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Chapter 4: Problem 115

Define the inverse cotangent function by restricting the domain of the cotangent function to the interval \((0, \pi),\) and sketch the graph of the inverse trigonometric function.

Short Answer

Expert verified
The inverse cotangent function is defined by restricting the domain of the cotangent function to the interval \((0, \pi)\). This results in a decreasing function. Its graph can be obtained by reflecting the graph of the cotangent function in this interval over the line \(y = x\), resulting in a curve that goes upwards, getting infinitely close to \(y = \pi\), but never touching it.

Step by step solution

01

Understanding the cotangent function in the interval \((0, \pi)\)

The cotangent function, \(\cot(x)\), is defined as the reciprocal of the tangent function, that is, \(\cot(x) = \frac{1}{\tan(x)}\). The graph of the cotangent function in the interval \((0, \pi)\) starts from positive infinity at \(x = 0\) and decreases to negative infinity as \(x\) approaches \(\pi\). The function is undefined at \(x = \pi\).
02

Defining the inverse cotangent function

The inverse cotangent function, denoted as \(\cot^{-1}(x)\) or \(arccot(x)\), is the function that takes a cotangent value and tells you what angle the cotangent came from. It can be defined by restricting the domain of the cotangent function to the interval \((0, \pi)\), making it a decreasing function in that interval. This way, it passes the horizontal line test and thus ensures the existence of an inverse.
03

Sketching the graph of the inverse cotangent function

To sketch the graph of the inverse cotangent function, first sketch the graph of the cotangent function in the interval \((0, \pi)\). It goes straight from positive infinity down to negative infinity. Now, to obtain the graph of the inverse function, flip the graph over the line \(y = x\). The graph now starts from close to \(y = 0\) and increases until it gets close to \(y = \pi\), without touching these limits. This gives the graph of the inverse cotangent function.

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