Problem 107 - Algebra and Trigonometry Real Mathematics, Real People

Question

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 function.

Step-by-Step Solution

Answer

The inverse cotangent function, \(cot^{-1}(x)\), has a domain equal to the set of all real numbers and a range equal to the open interval \((0, \pi)\). The graph of \(cot^{-1}(x)\) is increasing and asymptotic to \(y = 0\) and \(y = \pi\) as \(x\) approaches \(\infty\) and \(-\infty\) respectively.

Step 1: Understand and restrict the domain

The cotangent function, \(cot(x)\), is periodic with period \(\pi\). Its vertical asymptotes occur when the tangent function equals 0, i.e., at \(x = 0\) and \(x = \pi\). Between \(0\) and \(\pi\), \(cot(x)\) decreases from \(\infty\) to \(-\infty\) whilst being defined for any \(x\) in this interval. Hence, the domain of \(cot(x)\) is restricted to the open interval \((0, \pi)\).

Step 2: Define the inverse cotangent function

The inverse cotangent function, denoted as \(cot^{-1}(x)\), is defined such that for any \(x\) in the range of the cotangent function, \(cot(cot^{-1}(x)) = x\). Importantly, the range of the inverse cotangent function is the interval \((0, \pi)\), which was the restricted domain of the cotangent function.

Step 3: Sketch the graph

To sketch the graph of \(cot^{-1}(x)\), first note that the range of \(cot^{-1}(x)\) is \(0 < y < \pi\). The function increases as \(x\) decreases. As \(x\) approaches \(\infty\), \(cot^{-1}(x)\) approaches 0, and as \(x\) approaches \(-\infty\), \(cot^{-1}(x)\) approaches \(\pi\). Thus, the graph is a reflection of the cotangent function's graph about the line \(y = x\), within the restricted domain \((0, \pi)\).

Key Concepts

Cotangent Function

The cotangent function, denoted as \( \cot(x) \), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the tangent function, i.e., \( \cot(x) = \frac{1}{\tan(x)} \). This means it can also be expressed as the ratio of the adjacent side to the opposite side in a right triangle, through the relation \( \cot(x) = \frac{\cos(x)}{\sin(x)} \).

Unlike sine or cosine, the cotangent function is undefined at points where \( \sin(x) = 0 \), due to division by zero. These points are typically where the function experiences vertical asymptotes. Since \( \sin(x) \) equals zero at integer multiples of \( \pi \), the function has vertical asymptotes at these values.

The cotangent function has a period of \( \pi \), meaning it repeats every \( \pi \) units. Over an interval of \( \pi \), \( \cot(x) \) starts from positive infinity and ends at negative infinity, decreasing continuously. This behavior is crucial for understanding the restricted domain when defining its inverse.

Restricted Domain

When dealing with inverse trigonometric functions, like the inverse cotangent, it's essential to restrict the domain of the original function to ensure it is one-to-one. A one-to-one function passes the horizontal line test, which means that each input corresponds to exactly one output, and vice versa.

For the cotangent function \( \cot(x) \), the domain is restricted to the open interval \((0, \pi)\). This restriction helps to eliminate any duplicates in the output values, enabling the function to have a unique inverse. During this restricted interval, \( \cot(x) \) is continuously decreasing, making it suitable for finding an inverse.

By selecting \((0, \pi)\) as the domain, we ensure that every value of \( \cot(x) \) has a unique output, which simplifies the process of defining \( \cot^{-1}(x) \). This carefully chosen domain excludes the undefined points and asymptotes while maintaining the natural decreasing behavior of the function.

Graph Sketching

To visualize and sketch the inverse cotangent function \( \cot^{-1}(x) \), we need to first understand the behavior of \( \cot(x) \) in the restricted domain \((0, \pi)\). In this interval, \( \cot(x) \) flows smoothly from positive infinity to negative infinity.

The graph of \( \cot^{-1}(x) \) will be a reflection of \( \cot(x) \) over the line \( y = x \). This results in \( \cot^{-1}(x) \) having a range where \( 0 < y < \pi \) and extending from negative to positive infinity. As \( x \to +\infty \), \( \cot^{-1}(x) \to 0 \), and as \( x \to -\infty \), \( \cot^{-1}(x) \to \pi \).

To sketch this graph confidently:

• Start by plotting key points corresponding to where \( \cot(x) \) has known values.
• Remember the reflection principle that flips the graph across the line \( y = x \).
• Consider drawing an asymptotic behavior: as \( x \) increases and decreases without bounds, \( \cot^{-1}(x) \) approaches its horizontal limits at \( 0 \) and \( \pi \) respectively.

Inverse Cotangent

The concept of the inverse cotangent function, \( \cot^{-1}(x) \), is essential in trigonometry for solving equations involving \( \cot(x) \). The inverse cotangent helps to find an angle whose cotangent is a given number.

Since the range of \( \cot(x) \) in the restricted domain \((0, \pi)\) is all real numbers, the domain of \( \cot^{-1}(x) \) is all real numbers, and its range becomes \((0, \pi)\).

When using \( \cot^{-1}(x) \), keep in mind:

• The function produces angles in the domain \((0, \pi)\).
• It's vital for solving right-angled triangle problems where you know the cotangent of an angle and need to find the angle itself.
• It simplifies complexity in many trigonometric identities and relationships.

These characteristics of \( \cot^{-1}(x) \) assure its importance in both theoretical and practical aspects of trigonometry, providing a solid foundation for further exploration in mathematics.