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Chapter 4: Problem 117

Define the inverse cosecant function by restricting the domain of the cosecant function to the intervals \([-\pi / 2,0)\) and \((0, \pi / 2],\) and sketch the graph of the inverse trigonometric function.

Short Answer

Expert verified
The inverse cosecant function is defined as the angle resulting from the restriction of the cosecant function to the intervals \([-\pi / 2,0)\) and \((0, \pi / 2]\). Its graph is the reflection of the graph of the restricted cosecant function in the line \(y = x\), with asymptotes at \(x = 0\) and \(\pi / 2\).

Step by step solution

01

Function Restriction

The inverse cosecant function, denoted as \(csc^{-1}(x)\) or \(arc csc(x)\), is the function that undoes the action of the cosecant function. It is defined only if the original cosecant function \(y = csc(x)\) is defined, which is only true for \(x\) belonging to the intervals mentioned, that is, \([-\pi / 2,0)\) and \((0, \pi / 2]\). Thus, to define \(csc^{-1}(x)\), we restrict the domain of \(csc(x)\) to these intervals.
02

Defining the Inverse Cosecant Function

The inverse coscecant function, \(csc^{-1}(x)\), is now defined as the angle resulting from the restriction of the cosecant function defined in step 1. This means that \(csc(csc^{-1}(x)) = x\) for \(x\) in the interval \([-\pi / 2,0)\) and \((0, \pi / 2]\).
03

Sketching the Graph

To sketch the graph of the inverse cosecant function, first sketch the graph of the restricted cosecant function in the given intervals. The graph of \(csc^{-1}(x)\) should be the reflection of the original cosecant graph in the line \(y = x\). The graph will have asymptotes at \(x = 0\) and \(\pi / 2\), and will illustrate the inverse nature of the function.

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