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Chapter 7: Problem 100

Show that the range of the cotangent function is the set of all real numbers.

Short Answer

Expert verified
The range of the cotangent function is all real numbers, \( (-鈭�, +鈭�). \)

Step by step solution

01

Understand the cotangent function

The cotangent function is defined as \(\text{cot}(x) = \frac{\text{cos}(x)}{\text{sin}(x)}\). It is the reciprocal of the tangent function.
02

Determine the domain of the cotangent function

The cotangent function is undefined when \(\text{sin}(x) = 0\), that is at \(x = k\frac{\text{蟺}}{2}\) where \(k\) is any integer. Therefore, the domain of \(\text{cot}(x)\) is \(x eq k\frac{\text{蟺}}{2}\).
03

Analyze the behavior of the cotangent function

Observe the behavior of \(\text{cot}(x)\) as \(\text{sin}(x)\) approaches zero. As \(\text{sin}(x)\) approaches zero, the value of \(\text{cot}(x)\) tends to \(\frac{1}{0}\), resulting in vertical asymptotes at \(x = k\frac{\text{蟺}}{2}\). Between these vertical asymptotes at \(x = (k\text{蟺})\) and \(x = (k+1)\frac{\text{蟺}}{2}\), the function \(\text{cot}(x)\) covers all real numbers, descending from \(\text{+鈭瀩\) to \(\text{-鈭瀩\).
04

Conclude the range

Since \(\text{cot}(x)\) decreases continuously from \(\text{+鈭瀩\) to \(\text{-鈭瀩\) within each interval between its vertical asymptotes, the range of \(\text{cot}(x)\) for all \((x eq k\frac{\text{蟺}}{2})\) is \(\text{(-鈭�, +鈭�)}\). Therefore, the range of the cotangent function is all real numbers.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Range of Cotangent
The cotangent function, represented as \(\text{cot}(x) = \frac{\text{cos}(x)}{\text{sin}(x)}\), can take any real number as its output. When you look at a graph of \(\text{cot}(x)\), you鈥檒l notice that it continuously decreases from \(+鈭瀄) to \(-鈭瀄) within each interval between its vertical asymptotes.
Cotangent Domain
Understanding the domain of the cotangent function is crucial. Since \(\text{cot}(x) = \frac{\text{cos}(x)}{\text{sin}(x)}\), it is defined for all angles \( x \eq k\frac{\text{蟺}}{2}\), where \( k\) is any integer.
Vertical Asymptotes
Vertical asymptotes for cotangent occur at points where the function is undefined鈥攊n other words, where \( \text{sin}(x) = 0 \). This results in a division by zero, leading the function value towards \( \frac{1}{0} = 卤鈭�.\)

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