Problem 54 - 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) $$x=4 \sin 4 t, y=3 \sin 5 t$$

Step-by-Step Solution

Answer

Plot the parametric equations within the specified ranges to get a Lissajous figure with a complex loop structure.

Step 1: Understand the Equations

We are given the parametric equations \( x = 4 \sin(4t) \) and \( y = 3 \sin(5t) \). These equations represent a Lissajous figure, which is a graph generated using trigonometric functions to describe the paths in terms of a parameter \( t \).

Step 2: Set the Range for Parameter t

The problem specifies that the graph should be plotted for \( 0 \leq t \leq 6.5 \). This means we will evaluate and plot the equations for \( t \) values within this range.

Step 3: Define the Graphing Window

The graph should be displayed in the window \([-6.6, 6.6]\) along the x-axis and \([-4.1, 4.1]\) along the y-axis. This helps visualize the Lissajous figure across appropriate scales for these axis limits.

Step 4: Plot the Function

Using a graphing tool or calculator, plot the function by substituting various values of \( t \) from the specified range into the equations. Evaluate \( x \) and \( y \) for these \( t \)-values, and plot the corresponding points \((x, y)\).

Step 5: Analyze the Lissajous Figure

Once plotted, observe the resulting graph. It should exhibit a complex looping structure, characteristic of Lissajous figures, influenced by the ratio of frequencies in the sine functions (4 and 5 in this case).

Key Concepts

Parametric Equations

Parametric equations are a way to define a set of related quantities as functions of a common variable, often denoted as \( t \), which is the parameter. Instead of expressing \( y \) directly in terms of \( x \) as in traditional Cartesian coordinates, both \( x \) and \( y \) are defined separately in terms of \( t \). This approach provides an effective way to describe curves and complex paths that cannot be easily represented by a single function.

In the context of Lissajous figures, the equations \( x=4\sin(4t) \) and \( y=3\sin(5t) \) specify the horizontal and vertical positions based on the varying parameter \( t \). By altering \( t \) within the given range \( 0 \leq t \leq 6.5 \), we can trace different points throughout the plane, which collectively form the complete figure. The use of parametric equations simplifies the process of creating and analyzing such intricate patterns.

Trigonometric Functions

Trigonometric functions like sine and cosine are fundamental in describing oscillatory and periodic phenomena. In our example, the focus is on the sine function—a smooth, continuous wave that repeats over regular intervals.

For the Lissajous figure, the equations \( x=4\sin(4t) \) and \( y=3\sin(5t) \) involve sine functions. The term inside the sine, such as \( 4t \) or \( 5t \), determines the frequency—a measure of how often the cycle repeats as \( t \) increases. The amplitudes (4 for \( x \) and 3 for \( y \)) stretch the curves, amplifying their oscillations along their respective axes.

This use of trigonometric functions introduces periodicity, setting the stage for creating complex paths that loop and intertwine uniquely, as demonstrated in Lissajous figures.

Graphing Tools

Graphing tools are essential for visualizing mathematical equations, especially when dealing with parametric forms and intricate figures like Lissajous patterns. These tools range from graphing calculators to sophisticated software applications capable of plotting multidimensional graphs with finesse.

To plot the given parametric equations \( x=4\sin(4t) \) and \( y=3\sin(5t) \), a graphing tool substitutes various values of \( t \) within the specified range \( 0 \leq t \leq 6.5 \) into these equations, resulting in numerous coordinate points \((x, y)\). These points are then connected to form the complete Lissajous figure.

Effective graphing tools enable students and researchers to experiment with different settings, such as adjusting axes or scaling, helping to explore how changes in parameters alter the shape and form of the resulting graph.

Frequency Analysis

Frequency analysis in the context of Lissajous figures refers to the study and interpretation of the frequency components of the parametric equations. The equations \( x=4\sin(4t) \) and \( y=3\sin(5t) \) exhibit frequencies of 4 and 5, respectively, which are crucial for determining the figure's shape.

The frequency ratio (in this case, \( \frac{4}{5} \)) influences the number of loops or lobes observed in the graph. A rational ratio between these frequencies leads to a closed, repeating pattern—a hallmark of Lissajous figures used in various applications such as signal analysis.

Understanding this frequency interaction helps in predicting and analyzing the visual complexity of Lissajous figures, making frequency analysis a valuable aspect of studying these intriguing mathematical phenomena.