Problem 52 - 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) $$x=2 \sin 2 t, y=3 \cos 3 t$$

Step-by-Step Solution

Answer

Graph the parametric equations \(x = 2 \sin(2t), y = 3 \cos(3t)\) for \(0 \leq t \leq 6.5\) within the specified window.

Step 1: Understand the Components of the Lissajous Figure

A Lissajous figure is a graph produced by the combined motion of two perpendicular oscillatory motions, represented by trigonometric functions. In this case, the functions are:\[x(t) = 2 \sin(2t)\] and \[y(t) = 3 \cos(3t)\]. These functions describe how the curve is traced in the Cartesian plane as parameter \(t\) changes.

Step 2: Set the Parameter Range

We are given that the parameter \(t\) ranges from \(0\) to \(6.5\). This means you will calculate the x and y values at different points within this range to plot the figure.

Step 3: Plot the X-values

Calculate the x-values for different values of \(t\) using the equation \(x = 2 \sin(2t)\). For example, when \(t = 0\), \(x = 0\); when \(t = \frac{\pi}{2}\), \(x = 2\); when \(t = \pi\), \(x = 0\), and so on, in the range \([0, 6.5]\). Calculate more points to get a smooth curve.

Step 4: Plot the Y-values

Calculate the y-values for the same set of \(t\) values using \(y = 3 \cos(3t)\). For example, when \(t = 0\), \(y = 3\); when \(t = \frac{\pi}{2}\), \(y = 0\); when \(t = \pi\), \(y = -3\), and so on, in the range \([0, 6.5]\). Again, calculate enough points to ensure accuracy.

Step 5: Combine X and Y-values to Plot the Figure

Use the paired \((x, y)\) points calculated from Steps 3 and 4 to plot the figure. For each value of \(t\), use both \(x\) and \(y\) to get a coordinate point. Plot all points on the coordinate plane to form the Lissajous figure.

Step 6: Adjust the Viewing Window

Ensure that the graph is displayed within the specified window of \([-6.6, 6.6]\) for the x-axis and \([-4.1, 4.1]\) for the y-axis. This helps you visualize the entire figure properly within the given limits.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental tools in mathematics, especially when dealing with oscillatory motions, like those seen in Lissajous figures. They involve sine and cosine functions, which are periodic and define waves. Here, the functions are expressed parametrically as:

• \(x(t) = 2 \sin(2t)\) for the horizontal motion
• \(y(t) = 3 \cos(3t)\) for the vertical motion

Each function's amplitude, given by the coefficients 2 and 3 respectively, determines the maximum extent of the curve in the coordinate plane. The frequency of these functions, influenced by the coefficients inside the trigonometric terms, dictates how many oscillations occur within a specified interval. As \(t\) varies, these oscillations create the complex patterns known as Lissajous figures.

Parametric Equations

Parametric equations allow us to express a set of related quantities as functions of an independent parameter, often time, denoted here as \(t\). This approach is particularly useful in describing curves in the coordinate plane. Unlike regular equations that directly relate \(x\) and \(y\), parametric equations separate the motion into components.
For Lissajous figures, we define:

• \(x = 2 \sin(2t)\)
• \(y = 3 \cos(3t)\)

As \(t\) swings through its range from 0 to 6.5, these functions map out points on the curve. This method allows flexible and detailed representation of motion, especially the complex paths found in Lissajous figures.

Coordinate Plane

The coordinate plane is a two-dimensional plane where we can graphically represent equations, including parametric ones. It consists of two perpendicular axes:

• The x-axis (horizontal)
• The y-axis (vertical)

In the context of Lissajous figures, the coordinate plane becomes the stage where the separate motions of \(x(t)\) and \(y(t)\) synchronize to form intricate patterns. Remember that each axis range should be properly adjusted to see the entire design,
which is set here as \([-6.6, 6.6]\) for the x-axis and \([-4.1, 4.1]\) for the y-axis. This ensures that all points formed by the parametric equations are captured on the graph.

Plotting Techniques

Plotting techniques for Lissajous figures involve systematically calculating a series of points using the parametric equations over a given range of \(t\). Start by:

• Choosing values for \(t\) within the range \(0 \leq t \leq 6.5\).
• Calculating corresponding \(x\) and \(y\) values for each \(t\).

Each calculated pair \((x, y)\) represents a point on the graph. The precision of your figure depends on choosing enough points to accurately represent the oscillations. After calculating these pairs, plot them on the coordinate plane.
Ensure the entire figure fits within the specified window. Adjust the viewing window to \([-6.6, 6.6]\) by \([-4.1, 4.1]\) to achieve the correct display. This careful plotting lets the interconnected patterns of Lissajous figures fully emerge.