Problem 43 - Precalculus: Functions and Graphs

Question

Find the period and sketch the graph of the equation. Show the asymptotes. $$y=\csc \frac{1}{3} x$$

Step-by-Step Solution

Answer

Period: \(6\pi\); Asymptotes: \(x = 3n\pi\), where \(n\) is an integer.

Step 1: Understand the Cosecant Function

The function given, \(y = \csc \left( \frac{1}{3} x \right)\), is the cosecant function. It is the reciprocal of the sine function. The basic form is \(\csc(x) = \frac{1}{\sin(x)}\). In this form, we need to determine the period of the function.

Step 2: Determine the Period of the Function

The standard period of the cosecant function \(\csc(x)\) is \(2\pi\). However, due to the transformation \(\csc\left(\frac{1}{3} x\right)\), we apply the formula \( \frac{2\pi}{b} \), where \(b\) is the coefficient of \(x\). In this case, \(b = \frac{1}{3}\). Therefore, the period is \[\text{Period} = \frac{2\pi}{\frac{1}{3}} = 6\pi.\]

Step 3: Identify the Asymptotes

The vertical asymptotes of \(y = \csc\left(\frac{1}{3} x\right)\) occur where \(\sin\left(\frac{1}{3} x\right) = 0\). These occur at \(\frac{1}{3} x = n\pi\), where \(n\) is an integer. Solving for \(x\), we get \(x = 3n\pi\). So, there are vertical asymptotes at every \(3n\pi\).

Step 4: Sketch the Graph

Draw a set of coordinate axes. Since the period is \(6\pi\), draw one period from \(0\) to \(6\pi\). Mark the asymptotes at \(x = 0, 3\pi, 6\pi, \dots\). Between these asymptotes, the graph of the cosecant function will exhibit two halves of its characteristic 'U' shaped curves, opening upwards and downwards, touching the points midway between each pair of asymptotes.

Key Concepts

Understanding the Cosecant Function

The cosecant function, denoted as \( y = \csc(x) \), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the sine function, meaning \( \csc(x) = \frac{1}{\sin(x)} \). This function is unique because it is undefined wherever the sine function is zero, leading to the creation of vertical asymptotes at those points.
The graph of the cosecant function features a repeating pattern of curves, known for their distinctive 'U' shape, either opening upwards or downwards. These curves do not cross the x-axis because the function is undefined at zero, which is reflected by the presence of asymptotes.

• **Reciprocal of Sine**: The cosecant function highlights the inverse relationship with the sine.
• **U-shaped Graph**: The graph does not touch the x-axis due to asymptotes corresponding to the zeroes of sine.
• **Periodicity**: Similar to sine and other trigonometric functions, the cosecant function repeats its pattern over a specific interval known as the period.

Period of Trigonometric Functions

Each trigonometric function has a characteristic period that represents the length of one complete cycle of the function's graph. For the basic cosecant function \( \csc(x) \), the standard period is \( 2\pi \). This means the graph pattern repeats every \( 2\pi \) units along the x-axis.
When the function is transformed, such as in \( \csc\left( \frac{1}{3} x \right) \), the period changes. To find the new period, we use the formula \( \frac{2\pi}{b} \), where \( b \) is the coefficient of \( x \). For the given function, \( b = \frac{1}{3} \), so the period becomes \[ \frac{2\pi}{\frac{1}{3}} = 6\pi. \]
Adjusting our understanding of a function's period according to its transformation is crucial for graphing and analyzing trigonometric functions correctly.

• **Basic Period**: Cosecant by default repeats every \( 2\pi \).
• **Transformation Effect**: Scaling or shifting the function alters its period.
• **Formula Use**: Use \( \frac{2\pi}{b} \) to calculate the modified period.

Vertical Asymptotes

Vertical asymptotes are imaginary lines where a function approaches but never touches or crosses. In the context of the cosecant function, asymptotes occur at the points where the sine function, its reciprocal, equals zero.
For \( y = \csc\left( \frac{1}{3} x \right) \), the vertical asymptotes are found where \( \sin\left( \frac{1}{3} x \right) = 0 \), which occurs at integer multiples of \( \pi \). Solving \( \frac{1}{3} x = n\pi \) for \( x \), where \( n \) is an integer, we find \( x = 3n\pi \).
This leads to series of vertical lines at \( x = 0, 3\pi, 6\pi, \ldots \), showing each point where the graph sharply moves upwards or downwards towards infinity.

• **Points of Discontinuity**: Resulting from sine equaling zero.
• **Reflecting Infinite Behavior**: Cosecant's values become very large around these lines.
• **Importance in Graphing**: Asymptotes guide the placement of the curve segments of the graph.