Problem 52 - Precalculus: Functions and Graphs

Question

Graph the Lissajous figure in the viewing rectangle \([-1,1]\) by \([-1,1]\) for the specified range of \(t\). $$x(t)=\sin (4 t), \quad y(t)=\sin (3 t+\pi / 6) ; \quad 0 \leq t \leq 6.5$$

Step-by-Step Solution

Answer

Plot the parametric equations \(x(t)=\sin(4t)\) and \(y(t)=\sin(3t+\pi/6)\) from \(t = 0\) to \(t = 6.5\) in the given rectangle.

Step 1: Understand the Functions

The given functions for this problem are parametric equations: \(x(t) = \sin(4t)\) and \(y(t) = \sin(3t + \pi/6)\). These equations describe the movements in the x and y coordinates as the parameter \(t\) changes from 0 to 6.5.

Step 2: Determine the Range for t

The range for the parameter \(t\) is given as \(0 \leq t \leq 6.5\). This will be the interval over which you calculate the values of \(x(t)\) and \(y(t)\) to plot the curve.

Step 3: Evaluate x(t) and y(t) at Key Points

Evaluate the functions at key points within the interval, such as starting, ending points, and points where a full period of the sine functions might occur. Since sine has a period of \(2\pi\), evaluate the functions at \(rac{n\pi}{2}\) increments where \(n\) is an integer.

Step 4: Plot the Points in the Viewing Rectangle

Translate the calculated values of \(x(t)\) and \(y(t)\) onto the coordinate system bounded by \([-1, 1]\) for both x and y axes. Because we are using sine functions, values will not exceed these bounds.

Step 5: Connect the Dots Smoothly

After plotting the points, connect them in a smooth curve to form the Lissajous figure. Ensure that the transitions between the points reflect the smooth continuous nature of the sine functions involved.

Step 6: Analyze the Figure

The graph should demonstrate the interaction between the frequencies of \(4\) and \(3\), showing loops and intersections unique to Lissajous figures. Verify if the plotted figure fits within the viewing rectangle \([-1,1]\) by \([-1,1]\).

Key Concepts

Parametric Equations

Parametric equations are a powerful tool in mathematics used to describe a set of related quantities as functions of an independent variable, often denoted as the parameter \( t \). Unlike standard Cartesian equations, which directly relate \( x \) and \( y \) in a single equation, parametric equations define each coordinate separately. This is done using two distinct equations: one for \( x(t) \) and another for \( y(t) \).
These equations are especially useful for tracing complex curves and paths that might not be easily described by a single function due to their loops and varying slopes.

• **Purpose of Parametric Equations:** They allow us to represent complex curves such as circles, ellipses, and in our case, Lissajous figures.
• **Application:** When the parameter \( t \) varies over a specific interval, it determines the shape and direction of the curve in a coordinated way.

In this problem, the given parametric equations are \( x(t)=\sin(4t) \) and \( y(t)=\sin(3t + \frac{\pi}{6}) \). These equations are used to unfold the pathway of the Lissajous figure within a square viewing window.

Trigonometric Functions

Trigonometric functions, like sine and cosine, are fundamental in both mathematics and physics due to their periodic nature. They describe smooth, wave-like patterns that repeat over regular intervals, known as periods.
Sine functions specifically can model oscillations and cycles due to their periodic behavior.

• **Sine Function Characteristics:** The sine function \( \sin(\theta) \) has a period of \( 2\pi \), meaning it repeats its values every \( 2\pi \) units.
• **Amplitude:** This function's amplitude is the maximum distance it reaches from zero, typically ranging from -1 to 1.
• **Phase Shift:** Modifications such as \( \sin(3t + \frac{\pi}{6}) \) introduce a phase shift due to the additional \( \frac{\pi}{6} \). This shifts the graph horizontally, altering when the function reaches its peak or trough.

In our Lissajous equation, the use of sine functions with different coefficients (\( 4 \) and \( 3 \)) creates a mix of frequencies, resulting in complex loops. The interaction between these frequencies is what forms the distinct Lissajous figure shapes, characterized by their symmetries and intersections.

Graphing Techniques

Graphing complex figures such as Lissajous figures involves translating parametric equations into a visual curve on a coordinate plane. This process involves several key techniques that ensure an accurate and smooth representation of the curve.

• **Key Points Evaluation:** Calculate the values of \( x(t) \) and \( y(t) \) at strategic points within the parameter's range. These points often occur at intervals of \( \frac{n\pi}{2} \) to capture the essence of the curve's cycle, especially since sine functions complete a cycle every \( 2\pi \).
• **Range Consideration:** Here, the viewing window is limited to \([-1,1]\) by \([-1,1]\). Since sine functions stay within these ranges naturally, any calculated point will fit within the boundary.
• **Connecting Points:** After plotting, connect the dots with smooth, continuous lines. This approach respects the continuity of trigonometric functions without abrupt changes.

To accurately graph the Lissajous figure, understanding how to interpret and plot these points ensures that the curve maintains the intended shape, reflecting the interplay of its trigonometric components.