Question

Pick a integer value for \(n,\) where \(n \neq 2,3,\) and use technology to plot \(r=\sin \left(\frac{m}{n} \theta\right)\) for three different integer values of \(m\). Sketch these and determine a minimal interval on which the entire graph is shown.

Short answer

The minimal interval is from \(0\) to \(8\pi\).

Step-by-step solution

Understanding the Problem

We need to pick an integer value for \( n \) such that \( n eq 2 \text{ or } 3 \). Then, we need to plot the polar equation \( r = \sin \left(\frac{m}{n} \theta\right) \) for three different integer values of \( m \). Finally, we need to identify a minimal interval for \( \theta \) over which the entire graph is displayed.

Choosing a Value for n

Select an integer value for \( n \). Let's choose \( n = 4 \). This choice satisfies the requirement that \( n eq 2, 3 \).

Selecting Values for m

Choose three integer values for \( m \). Let's pick \( m = 1, 2, 3 \). These selections will allow us to observe different patterns in the graph.

Writing the Polar Equations

Now we construct the polar equations using the chosen values:- For \( m = 1 \), the equation is \( r = \sin \left(\frac{1}{4} \theta\right) \).- For \( m = 2 \), the equation is \( r = \sin \left(\frac{2}{4} \theta\right) \).- For \( m = 3 \), the equation is \( r = \sin \left(\frac{3}{4} \theta\right) \).

Plotting with Technology

Use graphing technology, such as a graphing calculator or software, to plot each equation. View how the graphs appear for \( \theta \) in the interval from \( 0 \) to \( 2\pi \) and beyond to check for repetition.

Identifying the Minimal Interval

Observe the plots to determine where the graphs begin repeating. For each \( m \), note the interval:- \( m = 1 \): The graph repeats after \( 8\pi \) since the pattern completes a full cycle every \( 8\pi \).- \( m = 2 \): The graph repeats after \( 4\pi \).- \( m = 3 \): The graph repeats after \( 8\pi \).The minimal interval that encompasses all full cycles for these values of \( m \) is \( 0 \) to \( 8\pi \).

Conclusion

The polar plots for the equations represent various petal patterns based on the values of \( m \). The minimal interval on which the entire graph for these equations is displayed is from \( 0 \) to \( 8\pi \).

Key concepts

Graphing Technology in Polar Equations

When it comes to visualizing polar equations, graphing technology like calculators or specialized software can be extremely helpful. Polar equations, such as the one given in the exercise, often produce intricate patterns that are not only beautiful but also very specific in shape and symmetry.
Using a graphing tool allows you to quickly draw these graphs and visually explore their properties. When we consider an equation like \(r = \sin(\frac{m}{n} \theta)\), we can alter the values of \(m\) and \(n\) and see immediate changes in the graph. This dynamic interaction helps deepen our understanding of how different parameters affect the polar graph.
Graphing technology also aids in identifying shapes and making precise measurements of intervals which would be very challenging to achieve manually. This becomes especially important when determining the minimal interval that displays the complete pattern of a graph.

Interval Identification in Polar Graphs

Identifying the correct interval over which a polar graph completes its full cycle is crucial for understanding its behavior. In the given exercise, this was achieved by observing the plots of the polar equations for values of \(m\) and \(n\).
The goal is to find the smallest angle \(\theta\) over which the petal pattern of the graph repeats. For different integer values of \(m\), the interval changes, impacting how many petals appear in one complete cycle.
For example, when \(n = 4\) and \(m = 1\), the graph completes a cycle at \(8\pi\). This tells us how far we need to go around the circle to see all distinct petals of the polar graph. Recognizing these intervals forms the basis of understanding symmetry in polar coordinates. This step is not only key to solving the exercise but also in appreciating the cyclical nature of trigonometric polar graphs.

The Role of Integer Values

In the context of polar coordinates, integer values play an important role in determining the characteristics of the graph. By adjusting integer values of \(m\) and \(n\) in the equation \(r = \sin(\frac{m}{n} \theta)\), we can directly influence the number and orientation of petals.
Selecting different integers for \(m\) results in different petal patterns appearing on the graph, as seen with \(m = 1, 2, 3\). This choice impacts how the equation manipulates the angle \(\theta\) and affects the symmetry and structure of the graph.
Understanding the effect of integers helps in forming hypotheses and interpreting the shapes seen in polar graphs. Each pair of \(m\) and \(n\) contributes to unique artistic and mathematical outcomes.

Understanding Petal Patterns in Polar Graphs

Petal patterns are a fascinating feature of polar graphs, particularly those involving sinusoidal functions like \(r = \sin(\frac{m}{n} \theta)\). These patterns vary based on the ratio of \(m\) to \(n\).
A graph's petals emerge as the function oscillates, tracing paths outward from the pole (origin of the polar graph). The number of petals is linked to the values of \(m\) and \(n\).
When \(m = n\), the graph typically forms \(n\) petals, while non-equal ratios may reveal complex overlapping patterns. In our exercise example, \(n = 4\) with \(m = 3\) results in a beautiful arrangement that is both symmetrical and repetitive, showcasing the elegance of mathematical symmetry.
Recognizing these patterns helps predict how changes to the formula will alter the appearance of polar plots, grounding students in both visual and numerical understanding.