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To solve for the other hyperbolic functions when given that \( \cosh(x) = \frac{41}{40} \) and \( x > 0 \), let's proceed as follows:

### 1. Recall the Hyperbolic Identities
We know that:
\[
\cosh^2(x) - \sinh^2(x) = 1
\]
Using this identity, we can solve for \( \sinh(x) \).

### 2. Solve for \( \sinh(x) \)
Given \( \cosh(x) = \frac{41}{40} \), plug this into the identity:
\[
\left(\frac{41}{40}\right)^2 - \sinh^2(x) = 1
\]
\[
\frac{1681}{1600} - \sinh^2(x) = 1
\]
\[
\sinh^2(x) = \frac{1681}{1600} - \frac{1600}{1600} = \frac{81}{1600}
\]
\[
\sinh(x) = \sqrt{\frac{81}{1600}} = \frac{9}{40}
\]

Thus, \( \sinh(x) = 0.225 \), which matches the value in the table.

### 3. Calculate \( \tanh(x) \)
The hyperbolic tangent function is defined as:
\[
\tanh(x) = \frac{\sinh(x)}{\cosh(x)}
\]
Substituting the values:
\[
\tanh(x) = \frac{\frac{9}{40}}{\frac{41}{40}} = \frac{9}{41} \approx 0.2195
\]

### 4. Calculate \( \coth(x) \)
The hyperbolic cotangent function is defined as:
\[
\coth(x) = \frac{\cosh(x)}{\sinh(x)}
\]
Substitute the values:
\[
\coth(x) = \frac{\frac{41}{40}}{\frac{9}{40}} = \frac{41}{9} \approx 4.5556
\]

### 5. Calculate \( \text{sech}(x) \)
The hyperbolic secant function is defined as:
\[
\text{sech}(x) = \frac{1}{\cosh(x)}
\]
Using \( \cosh(x) = \frac{41}{40} \):
\[
\text{sech}(x) = \frac{1}{\frac{41}{40}} = \frac{40}{41} \approx 0.9756
\]

### 6. Calculate \( \text{csch}(x) \)
The hyperbolic cosecant function is defined as:
\[
\text{csch}(x) = \frac{1}{\sinh(x)}
\]
Using \( \sinh(x) = \frac{9}{40} \):
\[
\text{csch}(x) = \frac{1}{\frac{9}{40}} = \frac{40}{9} \approx 4.4444
\]

### Summary of Values
- \( \sinh(x) = 0.225 \)
- \( \tanh(x) \approx 0.2195 \)
- \( \coth(x) \approx 4.5556 \)
- \( \text{sech}(x) \approx 0.9756 \)
- \( \text{csch}(x) \approx 4.4444 \)

Would you like more details on any step, or have further questions?

---

1. What is the general form of hyperbolic identities?
2. How does the identity \(\cosh^2(x) - \sinh^2(x) = 1\) relate to trigonometric identities?
3. Can the hyperbolic functions be applied to complex numbers?
4. How do hyperbolic functions relate to exponential functions?
5. What are some real-world applications of hyperbolic functions?

**Tip:** Remember that hyperbolic functions can simplify the analysis of hyperbolic shapes and are widely used in engineering and physics, especially in the study of catenary curves.

If cosh(x) = 41/40 and x > 0, find the values of the other hyperbolic functions at x.

Find the values of sinh(x), tanh(x), coth(x), sech(x), and csch(x) given that cosh(x) = 41/40 and x > 0. This detailed solution covers hyperbolic identities, useful formulas, and provides a step-by-step approach suitable for students in Grades 11-12.

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