Problem 34 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The period of \( y=\csc \frac{1}{2}x \) is \( 4\pi \).

Step 1: Understand the Basic Cosecant Function

The base cosecant function, a reciprocal of the sine function, is given as \( y = \csc x \). Typically, its period is \( 2\pi \) because it inherits the period of the sine function, \( y = \sin x \). Cosecant functions have undefined values where the sine function is zero.

Step 2: Identify the Modified Function

The given function is \( y = \csc \frac{1}{2} x \). This modifies the argument of the cosecant function. To find the new period, consider this modification as affecting the frequency of the base sine function.

Step 3: Determine the New Period

To find the period of \( y = \csc \frac{1}{2} x \), calculate how the modification affects the period of its reciprocal function, sine. The period of \( \sin bx \) is given by \( \frac{2\pi}{b} \). Here, \( b = \frac{1}{2} \), so the period becomes \( \frac{2\pi}{\frac{1}{2}} = 4\pi \). Thus, the period of the cosecant function is extended to \( 4\pi \).

Step 4: Sketch the Graph

With the period determined as \( 4\pi \), plot the graph of \( y = \csc \frac{1}{2} x \). Identify key points: the function is undefined where the sine function, \( \sin \frac{1}{2} x \), is zero such as \( x = 0, 4\pi, 8\pi, \dots \). Draw vertical asymptotes at each of these points. The function has a repeating pattern that spans one period between \( x = 0 \) and \( 4\pi \). Between these asymptotes, the graph will have branches approaching the asymptotes from above and below from \( \pi \) to \( 3\pi \), reflecting the reciprocal nature of the sine wave.

Key Concepts

Cosecant Function

The cosecant function, denoted as \( y = \csc x \), is a trigonometric function that is the reciprocal of the sine function. This means that for any angle \( x \), the cosecant of \( x \) is defined as \( \csc x = \frac{1}{\sin x} \). As a result, the cosecant function is undefined wherever the sine function equals zero.
Because the sine function has zeros at \( x = k\pi \) where \( k \) is an integer, you will find vertical asymptotes in the graph of the cosecant at these points. This is a crucial concept when plotting the graph, as these asymptotes guide the behavior of the cosecant graph.

The graph of the cosecant function appears as a series of repeating U-shaped and inverted-U-shaped curves between the asymptotes. Between each pair of vertical asymptotes, the graph curves away and approaches the asymptotes without ever touching them.

Period of a Function

The period of a function refers to the length of the interval over which the function's values repeat. Trigonometric functions, like the sine, cosine, and their reciprocals, inherently possess periodic behavior.
For the sine function, \( y = \sin x \), the period is \( 2\pi \). That means after every interval of \( 2\pi \), the sine function's output values repeat. Consequently, this also influences the cosecant function \( y = \csc x \) to initially have a period of \( 2\pi \).

• Modifying the function via transformations, such as amplifications and changes in frequency (e.g. \( y = \csc(\frac{1}{2}x) \)), directly affects its period.
• For a function of the form \( \sin(bx) \), the period becomes \( \frac{2\pi}{|b|} \). When applied to the cosecant function, this determines new periods based on its relationship to the sine function.

Graphing Trigonometric Functions

Graphing trigonometric functions involves series of steps to capture their repeating patterns, amplitude, and phase shifts. Taking into account vertical asymptotes, intervals between them, and sections where the function is defined and approaches the asymptotes are essential.

For \( y = \csc \frac{1}{2} x \):

• The function \( \sin \frac{1}{2} x \) has zeros at multiples of \( 4\pi \). Hence, \( y = \csc \frac{1}{2} x \) is undefined at these points, forming vertical asymptotes.
• Graph this by plotting asymmetric branches of the curves between the asymptotes.
• The key points to remember are the vertical stretches and the behavior approaching the asymptotes for each period of the function.

Graphing such functions not only rely on theoretical calculations but also visual guidelines established by plotting key points.

Sine Function

The sine function \( y = \sin x \) serves as a fundamental building block in trigonometry. It exhibits periodic behavior with a well-defined wave-like pattern.
Key characteristics include:

• The fundamental period of \( 2\pi \), where its output values cycle and repeat.
• Zeros at regular intervals of \( x = k\pi \), useful in determining undefined points for the cosecant function.
• Maximum and minimum value, \( 1 \) and \(-1 \), which dictate the peaks of the wave.

Understanding the sine function is crucial when dealing with its reciprocal functions, such as cosecant. Transformations applied to \( \sin x \) impact the associated cosecant function and influence how you explore the graph and transformations like period modifications.