Using a CBR or Motion Detector to Record a Secant/Cosecant Graph

Students often miss the connection between sine/cosecant or cosine/secant graphs. Quite by accident, we were working on an upcoming lab using an accelerometer and a motion graph. I was using a spring to provide harmonic motion and the students said "I bet I know what the graph looks like." So, I moved the apparatus to the overhead calculator and recorded the motion. Sure enough, they were right. It was a sinusoid. Just so happens, the day before, we were looking at the graph of secant and cosecant functions. On a whim I said, "How do I get a graph of secant or cosecant from this graph? The students thought for a minute and one said "Take its reciprocal."

Now that you know the story of this activity, I will develop it fully as it has a few quirks along the way. It also lead to another challenging question. As we started graphing tangent, a student asked the follwing question. "How would you explain the graph of the tangent function over time?" Since you need time to ponder that one, let me finish the first explanation and I'll answer this one later on.

Materials

1 - Spring with a mass attached to provide harmonic motion.
1 - Motion Detector and CBL or a Ranger (CBR)
1 - TI-8? calculator

Data Collection

This activity requires the program GENDIST which is part of the generic.83g package. Instructions for downloading and installing these files are also available. The file LINKCK also needs to be installed on the calculator for GENDIST to run properly. The program measures a specified distance from the motion detector as the origin (zero distance). All distance is referenced to this point with motion away from the detector being positive. In this case, the spring with a mass hanging on it in a neutral position is the refence point. When the spring is pulled down and released, its harmonic motion will be referenced from the neutral position.

In the image above, a spring has a mass attached. The motion detector is positioned so that when the spring is pulled down and released, the distance to the detector is more than 0.5 meters below the mass as shown below.

Allow the mass on the spring to reach its neutral position. Run the program GENDIST. Press ENTER to record the distance to the neutral mass on the spring. Enter the number of points to be collected (99) and the time (0.3 seconds). The values can be adjusted based on the speed of the motion of the spring. Pull the mass straight down and release it. Press ENTER on the calculator to record the motion of the mass on the end of the spring. A graph similar to the one below should be displayed. The time (sec) is stored in L1 and graphed on the x-axis. Distance (m) from the neutral point is stored in L2 and graphed on the y-axis.

Note the sinusoidal or cosinusoidal nature of the graph. Since csc(x) = 1 / sin(x) and the data in L2 is sinusoidal, let's see what happens if we graph the reciprocal of L2 (1/L2). Press Clear to return to the Home screen. Once at the home screen, enter 1/L2 STO L3 as shown in the image below.

Pressing ENTER will store the reciprocal of the data in L2 into L3. Next, set change the stat plot to show the graph of L1 vs. L3. Press Stat Plot (2nd Y=) and set up the stat plot as shown below.

Because the numbers will be very large with the reciprocals, you will not want to use ZoomStat. Instead, press the Window button and change the Y-values of the graphing window. (Time on the x-axis is unchanged). Set the Ymin and Ymax to perhaps -100 and 100 as shown below.

Pressing the graph button, displays a graph similar to the following.

At this point, students wanted to know where the vertical lines were that appear when the graph of y=sec(x) or y=1/cos(x) is shown a function graph in connected mode. If the function graph is changed to dot mode, it will appear as above. Frequently, students call these lines asymptotes. They do indeed approximate asymptotes, but are not a true asymptote. The lines connect the bottom dot with the dot at the top of the screen when in connected mode. A stat plot can be set up in connect line form by changing the stat plot as shown below.

The result of changing to an xyline graph is shown below.

A Few Quirks

Normally, the graphs of y = sin(x) and y = csc(x) graphed in the same window appear as:

If you turn on a second stat plot showing the sinusoidal data in L1 and L2, one will note that this does not happen in this case. In fact, due to the scale factor, the sinusoidal data may not even show up on the screen.

The explanation for this lies in taking the reciprocal of the data in L2 and storing it in L3. In reality, the data stored in L2 is a sinusoid with the equation L2= k + a sin(d(x-h)). Since the data was recorded around the neutral position of the spring, the midline of the graph is zero. Hence, the value of k is equal to 0. The model becomes L2 = a sin(d(x-h)). By taking the reciprocal, the following mathematical steps can be made:

This means that if we find the value of "a" for the sinusoidal data and take the reciprocal as noted above, the vertical stretch for the cosecant graph would be the reciprocal of "a" or "1/a".

Finding the sinusoidal regression for the sinusoidal graph orignally collected, one would find equation y = 0.05 sin(5.91(x - 0.71)). This means that the graph of the cosecant function plotted by using 1/L2 would be 1/ ( 0.05 sin(5.91(x - 0.71)) ) or y = 20 csc(5.91(x-0.71)).

Explaining the Tangent Graph

Earlier, I mentioned that a student asked how we could get the graph of a tangent function to fit with the "real world." After thinking a brief time, I recognized that tangent is the change in y over the change in x or slope. Hence, if the angle is on the x-axis and the y-value is the tangent function, the graph would show the change in the slop of a line as the angle changes. Using a meter stick, I held it up vertically and noted that the slope was undefined. By moving the top of the meter stick slowly counter-clockwise, one will note that the slope is very steep and negative. As movement continues, the slope approaches zero and is zero when the stick is horizontal or has an angle of zero. Progressing passed horizontal, the slope is positive and increasing until the meter stick becomes vertical again and the slope is undefined. Note the paired diagrams below.

Starting with a vertical line and the tip pointing straight down towards -90 degrees or -pi/2 radians, the slope is undefined. The resulting no points on the graph as shown at the right.

Moving from -pi/2 (-90 degrees) to -pi/3 (-60 degrees), the slope of the line is negative with a very large absolute value since the line has a greater slant. The beginning of one period of the tangent graph is shown at the right.

Moving towards -pi/4 (45 degrees), the slant of the line continues to decrease and the slope approaches -1. Hence, the last point on the graph is (-pi/4, -1).

Continuing to turn the line counter-clockwise, the angle aproaches -pi/6 (30 degrees). The slant of the line continues to decrease and the graph approaches zero.

When the line reaches horizontal (0 radians and 0 degrees), the slope is 0 and the tangent is 0. Hence, the last point on the graph is (0, 0).

As the line continues past horizontal and approaches pi/6 (30 degrees), the slant of the line increases and the slope is positive.

At pi/4 (45 degrees), the slope of the line is 1 and the last point on the tangent graph at the right is (pi/4, 1).

Moving past pi/4 towards pi/3 (60 degrees), the slant of the line continues to increase and begins to increase at a faster rate.

Continuing towards pi/2 (90 degrees), the slope of the line continues to increase rapidly approaching infinity. Once the line reaches pi/2, the slope is undefined and the cycle begins once again. Hence, the period of tangent is pi (from -pi/2 to pi/2).

If you are an educator or interested in this activity and wish to have more information, send an e-mail to jrynear@lps.org or jwelker@lps.org.
 
© 2004-2008, Jerel L. Welker
Page Updated: August 19, 2008