Problem 18 - Algebra 2 and Trigonometry

Question

The graphs of which two trigonometric functions have an asymptote at \(x=0 ?\)

Step-by-Step Solution

Answer

The cotangent and cosecant functions have asymptotes at \(x=0\).

Step 1: Understand Asymptotes

An asymptote is a line that a graph approaches but does not touch. In trigonometry, certain functions have vertical asymptotes at specific points where the function is undefined.

Step 2: Identify Trigonometric Functions with Asymptotes

The primary trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant. Among these, only tangent, cotangent, secant, and cosecant can have vertical asymptotes, as sine and cosine are continuous and defined for all real numbers.

Step 3: Analyze Behavior Near \( x=0 \)

The tangent function, \( \tan(x) = \frac{\sin(x)}{\cos(x)} \), is undefined when \( \cos(x)=0 \). The cosecant function, \( \csc(x) = \frac{1}{\sin(x)} \), is undefined when \( \sin(x)=0 \). At \( x=0 \), \( \cos(0)=1 \) and \( \sin(0)=0 \). Therefore, tangent remains defined, but cosecant is undefined, indicating a vertical asymptote at \( x=0 \).

Step 4: Consider Other Functions for Asymptotes

The secant function \( \sec(x) = \frac{1}{\cos(x)} \) is undefined when \( \cos(x)=0 \), which does not occur at \( x=0 \). The cotangent function, \( \cot(x) = \frac{\cos(x)}{\sin(x)} \), is undefined when \( \sin(x)=0 \). Since \( \sin(0)=0 \), cotangent also has a vertical asymptote at \( x=0 \).

Key Concepts

Tangent Function Asymptotes

When studying the behavior of the tangent function, it's important to identify where vertical asymptotes occur. The tangent function is represented by the formula \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). This function is undefined when the denominator equals zero. Therefore, vertical asymptotes are present when \( \cos(x) = 0 \).

Key properties of the tangent function include:

• Periodic nature with a period of \( \pi \); it repeats itself every \( \pi \) units along the x-axis.
• Asymptotes appear at odd multiples of \( \frac{\pi}{2} \), specifically when \( x = \frac{\pi}{2} + n\pi \), where \( n \) is an integer.

Since \( \cos(0) = 1 \), the tangent function is defined at \( x=0 \), not producing an asymptote there. Understanding these elements helps with graphing the tangent function and predicting its behavior at critical points.

Cosecant Function Asymptotes

The cosecant function, expressed as \( \csc(x) = \frac{1}{\sin(x)} \), is notorious for having vertical asymptotes. These occur at points where \( \sin(x) = 0 \), since the function becomes undefined.

Some facts about the cosecant function include:

• It is undefined and thus exhibits asymptotes at integer multiples of \( \pi \) (e.g., \( x = n\pi \) where \( n \) is an integer).
• It has no continuity across the x-axis, contrasting the sine function’s continuity.

At \( x=0 \), \( \sin(0) = 0 \), leading to a vertical asymptote for the cosecant function. This characteristic clearly distinguishes it from sine and provides a critical aspect for sketching its graph.

Cotangent Function Asymptotes

The cotangent function, denoted by \( \cot(x) = \frac{\cos(x)}{\sin(x)} \), similarly shows vertical asymptotes where \( \sin(x) = 0 \). This occurs because division by zero is undefined.

Characteristic properties of the cotangent function include:

• Periodic properties with a period of \( \pi \), similar to the tangent function.
• Asymptotes are present at integer multiples of \( \pi \) (e.g., \( x = n\pi \) where \( n \) is an integer).

Thus, since \( \sin(0) = 0 \), the cotangent function also has an asymptote at \( x=0 \). Recognizing these aspects is essential for understanding the graphing of cotangent functions and identifying key discontinuities.