Problem 16 - Algebra and Trigonometry

Question

Find the period and graph the function. $$ y=\frac{1}{2} \csc x $$

Step-by-Step Solution

Answer

The period is \(2\pi\).

Step 1: Understand the csc function

The function given is \(y = \frac{1}{2} \csc x\). Cosecant, \(\csc x\), is the reciprocal of sine, \(\sin x\), so \(\csc x = \frac{1}{\sin x}\). This function is undefined whenever \(\sin x = 0\).

Step 2: Find the period of \(\csc x\)

The period of the sine function \(\sin x\) is \(2\pi\). Therefore, the period of \(\csc x\), which is its reciprocal, is also \(2\pi\).

Step 3: Consider the effect of \(\frac{1}{2}\)

The scalar \(\frac{1}{2}\) does not affect the period of the function; it only vertically stretches the graph by a factor of \(\frac{1}{2}\). Thus, the period remains \(2\pi\).

Step 4: Graph the function

To graph \(y = \frac{1}{2} \csc x\), identify key points and asymptotes from where \(\csc x\) is undefined (i.e., where \(\sin x = 0\), such as \(x = 0, \pi, 2\pi\), etc.). This function has vertical asymptotes at these points. Over one period \(0 < x < 2\pi\), the graph will exhibit two branches opening upwards between these asymptotes, at \((\frac{\pi}{2}, 2)\) and open downwards at \((\frac{3\pi}{2}, 2)\), showing that \(\frac{1}{2} \) is the factor stretching the standard \(\csc x\) graph.

Key Concepts

Cosecant Function

The cosecant function, denoted as \( \csc x \), is an important trigonometric function that is the reciprocal of the sine function. This means that \( \csc x = \frac{1}{\sin x} \). Because it is based on the sine function, \( \csc x \) is undefined wherever the sine function equals zero. In simpler terms:

• \( \csc x \) is undefined at angles where \( \sin x = 0 \), such as at \( x = 0, \pi, 2\pi, 3\pi, \ldots \)
• At these undefined points, the graph of the cosecant function exhibits vertical asymptotes.
• The cosecant function has two parts within one period, moving to infinity in either the positive or negative y-direction between its asymptotes.

By understanding its relationship with the sine function, students can effectively graph and interpret \( \csc x \) with ease.

Periodicity

Periodicity refers to the repeating nature of trigonometric functions over intervals. For the sine function, \( \sin x \), this interval is \( 2\pi \), meaning every \( 2\pi \) units along the x-axis the function repeats its values and graph shape. Since the cosecant function, \( \csc x \), is derived from \( \sin x \) by taking its reciprocal:

• The period of \( \csc x \) is also \( 2\pi \).
• This period means that \( \csc x \) repeats its pattern every \( 2\pi \) units on the x-axis.
• Though the vertical asymptotes and other features may change due to modifications, the overall period remains consistent unless there is a horizontal scaling involved.

Understanding periodicity helps when predicting the behavior of trigonometric graphs beyond the initial span, confirming that features like asymptotes and peaks recur as their base function continues.

Graphing Trigonometric Functions

Graphing trigonometric functions like \( y = \frac{1}{2} \csc x \) involves recognizing features such as asymptotes and peaks. Here's how to approach graphing \( \csc x \):

• Identify key points where \( \csc x \) is undefined, like \( x = 0, \pi, 2\pi, \ldots \), marking vertical asymptotes.
• The transformation \( \frac{1}{2} \) affects the graph by compressing it vertically, moving its peaks closer to the x-axis without changing period or asymptote positions.
• Over one period, \( 0 < x < 2\pi \), the graph exhibits branches that approach these asymptotes from above or below based on the costructure \( \frac{1}{2} \).
• The graph doesn't cross its asymptotes but instead reflects the endpoints between them, forming a sort of U-shape pattern repeated in each period.

By breaking the graphing process into these steps, students can better visualize and understand how various factors like coefficients impact the standard graph's appearance.