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.