Problem 53 - A Graphical Approach to Precalculus with Limits

Question

The screen shown to the right is an example of a Lissajous figure. Lissajous figures occur in electronics and may be used to find the frequency of an unknown voltage. Graph each Lissajous figure for \(0 \leq t \leq 6.5\) in the window \([-6.6,6.6]\) by \([-4.1,4.1]\). (GRAPH CANNOT COPY) $$x=3 \sin 4 t, y=3 \cos 3 t$$

Step-by-Step Solution

Answer

Graph the Lissajous figure in the given window using parametric plotting.

Step 1: Understanding the Functions

The given parametric equations are \(x = 3 \sin 4t\) and \(y = 3 \cos 3t\). These represent the movement along the x-axis and y-axis, respectively, as a function of time \(t\). The coefficients 4 and 3 in the sine and cosine functions indicate the frequencies of these oscillations.

Step 2: Identifying the Graphing Window

We are instructed to graph these functions for \(0 \leq t \leq 6.5\) and within the viewing window of \([-6.6,6.6]\) by \([-4.1,4.1]\). This means we will vary \(t\) in this interval to plot the points \((x, y)\) and observe how they form the Lissajous figure within the given range.

Step 3: Analyzing the Lissajous Figure Parameters

The frequencies of the sine and cosine functions are different (4 and 3), which is key to forming a Lissajous figure. Since \(3:4\) is not an integer ratio, the figure will close after several oscillations and will not be a simple ellipse or line.

Step 4: Plotting the Lissajous Figure

To manually determine some coordinate points, calculate values for \(t\) at various intervals like 0, \(\pi/2\), \(\pi\), ..., up to 6.5. Substitute each \(t\) into the equations for \(x\) and \(y\). This will give discrete points that can be plotted. For example, when \(t=0\), \(x=0\) and \(y=3\) which is one point on the graph. Continue sampling to capture the curve.

Step 5: Interpreting the Graph

Once you've plotted enough points from step 4, connect them smoothly considering the periodic nature of sine and cosine. Lissajous figures are characterized by their loops and crossings, resulting from the different frequencies affecting the path's periodicity and symmetry within the window.

Key Concepts

Parametric Equations

Parametric equations are a fascinating tool in mathematics, providing a way to describe complex curves in a plane more conveniently. Rather than expressing one variable, say \(y\), directly in terms of another, \(x\), parametric equations use a third parameter, often time \(t\), to define both \(x = f(t)\) and \(y = g(t)\). This approach allows for more flexibility in graphing curves, especially those not easily described with a single equation.
For example, in our Lissajous figure, we have the equations \(x = 3 \sin 4t\) and \(y = 3 \cos 3t\). Here:

• The parameter \(t\) serves as the time variable, helping us track where the particle is on the curve at any given moment.
• The functions \(3 \sin 4t\) and \(3 \cos 3t\) define the path on the x-axis and y-axis, respectively.
• Each value of \(t\) gives a unique pair \((x, y)\), representing a single point in the plane.

Parametric representations are quite powerful, particularly in animation and physics, where they allow for smooth and continuous plotting of paths.

Oscillations

Oscillations describe the repetitive variation, typically in time, of some measure about a central value. In mathematics and physics, they often appear in waveforms and frequency analysis. For Lissajous figures:

• The functions \(\sin\) and \(\cos\) inherently carry an oscillatory nature, as they repeat values in a periodic fashion.
• In our equations, the coefficients 4 and 3 govern the frequency of these oscillations. The term 'frequency' mentions how many cycles fit into a fixed interval.

When different frequencies are applied to the x and y movements, like in our case, interesting patterns emerge, such as loops and crossings.
These patterns form because the ratio of frequencies, \(3:4\) for \(y\) and \(x\), dictates the complexity of the shape. A mismatch in frequency makes new sections overlap non-coincidentally, pivotal for creating Lissajous curves. It's this interplay or synchrony of oscillations that results in the beautiful symmetry and intricate designs characteristic of Lissajous figures.

Graphing Window

The graphing window is an essential part of plotting parametric equations, as it sets the limits within which we can view the curve. It essentially functions as the boundary box or frame for our graph.

For this exercise, the given graphing window is

• Horizontal range: \([-6.6, 6.6]\)
• Vertical range: \([-4.1, 4.1]\)

This ensures that the Lissajous figure fits well into the screen space when plotting for \(t\) in the interval \([0, 6.5]\). A well-chosen window size allows the entire curve to be displayed clearly, maintaining aspect ratios enough not to distort perceived symmetries and characteristics.
The window also factors in how dynamic changes in the parametric equation plot happen. Adjusting the graphing window can help if any part of the curve isn’t visible or if details need emphasis. Understanding how to set an appropriate window is a fundamental skill in graphing that assists in generating accurate visual representations of equations.