Question

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

Short answer

The range of the cotangent function is all real numbers, \( (-∞, +∞). \)

Step-by-step solution

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.

Determine the domain of the cotangent function

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

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 \(x = k\frac{\text{π}}{2}\). Between these vertical asymptotes at \(x = (k\text{π})\) and \(x = (k+1)\frac{\text{π}}{2}\), the function \(\text{cot}(x)\) covers all real numbers, descending from \(\text{+∞}\) to \(\text{-∞}\).

Conclude the range

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

Key concepts

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’ll notice that it continuously decreases from \(+∞\) to \(-∞\) 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 \( x \eq k\frac{\text{π}}{2}\), where \( k\) is any integer.

Vertical Asymptotes

Vertical asymptotes for cotangent occur at points where the function is undefined—in other words, where \( \text{sin}(x) = 0 \). This results in a division by zero, leading the function value towards \( \frac{1}{0} = ±∞.\)