Problem 32 - Algebra 2

Question

Graph each function in the interval from 0 to 2\(\pi\) $$ y=\csc 2 \theta $$

Step-by-Step Solution

Answer

The graph of the function \(y=\csc(2\theta)\) will have two full periods between \(0\) and \(2\pi\). It consists of shape like 'U' or inverted 'U' connecting maximum and minimum points of the sine wave to the asymptotes, with vertical asymptotes at multiples of \(pi/2\).

Step 1: Express Cosecant in Terms of Sine

We know that the function cosecant is the reciprocal of the sine function i.e., \(\csc(\theta) = \frac{1}{\sin(\theta)}\). Therefore, we can write the given function as \(y = \frac{1}{\sin(2\theta)}\).

Step 2: Sketch the Graph of \(\sin(2\theta)\)

Now, consider the parent function \(y= \sin(\theta)\). We know that this graph completes one full period from \(0\) to \(2\pi\). However, in the case of \(y=\sin(2\theta)\), the graph will complete two full periods in the same range, due to the coefficient of \(2\) attached with \(\theta\). Because the period of the function is halved, the standard \(y= \sin(\theta)\) graph is compressed horizontally by a factor of \(2\).

Step 3: Draw the Cosecant Function

We are graphing \(y= \csc(2\theta)\) i.e., \(y = \frac{1}{\sin(2\theta)}\). Cosecant function is undefined wherever sine function is \(0\) which occurs at multiples of \(\pi\). Hence, we cannot plot \(y= \csc(2\theta)\) at multiples of \(\pi/2\), making these points vertical asymptotes. We use the rest of the locations of the sine function to draw the cosecant curve connecting maximum and minimum points of the sine wave to the asymptotes.

Key Concepts

Cosecant

The cosecant function, often represented as \( \csc(\theta) \, \) is a less commonly used trigonometric function but holds significant importance due to its relationship with the sine function. Cosecant is essentially the reciprocal of the sine function:

• \( \csc(\theta) = \frac{1}{\sin(\theta)} \, \)

This means that whenever sine is zero, cosecant becomes undefined, which is an essential feature when you are graphing trigonometric functions like in our example \( y = \csc(2\theta) \. \) The graph of the cosecant function will show vertical asymptotes at every point where the sine function equals zero, resulting in a curve with distinct up-and-down "arms" between these asymptotes. Understanding this reciprocal relationship helps in predicting the cosecant graph's behavior when comparing it directly to the sine wave its derived from.

Sine Function

The sine function is one of the fundamental trigonometric functions, symbolized by \( y = \sin(\theta) \. \) Its graph is a smooth, repetitive wave that completes a full cycle or period over the interval from \( 0 \) to \( 2\pi \, \) known as the sine curve:

• Peaks at \( y = 1 \) and valleys at \( y = -1 \,. \)
• Crosses zero at multiples of \( \pi \), specifically \( 0, \pi, \, \) and \( 2\pi \. \)
• The sine function is periodic, repeating the same cycle again and again.

To transform this simple sine wave to a more complex function like \( y = \sin(2\theta) \, \) we strategically manipulate different aspects such as amplitude, period, and phase to achieve desired outcomes such as compression or stretching in the graph.

Graphing Trigonometric Functions

Graphing trigonometric functions, like the ones involving cosine or sine, involves understanding some specific traits that dictate the overall shape and dimensions of the graph. To successfully plot a function such as \( y = \sin(2\theta) \, \) or its reciprocal \( y = \csc(2\theta) \, \) it's important to know that:

• The presence of a coefficient before \( \theta \, \) like \( 2 \, \) will affect the graph's period. In other words, \( y = \sin(2\theta) \, \) completes its cycle faster as the coefficient is greater than 1.
• For \( y = \csc(2\theta) \, \) vertical asymptotes occur whenever the sine function equals zero. Graph these asymptotes at every multiple of \( \pi/2 \, \) to indicate undefined regions of the graph.
• The part of the sine curve that lies above the x-axis will correspond to upward arms of the cosecant graph, while the section below the x-axis forms the downward arms.

Each plotted position of either \( y = \sin(2\theta) \, \) or its related cosecant function maps these characteristics to accurately form their graphical representation.

Periodicity

Periodicity in trigonometric functions refers to the repeating nature of their graphs. Each function like sine or cosecant will repeat its pattern over a fixed interval known as the period. In a function like \( y = \sin(2\theta) \, \) the "\( 2 \, \)" before \( \theta \, \) impacts this periodic nature:

• For standard sine, one full cycle is \( 2\pi \, \). However, in \( y = \sin(2\theta) \, \), the period is truncated to \( \pi \. \)
• The cycle's repetition is quicker, completing twice within the usual range of \( 2\pi \, \).
• For its reciprocal, \( y = \csc(2\theta) \, \), it maintains the same period characteristics as the transformed sine function.

Understanding periodicity is crucial to predicting where the cycles start and end, alongside knowing how these consistent cycles behave over particular intervals. This pattern repetition is fundamental to graphing these functions effectively within specified domains.