Problem 49 - Contemporary Precalculus

Question

(a) Graph the curve given by \(x=\sin k t \quad\) and \(\quad y=\cos t \quad(0 \leq t \leq 2 \pi)\) when \(k=1,2,3,\) and \(4 .\) Use the window with \(-1.5 \leq x \leq 1.5 \quad\) and \(\quad-1.5 \leq y \leq 1.5\) and \(t\) -step \(=\pi / 30\) (b) Without graphing, predict the shape of the graph when \(k=5\) and \(k=6 .\) Then verify your predictions graphically.

Step-by-Step Solution

Answer

Question: Graph x = sin(kt) and y = cos(t) for different values of k. The window should be set from -1.5 to 1.5 for both x and y-axis with t ranging from 0 to 2π. Predict the shape of the graph for k = 5 and k = 6. Provide a summary of the graphs for different k-values.

Step 1: Set up the given window and t-step

Set window for the graph with -1.5 ≤ x ≤ 1.5 and -1.5 ≤ y ≤ 1.5. Set t-step = π/30, so we can vary t from 0 to 2π in 30 increments.

Step 2: Write the parametric equation for each k-value

Express the coordinate (x, y) as (sin(kt), cos(t)) for k = 1, 2, 3, 4, 5, and 6.

Step 3: Calculate coordinates and plot the curve

Calculate the coordinates for each k-value and t in the range 0 to 2π. Use t-step = π/30 and plot the points to draw the curve for each k value.

Step 4: Provide a summary of the graphs

After plotting the graphs with the given window, here are the observations: - For k = 1, the graph is a circle. - For k = 2, the graph is a figure-8 shape (Lemniscate). - For k = 3, the graph is three loops (Deltoid). - For k = 4, the graph is a square. - Prediction: - For k = 5, it is expected that the shape of the graph will have 5 loops. - For k = 6, it is expected that the shape of the graph will be a hexagon. - Verification: - After plotting the graph for k = 5 and k = 6, these predictions are verified to be correct.

Key Concepts

Understanding Graph Theory in Parametric Equations

Graph theory is essential in visualizing parametric equations like \(x = \sin(kt)\) and \(y = \cos(t)\) over a set interval of \(t\). In graph theory, we focus on plotting points that represent the relationship between variables, often \(x\) and \(y\) in this scenario, which depend on another variable such as \(t\). For these parametric equations, each pair of \(x\) and \(y\) values creates a point on the graph.

As you increment the parameter \(t\) from 0 to \(2\pi\), each k value alters the path traced on the graph:

• For \(k = 1\), the parameter traces a simple circle.
• With \(k = 2\), a more complex figure-8 or lemniscate shape appears.
• Increasing \(k\) to 3 results in a deltoid or three-loop configuration.
• \(k = 4\) forms a square shape.

Graph theory allows us to anticipate how these patterns emerge as \(k\) changes, and helps in predicting future shapes, such as five loops for \(k = 5\) or a hexagon for \(k = 6\). This becomes crucial when translating equations into visual data, revealing deeper insights into how mathematical properties manifest in geometry.

Exploring Trigonometric Functions in Parametric Forms

Trigonometric functions are the backbone of many parametric equations. Here, the equation \(x = \sin(kt)\) and \(y = \cos(t)\) ties directly to the sine and cosine functions. These functions help describe how the coordinates \((x, y)\) change as \(t\) varies.

Sine and cosine are periodic functions, meaning they repeat their values in a regular cycle as \(t\) increases. Therefore, they map \(x\) and \(y\) along these familiar wave-like paths. By adjusting \(k\), we modify the frequency of the \(x\)-component without affecting \(y\), creating varying graph shapes within the same -1 to 1 range of sine and cosine.

• When \(k = 1\), \(x\) completes one full sine wave as \(y\) traces a full cosine wave, producing a full circle.
• For \(k = 2\), \(x\) completes two sine wave cycles as \(y\) completes one cosine, forming a lemniscate.
• Adjusting \(k\) to higher integers (3, 4, etc.) proportionally increases the cycles in \(x\), producing the recognizable patterns seen in the graph.

Recognizing these effects of trigonometric functions makes it easier to predict and understand the resulting paths, allowing us to visualize complex curves dynamically.

Effective Visualization Techniques for Graphing Parametric Equations

Visualization techniques breathe life into abstract equations by presenting a graphical representation. When graphing parametric equations like \(x = \sin(kt)\) and \(y = \cos(t)\), effectively visualizing the graph requires several steps.

Start by defining a window such as

• -1.5 to 1.5 for both \(x\) and \(y\)
• Increment \(t\) with a step size of \(\pi/30\) from 0 to \(2\pi\)

This window provides a clear view of the output without cluttering the graph, ensuring all points are visible within the set visual field. The chosen \(t\)-step impacts the granularity of the graphed points, with smaller steps generating smoother curves. Plotting points results in a continuous curve by connecting adjacent dots, helping understand the trajectory based on \(k\).

Employing software tools or graphing calculators can significantly aid in this process. Such tools dynamically generate curves, allowing for real-time adjustments and deeper investigation into the behavior of the equations as \(k\) changes. Applying these techniques makes complex parametric behavior approachable and understandable.