Question

Beautiful curves Consider the family of curves $$\begin{array}{l}x=\left(2+\frac{1}{2} \sin a t\right) \cos \left(t+\frac{\sin b t}{c}\right) \\\y=\left(2+\frac{1}{2} \sin a t\right) \sin \left(t+\frac{\sin b t}{c}\right)\end{array}$$ Plot a graph of the curve for the given values of \(a, b,\) and \(c\) with \(0 \leq t \leq 2 \pi\). $$a=18, b=18, c=7$$

Short answer

Question: Plot the graph of the given family of curves considering the values of a=18, b=18, and c=7 in the interval of t from 0 to 2π: $$ \begin{array}{l} x(t) = \left(2+\frac{1}{2} \sin{a t}\right) \cos\left(t+\frac{\sin{b t}}{c}\right) \\ y(t) = \left(2+\frac{1}{2} \sin{a t}\right) \sin\left(t+\frac{\sin{b t}}{c}\right) \end{array} $$ Answer: To plot the graph, follow these steps: understand the given parametric equations, substitute the given values of a, b, and c, create a range for t in the interval [0, 2π], evaluate x(t) and y(t) for each value of t in the range and then plot the graph using a graphing software or calculator. The resulting graph will be an intricate and beautiful curve.

Step-by-step solution

Understand the equations

The given equations define a parametric curve in the plane, with x and y being functions of the parameter t: $$ \begin{array}{l} x(t) = \left(2+\frac{1}{2} \sin{a t}\right) \cos\left(t+\frac{\sin{b t}}{c}\right) \\ y(t) = \left(2+\frac{1}{2} \sin{a t}\right) \sin\left(t+\frac{\sin{b t}}{c}\right) \end{array} $$ By varying t, we can obtain different points on the curve that we will plot.

Substitute given values for a, b, and c

Given the values of a, b, and c, substitute them in the equations like this: $$ \begin{array}{l} x(t) = \left(2+\frac{1}{2} \sin{18 t}\right) \cos\left(t+\frac{\sin{18 t}}{7}\right) \\ y(t) = \left(2+\frac{1}{2} \sin{18 t}\right) \sin\left(t+\frac{\sin{18 t}}{7}\right) \end{array} $$ Now we will use these equations to compute x(t) and y(t) values.

Create a range for t

As given, the range of t is from 0 to 2π. We now consider a range for t values in the interval [0, 2π]. For better visualization of the curve, choose a significant number of values within this range.

Evaluate x(t) and y(t) values

For each value of t in the range, evaluate the respective x(t) and y(t) values using the equations from step 2.

Plot the graph

Now, using the calculated x(t) and y(t) values, plot the graph of the curve. Use a graphing software, or a graphing calculator to plot the curve. Input the parametric equations for x(t) and y(t) and set the range of t from 0 to 2π. Then, plot the curve. The graph will be a beautiful, intricate curve reflecting the given parametric equations and the provided values for a, b, and c.

Key concepts

Graph Plotting

Graph plotting is a visual way to understand how equations behave. It is especially useful for parametric curves because it helps you see the relationship between two functions.
To plot a graph, you first take continuous data points from your equations. The values of the parameter, like \(t\), determine these data points. In our exercise, \(t\) goes from 0 to \(2\pi\). Collect enough points to show a smooth and accurate curve.

• Using a range and calculating many points makes sure the curve is well-drawn.
• Software or calculators can quickly compute and draw these graphs.

When finished, each plotted point connects to form a visual representation of the relationships shown by your equations. This specific curve can show periodic patterns or other interesting shapes based on the parametric form.

Parametric Equations

Parametric equations express complex mathematical relationships using parameters. Here, we have equations for \(x(t)\) and \(y(t)\), where \(t\) is a varying parameter. This means for every \(t\), you have a distinct \(x\) and \(y\), creating a point that belongs to the curve.
Unlike standard equations like \(y = x^2\), parametric equations can offer more sophisticated structures. They are great for modeling motion paths or waves.

• Each parametric equation can change separately, giving diverse and complex shapes when plotted.
• These equations often involve trigonometric functions because trigonometry provides periodic functions useful in parametric representations.

Understanding how parametric equations work helps in creating and interpreting vivid and complex curves like those you might encounter in engineering or computer graphics.

Trigonometric Functions

Trigonometric functions like sine and cosine feature prominently in our parametric equations. These functions introduce periodic behavior, which is essential for creating repeating patterns in graphs.
When you use trigonometric functions, your curve can become more dynamic, creating waves or oscillations based on the input values.

• The sine \(\sin\) and cosine \(\cos\) functions create peaks and valleys in your equation.
• The trigonometric part \( \sin(at) \) adjusts the periodic cycle based on the amplitude and frequency.

By tweaking the coefficients of \( \sin \) and \( \cos \), you influence the shape and symmetry of your graph. In our problem, the use of these functions with parameters like \(a, b,\) and \(c\) results in a visually appealing and complex design.