Problem 1 - Algebra 2 and Trigonometry

Question

If tan \(x\) increases for all values of \(x\) for which it is defined, explain why cot \(x\) decreases for all values of \(x\) for which it is defined.

Step-by-Step Solution

Answer

As \( \tan(x) \) increases, \( \cot(x) \), being the reciprocal, decreases for all defined values of \( x \).

Step 1: Understand the Relationship

The tangent function, \( \tan(x) \), is the ratio of \( \sin(x) \) to \( \cos(x) \). The cotangent function, \( \cot(x) \), is the reciprocal of the tangent function: \( \cot(x) = \frac{1}{\tan(x)} \). This means when \( \tan(x) \) increases, \( \cot(x) \) should do the opposite since it is its reciprocal.

Step 2: Analyze the Behavior of \( \tan(x) \)

The function \( \tan(x) \) is increasing in every interval where it is defined, specifically between the vertical asymptotes at \( x = \frac{\pi}{2} + k\pi \), where \( k \) is an integer. Understanding that \( \tan(x) \) increases continuously implies the values \( \tan(x) = a \) transform to \( a' \) such that \( a' > a \).

Step 3: Examine the Effect on \( \cot(x) \)

Since \( \cot(x) = \frac{1}{\tan(x)} \), when the value of \( \tan(x) \) increases, it implies that \( \frac{1}{\tan(x)} \) decreases. For example, if \( \tan(x) \) moves from a smaller value to a larger value, the value of its reciprocal decreases (e.g. \( \frac{1}{2} \) decreases to \( \frac{1}{4} \) if it moves from 2 to 4).

Step 4: Conclude with the Decreasing Nature of \( \cot(x) \)

Thus, for all real values of \( x \) where both \( \tan(x) \) and \( \cot(x) \) are defined, \( \tan(x) \) being an increasing function results directly in \( \cot(x) \) being a decreasing function, due to their reciprocal relationship.

Key Concepts

Tangent and Cotangent Relationship

The tangent and cotangent functions are closely related in trigonometry. The tangent function, represented as \( \tan(x) \), and the cotangent function, represented as \( \cot(x) \), are reciprocal functions. This means that the cotangent is simply the reciprocal of the tangent: \( \cot(x) = \frac{1}{\tan(x)} \). Because of this reciprocal relationship, whenever the tangent function exhibits a particular behavior, the cotangent function showcases the opposite behavior.

For example, if \( \tan(x) \) increases, then \( \cot(x) \) decreases. This happens because increasing values in a function lead to decreasing values in its reciprocal and vice versa. Understanding this basic relationship helps in analyzing how these two essential trigonometric functions behave over their respective domains.

Reciprocal Functions

Reciprocal functions are functions where one function is the inverse, or reciprocal, of another. This is crucial in trigonometry, particularly when dealing with the tangent and cotangent functions. For a function \( f(x) \), its reciprocal is \( g(x) = \frac{1}{f(x)} \). For tangent and cotangent, this relationship is \( \cot(x) = \frac{1}{\tan(x)} \).

This means if you have \( \tan(x) = a \), then \( \cot(x) = \frac{1}{a} \). Hence, any change in the value of \( \tan(x) \) results in an inverse change in \( \cot(x) \). As the tangent value increases, its reciprocal remains valid by decreasing, ensuring both functions maintain their distinct characteristics over the intervals they're defined. This interplay is an essential aspect when solving trigonometric problems, as it underlines how reciprocal functions govern each other's rate of change.

Intervals of Increase and Decrease

To understand how a function behaves over its domain, it's essential to determine the intervals where it is increasing or decreasing. For \( \tan(x) \), this function is increasing on any interval where it is defined: specifically between each vertical asymptote. These occur at \( x = \frac{\pi}{2} + k\pi \), where \( k \) is any integer.

An increasing function means that as \( x \) advances, \( \tan(x) \) yields progressively larger values. Conversely, \( \cot(x) \), being the reciprocal of \( \tan(x) \), will display a decreasing nature within those same intervals. This is because as the value obtained from \( \tan(x) \) becomes larger, the outcomes for \( \cot(x) = \frac{1}{\tan(x)} \) shrink.

This dynamic displays the beauty of trigonometric relationships, as they allow us to predict and understand the behavior of complex functions purely through their reciprocal interactions. By visualizing these intervals of increase and decrease, one can grasp why \( \cot(x) \) decreases precisely because \( \tan(x) \) increases.