Problem 46 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The graph of the function \( y=\csc(\theta)-\frac{\pi}{2} \) on the interval from 0 to \( 2 \pi \) has vertical asymptotes at \( \theta = 0 \), \( \pi \), and \( 2 \pi \). The graph has the shape of a standard cosecant function, but shifted down by \( \frac{\pi}{2} \) units.

Step 1: Understand the Basic Shape of the Cosecant Function

The cosecant function, \( \csc(\theta) \), is the reciprocal of the sine function. \(\csc(\theta) = 1/\sin(\theta)\). It has vertical asymptotes where the sine function is zero. Within the given range, sine equals zero at \( \theta=0 \), \( \pi \), and \( 2\pi \). Therefore, the cosecant function will have vertical asymptotes at these values.

Step 2: Understand the Effect of Subtracting \( \frac{\pi}{2} \)

The minus \(\frac{\pi}{2}\) term shifts the graph of the function down by \(\frac{\pi}{2}\) units. This means that the asymptotes will still be at the same values of \( \theta \), but the 'height' of the curves between the asymptotes will be shifted down by \(\frac{\pi}{2}\).

Step 3: Sketch the Graph

Firstly, draw the vertical asymptotes at \( \theta=0 \), \( \pi \), and \( 2\pi \). Then sketch in the basic shape of the cosecant function between these asymptotes, making sure to shift it down by \(\frac{\pi}{2}\). Note that there should be a minimum point halfway between each pair of asymptotes, which corresponds to a maximum point of the sine function.

Key Concepts

Vertical Asymptote

The concept of a vertical asymptote is vital when understanding the graph of the cosecant function. A vertical asymptote occurs where a function's denominator equals zero. Since the cosecant function is the reciprocal of the sine function, \( \csc(\theta) = \frac{1}{\sin(\theta)} \), it has vertical asymptotes where the sine function is zero.
This happens because division by zero is undefined. Within the interval from 0 to \(2\pi\), the sine function equals zero at \(\theta = 0, \pi,\) and \(2\pi\). Consequently, the cosecant function will have vertical asymptotes at these same values of \(\theta\).

• Vertical asymptotes are represented as dashed lines on a graph since they are not actual part of the function.
• The curves of the cosecant function approach but never touch these vertical lines, indicating the points of discontinuity.

As you sketch the graph, these vertical asymptotes will help structure the general layout of the graph.

Shift Transformation

Shift transformations affect how functions are moved or transformed on the coordinate plane. In the exercise provided, the function \(y = \csc\theta - \frac{\pi}{2}\) involves a vertical shift downward by \(\frac{\pi}{2}\) units.
This vertical shift keeps the vertical asymptotes at \(\theta = 0, \pi,\) and \(2\pi\) unchanged. The effect of a vertical shift is straightforward.

• Every point on the original graph of \(\csc\theta\) is moved down by \(\frac{\pi}{2}\) units.
• This impacts the curve's peaks and valleys, not the asymptotes, because asymptotes are related to the zeros of the function which do not shift vertically.

This means the general form of the curve remains intact but appears lower on the y-axis. Understanding how this transformation interacts with the original function is key in graph plotting.

Sine Function

The sine function is a fundamental trigonometric function, with its oscillating wave pattern.
Given that the cosecant function \( \csc(\theta) \) is the reciprocal of the sine, knowing where the sine function is zero is critical. The equation for the sine function is \( \sin(\theta) \). Throughout its cycle from \( 0 \) to \( 2\pi \), it crosses the x-axis at \( \theta = 0, \pi,\) and \(2\pi\).
At these points, the sine value is zero, resulting in the vertical asymptotes for the cosecant function.

• The peaks and valleys of the sine function (its maximum and minimum points) correspond to the minimum and maximum points of the cosecant curve.
• In between these zeros, \(\sin(\theta)\) is either positive or negative, determining whether \(\csc(\theta)\) is above or below the x-axis.

The sine function's shape directly dictates the behavior and key features of the cosecant function.

Reciprocal Trigonometric Function

Reciprocal trigonometric functions are derived by taking the reciprocal of basic trigonometric functions. The cosecant function \(\csc(\theta)\) is specifically the reciprocal of the sine function.
This means \(\csc(\theta) = \frac{1}{\sin(\theta)}\). The relationship between reciprocal functions leads to several key features:

• When \(\sin(\theta)\) is zero, \(\csc(\theta)\) does not exist, due to division by zero, thus causing vertical asymptotes.
• When \(\sin(\theta)\) is at its maximum or minimum, \(\csc(\theta)\) takes on its minimum or maximum values, respectively.

Reciprocal trigonometric functions exhibit a different graph shape compared to their basic counterparts. Understanding these interactions helps you plot and interpret the graph of \(y = \csc(\theta) - \frac{\pi}{2}\). By knowing the behavior of the sine function, predicting the features of the cosecant function becomes manageable.