Problem 109 - Algebra and Trigonometry

Question

Graph \(y=2 \sin \frac{1}{2} x .\) Then use the graph to obtain the graph of \(y=2 \csc \frac{1}{2} x .\) (Section \(5.6,\) Example 4 )

Step-by-Step Solution

Answer

First sketch the graph of \(y=2 \sin \frac{1}{2} x\), which will oscillate between -2 and 2, and will have a period of \(4\pi\). Then sketch the graph of \(y=2 \csc \frac{1}{2} x\), which will be undefined wherever the sine function is zero, and will have 'U' or 'n' shapes at the peak and trough points of the sine function. The intersections of these two graphs occur at the maximum and minimum points of the sine function, which help in relation to reciprocal functions.

Step 1: Graph \(y=2 \sin \frac{1}{2} x\)

Start by graphing the sine function. The function \(y=2 \sin \frac{1}{2} x\) has an amplitude of 2 (the '2' in front of the sine) and a period of \(4\pi\), because the period of sine function \(y=\sin Bx\) is given by \(\frac{2\pi}{B}\), here \(B=\frac{1}{2}\). So, it would oscillate between -2 and 2 in its y-value and complete a full oscillation every \(4\pi\). Plot accordingly.

Step 2: Identify critical points

Identify the points where \(y=2 \sin \frac{1}{2} x = 0\), because those will be points of discontinuity for the cosecant function. Also, note the maximum and minimum points on the sine wave. These are the peaks and troughs of the wave, which will be important for graphing the cosecant function.

Step 3: Draw the cosecant function

Draw the cosecant function \(y=2 \csc \frac{1}{2} x\) as the reciprocal of the sine function. For every x-value where the sine function is equal to zero, draw a vertical asymptote since cosecant function will be undefined there. At the peak and trough of each wave on the sine graph, draw a 'U' or 'n' shape which corresponds to the behavior of the cosecant function. These shapes open downwards when the sine curve is at a peak (maximum), and upwards when the sine curve is at a trough (minimum).

Step 4: Finalize and analyze the graph

Complete drawing all the relevant 'U' or 'n' shapes by following the behavior of the sine function. Make sure all asymptotes are linear and vertical where the sine function touches the x-axis. These two graphs now aid in understanding the relationship between a function and its reciprocal function. Moreover, intersections of the sine and cosecant graphs only occur at maximum and minimum points of the sine function - points of sine's peak or trough.

Key Concepts

Amplitude

Amplitude in trigonometric graphs refers to the height of the wave. It’s the distance from the midline of the graph to its peak or trough. For the sine function, the amplitude is given by the coefficient in front of the sine.
In the function \(y = 2 \sin \frac{1}{2}x\), the amplitude is 2. This means our sine wave will reach a maximum value of 2 and a minimum of -2.

• This tells us how "tall" or "short" the waves will look.
• Amplitudes help set the boundaries for reciprocal functions, like cosecant, since they depend on the sine wave's peaks and troughs.

Understanding amplitude helps in predicting the overall shape and behavior of trigonometric graphs.

Period

The period of a trigonometric function is the horizontal length required for the function to complete one full cycle. For sine and cosine, it’s calculated using the formula \(\frac{2\pi}{B}\), where \(B\) is the coefficient of \(x\) inside the function.
For \(y = 2 \sin \frac{1}{2}x\), we find the period by:

• Identifying \(B = \frac{1}{2}\).
• Calculating \(\frac{2\pi}{\frac{1}{2}} = 4\pi\).

This tells us that our sine wave repeats every \(4\pi\) units along the x-axis.
Recognizing periods is crucial because it also dictates the spacing of features like asymptotes in reciprocal functions.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are the flipped version of the basic trig functions. For sine, its reciprocal is cosecant.
The cosecant function \(y = 2 \csc \frac{1}{2}x\) is defined wherever the sine function is non-zero because it’s \(\frac{1}{\text{sine}}\). This introduces new shapes:

• 'U' shapes form where the sine has peaks.
• 'n' shapes appear where the sine has troughs.

Understanding reciprocal functions helps in visualizing how they relate and differ from their original functions.

Vertical Asymptotes

Vertical asymptotes occur in functions where they have respectively undefined areas. For sine's reciprocal, cosecant, vertical asymptotes happen wherever sine equals zero.
In \(y = 2 \csc \frac{1}{2}x\), watch for:

• Points where \(\sin\) equals zero, causing \(\csc\) to be undefined.
• Drawing vertical lines at these x-values to represent the asymptote.

These asymptotes help structure the reciprocal graph and highlight the differences in behavior from trigonometric parent functions.