A calculator has a built-in sinh⁻¹ x function, but no csch⁻¹ x function. How do you evaluate csch⁻¹ 5 on such a calculator?

Verified step by step guidance

1. Understand the relationship between the hyperbolic cosecant inverse (csch⁻¹ x) and the hyperbolic sine inverse (sinh⁻¹ x). The formula to convert csch⁻¹ x to sinh⁻¹ x is: ${sinh⁻1}^{}\left(\frac{1}{x}\right)$. This means you can use sinh⁻¹(1/x) to evaluate csch⁻¹ x.
2. Rewrite the given problem csch⁻¹ 5 using the formula: ${sinh⁻1}^{}\left(\frac{1}{5}\right)$. This simplifies the problem to finding sinh⁻¹(1/5).
3. Use the calculator's built-in sinh⁻¹ x function to evaluate sinh⁻¹(1/5). First, calculate the value of 1/5, which is 0.2.
4. Input 0.2 into the calculator's sinh⁻¹ x function to find the result. The calculator will return the value of sinh⁻¹(0.2).
5. Interpret the result as the value of csch⁻¹ 5, since csch⁻¹ x is equivalent to sinh⁻¹(1/x). This completes the evaluation process.

Key Concepts

Inverse Hyperbolic Functions

Inverse hyperbolic functions, such as sinh⁻¹(x) and csch⁻¹(x), are the inverses of hyperbolic functions. The function csch⁻¹(x) is the inverse of the hyperbolic cosecant function, csch(x), which is defined as 1/sinh(x). Understanding how to convert between these functions is essential for evaluating csch⁻¹(x) using sinh⁻¹(x).

Relationship Between Hyperbolic Functions

Hyperbolic functions are related through identities that allow for conversions between them. For example, csch(x) can be expressed in terms of sinh(x). This relationship is crucial when using a calculator that only has sinh⁻¹(x), as it enables the evaluation of csch⁻¹(x) by manipulating the input to utilize the available function.

Calculating csch⁻¹(x) Using sinh⁻¹(x)

To evaluate csch⁻¹(5) using sinh⁻¹(x), one can use the identity csch⁻¹(x) = sinh⁻¹(1/x). Therefore, csch⁻¹(5) can be calculated as sinh⁻¹(1/5). This method allows the use of the available sinh⁻¹ function on the calculator to find the desired value.