) in terms of exponentials**:
   \[
   \sinh(x) = \frac{e^x - e^{-x}}{2}, \quad \cosh(x) = \frac{e^x + e^{-x}}{2}
   \]
   Substituting these into the equation:
   \[
   3\left(\frac{e^x - e^{-x}}{2}\right) + \frac{e^x + e^{-x}}{2} = 0
   \]
   Multiply everything by 2 to eliminate the denominators:
   \[
   3(e^x - e^{-x}) + (e^x + e^{-x}) = 0
   \]

3. **Simplify the equation**:
   Combine like terms:
   \[
   3e^x - 3e^{-x} + e^x + e^{-x} = 0
   \]
   This simplifies to:
   \[
   4e^x - 2e^{-x} = 0
   \]

4. **Multiply through by \( e^x \)** to eliminate the negative exponent:
   \[
   4e^{2x} - 2 = 0
   \]
   Solve for \( e^{2x} \):
   \[
   4e^{2x} = 2 \quad \Rightarrow \quad e^{2x} = \frac{1}{2}
   \]

5. **Take the natural logarithm**:
   \[
   2x = \ln\left(\frac{1}{2}\right) = -\ln(2)
   \]
   Solve for \( x \):
   \[
   x = -\frac{\ln(2)}{2}
   \]

6. **Find \( \tanh(x) \)**:
   Now, recall that:
   \[
   \tanh(x) = \frac{\sinh(x)}{\cosh(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}}
   \]
   Substituting \( x = -\frac{\ln(2)}{2} \) into this expression for \( \tanh(x) \), we get:
   \[
   \tanh\left(-\frac{\ln(2)}{2}\right)
   \]
   Since the exact value can be computed, the result is \( -\frac{1}{3} \).

Thus, \( \tanh(x) = -\frac{1}{3} \).

---

Would you like further clarification or details on any step? Here are some related questions:

1. How do the hyperbolic functions \( \sinh(x) \) and \( \cosh(x) \) relate to exponential functions?
2. Can you explain why multiplying by \( e^x \) eliminates the negative exponent?
3. How can we derive the natural logarithm from exponential equations?
4. What are some real-world applications of hyperbolic functions like \( \tanh(x) \)?
5. How is the solution process similar for other transcendental equations?

**Tip:** Hyperbolic functions are closely related to their trigonometric counterparts but involve exponential expressions.

This page provides a detailed step-by-step solution to solving for tanh(x) in the equation 6sinh(x) + 2cosh(x) = 0. Learn how to work with hyperbolic functions and their relationship to exponential functions, solve equations using logarithms, and find the value of tanh(x). Suitable for undergraduate mathematics students.

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