Problem 40 - Precalculus with Calculus Previews

Question

Use a graphing utility to obtain the graph of the given set of parametric equations. $$ x=6 \cos 3 t, y=4 \sin 2 t, 0 \leq t \leq 2 \pi $$

Step-by-Step Solution

Answer

Use a graphing tool for x=6cos3t, y=4sin2t, with 0≤t≤2π, to visualize the shape.

Step 1: Define the Parametric Equations

We are given two equations: one for x, which is \( x = 6 \cos 3t \), and one for y, which is \( y = 4 \sin 2t \). These are parametric equations in terms of \( t \), where \( t \) represents a parameter.

Step 2: Understand the Range of the Parameter

The parameter \( t \) ranges from 0 to \( 2\pi \). This means that as \( t \) varies within this interval, the equations will generate corresponding points \( (x, y) \) which can be plotted.

Step 3: Use a Graphing Utility

To visualize the parametric equations, input \( x = 6 \cos 3t \) and \( y = 4 \sin 2t \) into a graphing utility. Set the parameter \( t \) to vary from 0 to \( 2\pi \). Adjust the graphing window to ensure you can view the entire plot.

Step 4: Observe the Graph

Upon entering the equations into the graphing utility and graphing, observe the shape produced. The graph is likely to display a visually interesting pattern due to the different frequencies in \( \cos 3t \) and \( \sin 2t \).

Key Concepts

Graphing Utilities

Graphing utilities are incredibly useful tools when it comes to visualizing mathematical equations, especially parametric equations. By using a graphing calculator or an online graphing tool, you can quickly input equations and see their graphical representation.

To graph parametric equations like those given in the problem, simply enter both the x and y equations into your graphing instrument. It's essential to ensure that the parameter \( t \) is set correctly to explore its entire range, which is from 0 to \( 2\pi \) in this instance.

Here are a few tips for using graphing utilities effectively:

• Ensure your device is set to parametric mode.
• Adjust settings so the parameter adjusts smoothly over its range of values.
• Modify the viewing window to capture the complete curve pattern without cutoff.
• Check for options to animate the curve for better understanding.

Utilizing these tools can lead to better insight and understanding of how the equations behave as \( t \) changes.

Trigonometric Functions

Trigonometric functions like sine and cosine are key components in describing wave-like patterns and periodic behavior. In parametric equations, they are used to articulate x and y coordinates smoothly over an interval.

In the given exercise, the equations \( x = 6 \cos 3t \) and \( y = 4 \sin 2t \) embody trigonometric functions. Here’s a brief rundown of what each element does:

• \( \cos(3t) \) and \( \sin(2t) \) represent periodic functions that oscillate between -1 and 1.
• Multiplying \( \cos(3t) \) by 6 scales the amplitude of oscillation along the x-axis.
• Similarly, multiplying \( \sin(2t) \) by 4 modifies the amplitude along the y-axis.
• The coefficients (3 and 2) inside the functions alter their frequencies, producing more cycles within the same interval for \( t \).

By analyzing these components, you can predict the periodic nature and wave-like shape in the graph of the parametric equations.

Graph Interpretation

Interpreting the graph of parametric equations requires attention to how the parametric functions interact with each other. With the trigonometric frequencies being different, an intriguing curve will manifest. This occurs as the x-coordinate completes cycles at a different rate than the y-coordinate.

When you observe the graph from the utility:

• Notice the shape, which could be a series of loops or a closed curve. This is due to differing frequencies, such as the 3:2 frequency ratio implied by \( 3t \) and \( 2t \).
• Look for symmetry. Often, the equations like \( x = 6 \cos 3t \) and \( y = 4 \sin 2t \) can create symmetrical curves as sine and cosine are even and odd functions, respectively.
• Determine the orientation. With trigonometric functions, the direction in which the curve is drawn can provide insights into the function's behavior across the parameter range.

By piecing together these observations, one gains a coherent understanding of how the curve's structure emerges from the underlying parametric equations.