Problem 51 - Precalculus: Functions and Graphs

Question

Graph the Lissajous figure in the viewing rectangle \([-1,1]\) by \([-1,1]\) for the specified range of \(t\). $$x(t)=\sin (6 \pi t), \quad y(t)=\cos (5 \pi t) ; \quad 0 \leq t \leq 2$$

Step-by-Step Solution

Answer

To graph the figure, plot the points of \((x(t), y(t))\) for \(0 \leq t \leq 2\) with small increments in \(t\) to capture the complex loop pattern.

Step 1: Understanding the Lissajous Figure

Lissajous figures are parametric plots of two oscillatory functions. In this exercise, we have two functions dependent on time parameter \(t\). Our goal is to plot \(x(t) = \sin(6\pi t)\) and \(y(t) = \cos(5\pi t)\) for \(t\) ranging from 0 to 2.

Step 2: Identify Waveforms

The provided equations \(x(t) = \sin(6\pi t)\) and \(y(t) = \cos(5\pi t)\) suggest that the x-coordinate is oscillating with a frequency determined by \(6\pi\), and the y-coordinate by \(5\pi\). Each function represents a sinusoidal wave moving independently.

Step 3: Determine the Plotting Range

The values of \(x(t)\) and \(y(t)\) inherently lie between -1 and 1 due to the nature of sine and cosine functions. Hence, the viewing rectangle ([-1, 1]) by ([-1, 1]) will sufficiently display the variations in the plot.

Step 4: Plotting the Lissajous Figure

With the time parameter running from \(t = 0\) to \(t = 2\), calculate sequential pairs of \((x(t), y(t))\) and plot these on a Cartesian coordinate system. Adjust the calculation granularity to ensure a smooth curve is displayed within the given bounds.

Step 5: Conclude the Pattern

Given the discrepancy in frequency factors (6 for \(x\) and 5 for \(y\)), the resultant graph needs to be carefully traced to recognize repetitions and overall symmetry. The difference in frequency causes the distinct looping characteristic of Lissajous figures.

Key Concepts

Parametric Equations

Parametric equations allow us to represent curves and figures by finding a unique pair of equations that describe points' positions as a function of a parameter, commonly time \(t\). In the Lissajous figure example, the parameter \(t\) ranges from 0 to 2. Instead of using a traditional \(y = f(x)\) format, parametric equations enable plotting using two equations: \(x(t) = \sin(6\pi t)\) and \(y(t) = \cos(5\pi t)\).

• The parameter \(t\) determines both \(x\) and \(y\), linking them together in a dynamic plot.
• Each value of \(t\) results in a unique point \((x(t), y(t))\).
• This method is particularly useful for plotting complex curves like Lissajous figures.

Are parametric equations new to you? Think of them as painting a picture point by point based on time, which can create intricate designs beyond linear or polynomial graphs.

Sinusoidal Functions

At the heart of Lissajous figures are sinusoidal functions, which are smooth, repetitive oscillations like sine and cosine waves. In the given exercise, the two key sinusoidal functions are:- \(x(t) = \sin(6\pi t)\)- \(y(t) = \cos(5\pi t)\)These functions are periodic, meaning they repeat their values in regular intervals. Sinusoidal waves provide foundational roles in many areas, such as acoustics, electronics, and even biology.

• Sine Function: For \(x(t)\), the standard sine wave is stretched by the frequency \(6\pi\), creating a faster oscillation.
• Cosine Function: The y-coordinate uses the cosine function, shifted by frequency \(5\pi\), different from the sine's behavior but equally periodic.

Sinusoidal functions also exhibit certain properties:- **Amplitude:** The peak value reached by the wave; in this case, always 1 or -1.- **Frequency:** How often the wave repeats over a unit interval, influencing the plot appearance.Understanding these properties helps us predict and interpret how wave interactions form complex figures like the Lissajous curves.

Frequency Discrepancy

A vital aspect of Lissajous figures is the frequency discrepancy between the two sinusoidal functions. This discrepancy shapes the figure's looping patterns and symmetry.- For \(x(t) = \sin(6\pi t)\), the frequency is 6 cycles per unit interval.- For \(y(t) = \cos(5\pi t)\), the frequency is 5 cycles per unit interval.

• The difference in frequencies, \(6\) vs. \(5\), causes these functions to interact in unique ways, creating intricate loops.
• The ratio of frequencies (6:5 in this case) brings about symmetry and repetition in the plot.
• Lissajous figures often reflect these frequency ratios, appearing as closed loops.

This frequency difference can make Lissajous plots both beautiful and complex, requiring a careful choice of viewing parameters and ranges to visually capture these interactions.

Cartesian Coordinates

The Cartesian coordinate system forms the fundamental basis for plotting Lissajous figures. In this system, points are denoted by pairs of numbers representing their position relative to two perpendicular axes (x and y).To plot:- Identify each pair \((x(t), y(t))\) as a point in the xy-plane.- Locate where \(x(t) = \sin(6\pi t)\) and \(y(t) = \cos(5\pi t)\) intersect a grid bounded by the limits \([-1,1]\) for both axes.The rectangular bounds are:

• x-axis: Each value of \(x(t)\) corresponds to horizontal movement.
• y-axis: Each value of \(y(t)\) corresponds to vertical movement.

This coordinate system is straightforward yet powerful, allowing us to map complex patterns from mathematical formulas to a visual graph comfortably. Understanding this scheme enables better interpretation of Lissajous figures and other advanced graphics.