Problem 40 - Algebra and Trigonometry

Question

Use a graphing device to draw the curve represented by the parametric equations. $$ x=\sin 4 t, \quad y=\cos 3 t $$

Step-by-Step Solution

Answer

The graph of the parametric equations shows a complex looping pattern due to the differing frequencies of sine and cosine.

Step 1: Identify the Parametric Equations

The parametric equations provided are \( x = \sin 4t \) and \( y = \cos 3t \). These equations describe a curve in the plane where \( x \) and \( y \) are functions of the parameter \( t \).

Step 2: Set Up Your Graphing Device

Ensure your graphing calculator or software is in parametric mode. This mode is necessary for plotting equations where \( x \) and \( y \) are expressed as functions of a third variable, \( t \).

Step 3: Input Parametric Equations

Enter the equations \( x = \sin 4t \) and \( y = \cos 3t \) into your graphing device. Make sure each equation is entered in the respective fields designated for \( x(t) \) and \( y(t) \).

Step 4: Determine the Range for t

Select a range of values for \( t \) to see a complete graph of the curve. A common choice is \( t \) between \( 0 \) and \( 2\pi \), but you can choose a wider range if needed to fully visualize the pattern of the curve.

Step 5: Adjust Graphing Settings

Check the graphing window settings to ensure that both the x and y axes are visible and appropriately scaled. You may need to adjust them to fit the amplitudes of the sine and cosine functions.

Step 6: Graph the Parametric Curve

Once all settings are configured, use your graphing device to plot the curve. Observe the pattern that emerges, which should resemble a complex looping shape, due to the differing frequencies of the sine and cosine terms.

Step 7: Analyze the Graph

The plot should display a closed curve that repeats periodically, showing intersections and loops resulting from the relationship of the sine and cosine frequencies. Observe any symmetrical patterns or repeated sections in the graph.

Key Concepts

Graphing Parametric Curves

When dealing with parametric equations, we are essentially describing how both the x and y coordinates of a point change over time as a function of a third variable, often denoted as \( t \). This differs from traditional Cartesian coordinates where \( y \) is expressed directly in terms of \( x \). Instead, parametric equations give each coordinate a unique trajectory

• These equations offer flexibility in representing curves that are more complex or difficult to model using just x and y relations.
• In our exercise, the parametric equations are \( x = \sin 4t \) and \( y = \cos 3t \).

Graphing parametric curves can reveal intricate, beautiful, looping patterns as the parameter varies. The visualization of these curves provides insights into the nature of the underlying geometric figure. When plotted on a graphing calculator or software, students can witness an elegant unfolding of the curve, showcasing periodicity and intersections that occur due to the nature of sine and cosine.

Sine and Cosine Functions

Sine and cosine are fundamental trigonometric functions that describe oscillations and waves. These functions repeat values in a cyclical, or periodic, pattern every \(2π\). They are critical in creating curves from parametric equations.

• The sine function, \( \sin(4t) \), affects the horizontal position (x-coordinate) of each point on the graph, giving it an oscillating motion at a frequency determined by the coefficient 4.
• The cosine function, \( \cos(3t) \), adjusts the vertical position (y-coordinate) with its frequency determined by the coefficient 3.

These coefficients change how many cycles the function completes over \(2π\) and they create the loops observed in the resulting curve. This interaction between differing sine and cosine frequencies leads to complex paths, rich in symmetry and periodic behaviors.

Using Graphing Calculators

Graphing calculators are essential tools for visualizing the behavior of parametric equations. Most graphing calculators are capable of entering parametric mode, allowing users to plot graphs where both x and y are functions of \( t \).

• First, make sure the calculator is in the correct mode to input parametric equations.
• Enter \( x(t) = \sin 4t \) and \( y(t) = \cos 3t \) to set up your graph.
• Choose an appropriate range for \( t \). Typically, \( t = 0 \) to \( t = 2π \) is used, but a wider range can capture more details of the curve.
• Adjust window settings to ensure curves fit well within the display screen, accounting for the amplitude of sine and cosine.

With these settings correct, the calculator will reveal the parametric curve's nature, showing beautiful interactions of the two trigonometric functions.

Trigonometric Functions

Trigonometric functions extend beyond just sine and cosine, encompassing a family of functions used to analyze and describe rotations and waves. Their fundamental nature lies in angles and periodicity.

• Sine and cosine create circular and elliptical curves, perfect for representing oscillatory phenomena.
• These functions are periodic, meaning they show patterns that repeat indefinitely in a predictable manner.
• In the case of our parametric equations, adjusting the multiples of \( t \) for either sine or cosine changes the frequency, or how often the curves repeat themselves over a set interval.

Understanding these principles helps justify why parametric graphs with trigonometric functions tend to exhibit loops, circles, and other fascinating shapes, giving insight into the mathematical harmony between angles, lengths, and periodicity.