Question

Consider the family of curves $$\begin{aligned} &x=\left(2+\frac{1}{2} \sin a t\right) \cos \left(t+\frac{\sin b t}{c}\right).\\\ &y=\left(2+\frac{1}{2} \sin a t\right) \sin \left(t+\frac{\sin b t}{c}\right) \end{aligned}.$$ Plot the curve for the given values of \(a, b,\) and \(c\) with \(0 \leq t \leq 2 \pi\) (Source: Mathematica in Action, Stan Wagon, Springer, 2010 ; created by Norton Starr, Amherst College). $$a=7, b=4, c=1$$

Short answer

Question: Plot the curve given by the parametric equations \(x = (2 + \frac{1}{2}\sin{7t})\cos{(t+\sin{4t})}\) and \(y = (2 + \frac{1}{2}\sin{7t})\sin{(t+\sin{4t})}\) for \(0 \leq t \leq 2\pi\).

Step-by-step solution

Substitute the given values

First, substitute the given values of \(a, b\), and \(c\) into the parametric equations for \(x\) and \(y\). With \(a=7, b=4\), and \(c=1\), we get: $$\begin{aligned} &x=\left(2+\frac{1}{2} \sin 7t\right) \cos \left(t+\frac{\sin 4 t}{1}\right),\\ &y=\left(2+\frac{1}{2} \sin 7t\right) \sin \left(t+\frac{\sin 4 t}{1}\right) \end{aligned}.$$

Identify points of the curve

Now, we need to identify points on the curve by substituting different values of \(t\) in the range \(0 \leq t \leq 2\pi\). Preferably, we want to choose values of \(t\) at regular intervals to obtain a representative set of points. For simplicity, we can pick values of \(t\) at the following intervals: \(0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \pi, \frac{5\pi}{4}, \frac{3\pi}{2}, \frac{7\pi}{4}\), and \(2\pi\). Compute the corresponding \(x\) and \(y\) values for each of these \(t\) values.

Plot the curve

Finally, plot the curve using the points obtained in Step 2. You can use graphing software or tools like Desmos, GeoGebra, or even Microsoft Excel to create the plot. Verify that the plot appears to accurately represent the curve. After performing these steps, you will have a plot of the curve for the given values of \(a, b, c\) and the range \(0 \leq t \leq 2\pi\).

Key concepts

Curve Plotting

In the study of parametric equations, curve plotting is an essential technique. It involves representing a set of mathematical data points on a two-dimensional graph, usually with the horizontal axis for the independent variable and the vertical axis for the dependent variable. Curve plotting not only provides an easy visualization of complex relationships within equations but also helps in identifying crucial properties of curves like intersections, symmetry, and periodicity.

Here are some key points to consider when plotting curves:

• Predict the behavior: Understanding the basic shape and behavior of trigonometric functions will guide how the curve emerges as different values are substituted.
• Use consistent intervals: By plotting points at regular intervals within the given range, such as the multiples of \(\frac{\pi}{4}\), you ensure accurate representation.
• Tools for accuracy: Programs like Desmos or GeoGebra help plot these curves accurately, aiding in verifying manual calculations.

During the curve plotting process for our problem, we substitute known values into the parametric equations and calculate corresponding points to produce a precise plot for visual examination.

Mathematical Visualization

Mathematical visualization is a powerful tool for understanding abstract concepts. When we visualize parametric equations, it helps us grasp ideas that might be difficult to comprehend through numerical data alone. By transforming equations into graphical representations, complex behaviors and interactions between variables become more accessible to students.

The use of mathematical visualization includes:

• Understanding relationships: By visualizing how variables change over a range, one can observe the direct relationship and interaction between them.
• Highlighting dynamic changes: Parametric equations often depict curves that change shape and direction based on different parameters, making visualization critical in identifying these dynamics.
• Enhancing problem-solving: Discovering patterns and irregularities through visualization can offer insights into possible solutions and simplify problem-solving processes.

Visualizing the family of curves described in the exercise paints a picture of how parameters like \(a\), \(b\), and \(c\) affect the shape and orientation of the curve, facilitating a deeper understanding of the equation's behavior across different conditions.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental to understanding parametric equations. These functions model periodic phenomena and are crucial in plots involving waves and rotations.

In the given problem, trigonometric functions appear in expressions for both \(x\) and \(y\). These functions account for:

• Oscillation patterns: The sine function within \(x\) and \(y\) determines the oscillatory nature, essential in creating periodic curves.
• Amplitude variation: The coefficient of terms involving sine affects the amplitude of the wave, altering its height and depth across different \(t\) values.
• Frequency and phase: The multiplicative factors and additional terms in \(t\) (like \(\frac{\sin \)>inside cosine or sine functions) adjust the wave's frequency and phase, resulting in different stretching and shifting effects in plot outputs.

These trigonometric functions combine to present a comprehensive understanding of wave-like behaviors in curves, showing why certain curves repeat themselves and how they transition as different parameters change.