Problem 7 - Calculus Early Transcendentals

Question

A calculator has a built-in \(\sinh ^{-1} x\) function, but no \(\operatorname{csch}^{-1} x\) function. How do you evaluate \(\operatorname{csch}^{-1} 5\) on such a calculator?

Step-by-Step Solution

Answer

Answer: To find the value of \(\operatorname{csch}^{-1}(5)\), use the formula \(\operatorname{csch}^{-1}(x) = \sinh^{-1}\left(\frac{1}{x}\right)\) and input into the calculator as follows: $$\operatorname{csch}^{-1}(5) = \sinh^{-1}\left(\frac{1}{5}\right) = \sinh^{-1}(0.2)$$ Calculate the value using the calculator's built-in \(\sinh^{-1}(x)\) function to obtain the result.

Step 1: Find the relationship between csch and sinh

Recall that the hyperbolic cosecant function, \(\operatorname{csch}(x)\), is defined as the reciprocal of the hyperbolic sine function, \(\sinh(x)\): $$\operatorname{csch}(x) = \frac{1}{\sinh(x)}$$ Since we want to find \(\operatorname{csch}^{-1}x\), we can rewrite this equation in terms of the inverse functions: $$x = \frac{1}{\sinh(\operatorname{csch}^{-1}(x))}$$

Step 2: Solve for csch^(-1)(x) in terms of sinh^(-1)(x)

We now want to isolate \(\operatorname{csch}^{-1}(x)\) in the above equation. Firstly, take the reciprocal of both sides: $$\frac{1}{x} = \sinh(\operatorname{csch}^{-1}(x))$$ Now, apply the inverse hyperbolic sine function, \(\sinh^{-1}(x)\), to both sides of the equation: $$\sinh^{-1}\left(\frac{1}{x}\right) = \sinh^{-1}\left(\sinh(\operatorname{csch}^{-1}(x))\right)$$ Since applying the inverse function to the original function will cancel them out, we'll have: $$\sinh^{-1}\left(\frac{1}{x}\right) = \operatorname{csch}^{-1}(x)$$

Step 3: Evaluate csch^(-1)(5) using the obtained formula

Using the formula derived in Step 2, we can now evaluate \(\operatorname{csch}^{-1} 5\) using the calculator's built-in \(\sinh^{-1} x\) function: $$\operatorname{csch}^{-1}(5) = \sinh^{-1}\left(\frac{1}{5}\right)$$ Now, simply input into the calculator: $$\sinh^{-1}(0.2)$$ This will give the value of \(\operatorname{csch}^{-1}(5)\).

Key Concepts

Hyperbolic Functions

Hyperbolic functions are analogs of the trigonometric functions we often use but are based on hyperbolas rather than circles. The main hyperbolic functions are similar in behavior and properties to their trigonometric counterparts. The set includes functions like hyperbolic sine (\(\sinh(x)\)), hyperbolic cosine (\(\cosh(x)\)), and hyperbolic tangent (\(\tanh(x)\)).But there's more to them beyond just these core functions. Just like we have cosecant, secant, and cotangent in trigonometry, we have hyperbolic counterparts:

• Hyperbolic cosecant (\(\operatorname{csch}(x)\))
• Hyperbolic secant (\(\operatorname{sech}(x)\))
• Hyperbolic cotangent (\(\operatorname{coth}(x)\))

Hyperbolic functions have unique properties that make them useful in various fields such as calculus, complex numbers, and even in physical models like the shape of a hanging cable (catenary), nuclear physics, and special relativity. They are defined through exponential functions and can be related back to geometric interpretations involving hyperbolas.

Calculator Usage

Using a calculator effectively can simplify the process of dealing with inverse hyperbolic functions. Calculators often come equipped with various functions to make complex calculations simpler but may lack some specific buttons. For example, most calculators have a \(\sinh^{-1}(x)\) button but might not directly provide a \(\operatorname{csch}^{-1}(x)\) feature.To use your calculator to find \(\operatorname{csch}^{-1}(5)\), you can use a trick with the available function \(\sinh^{-1}(x)\):

• First, find the reciprocal of 5, which is 0.2.
• Use the calculator's \(\sinh^{-1}(x)\) function to find \(\sinh^{-1}(0.2)\).
• The result will provide you with the answer for \(\operatorname{csch}^{-1}(5)\).

This method takes advantage of the mathematical property that relates the inverse of hyperbolic sine to hyperbolic cosecant. Ensure your calculator is set to the correct mode (degree/radians) required for your context, though for hyperbolic functions, typically this isn't a concern as with trigonometric functions.

Inverse Functions

Inverse functions are a crucial concept in mathematics that allows us to 'reverse' a process. With trigonometric or hyperbolic functions, they serve to undo the effects of those original functions. For example, if \(f(x) = y\), an inverse function \(f^{-1}(y) = x\) will essentially retrieve the initial input, provided \(f(x)\) was one-to-one and onto.With hyperbolic functions, the inverse hyperbolic functions include:

• Inverse hyperbolic sine (\(\sinh^{-1}(x)\))
• Inverse hyperbolic cosine (\(\cosh^{-1}(x)\))
• Inverse hyperbolic tangent (\(\tanh^{-1}(x)\))

To solve \(\operatorname{csch}^{-1}(x)\), understanding the properties of these inverse functions is essential. By knowing that \(\operatorname{csch}^{-1}(x)\) is related to \(\sinh^{-1}(x)\) as shown in the solution, one can simplify calculations where specific functions are not native to all calculators. This knowledge eases the process of translating a given problem into a solvable equation using available resources.