Problem 47 - Algebra and Trigonometry

Question

Graph the family of polar equations \(r=1+\sin n \theta\) for \(n=1,2,3,4,\) and \(5 .\) How is the number of loops related to \(n ?\)

Step-by-Step Solution

Answer

The number of loops is equal to \(n\).

Step 1: Understanding Polar Equations

Polar equations describe curves in terms of the distance from the origin, denoted as \(r\), and the angle \(\theta\). In the given equation \( r = 1 + \sin n\theta \), \(n\) determines the frequency of the sine function, which influences the shape of the curve.

Step 2: Finding Symmetry

The equation \(r = 1 + \sin n\theta\) is symmetric with respect to the vertical axis (\(\theta = \frac{\pi}{2}\)) when \(n\) is odd. For even \(n\), the symmetry is present with respect to both the line \(\theta = \frac{\pi}{2}\) and the horizontal axis.

Step 3: Graphing for n = 1

For \(n=1\), the equation becomes \(r = 1 + \sin \theta\). This results in a single loop, resembling a limaçon with an inner loop that touches the origin.

Step 4: Graphing for n = 2

When \(n = 2\), the equation is \(r = 1 + \sin 2\theta\). This produces a curve with two loops, evident as a figure-eight pattern with more pronounced loops.

Step 5: Graphing for n = 3

For \(n=3\), the equation \(r = 1 + \sin 3\theta\) results in three loops. The shape appears like three petals radiating from the origin.

Step 6: Graphing for n = 4

With \(n = 4\), the equation \(r = 1 + \sin 4\theta\) forms four loops, creating a symmetric flower-like pattern with four distinct petals.

Step 7: Graphing for n = 5

Finally, for \(n = 5\), the equation \(r = 1 + \sin 5\theta\) generates five loops, resembling a starfish with five arms or petals.

Step 8: Conclusion on Loop Number Relation

The number of loops in the graph of \( r = 1 + \sin n\theta \) is exactly \(n\). As \(n\) increases, the number of loops in the polar plot also increases exactly by \(n\). This means each increment of \(n\) adds an additional loop to the pattern.

Key Concepts

Symmetry in Polar Graphs

When graphing polar equations like \( r = 1 + \sin n\theta \), it is important to observe the symmetry, as it can provide insights into the graph's shape and behavior. Symmetry in polar graphs refers to the consistent and repeating patterns around a particular axis.

For the equation \( r = 1 + \sin n\theta \), the presence of symmetry depends on whether \( n \) is even or odd.

• If \( n \) is odd, the graph will exhibit symmetry about the vertical line \( \theta = \frac{\pi}{2} \). This means the graph will look the same on either side of this line.
• When \( n \) is even, the symmetry is not only about the vertical line \( \theta = \frac{\pi}{2} \) but also about the horizontal axis. This dual symmetry means the figure can be mirrored across both axes, resulting in more balanced patterns.

Understanding symmetry helps predict the complete structure of polar graphs without needing to calculate and plot every single point.

Sine Function Frequency

In the given equation \( r = 1 + \sin n\theta \), the parameter \( n \) plays a critical role in determining the frequency of the sine function. Frequency in this context refers to how many complete cycles or waves the sine function goes through as \( \theta \) ranges from 0 to \( 2\pi \).

As \( n \) increases, the frequency of the sine wave also increases. This means:

• The graph completes more oscillations within the same range of \( \theta \).
• For each integer increase in \( n \), you see an additional loop in the polar graph.

This is because the sine function \( \sin n\theta \) completes \( n \) full cycles between \( \theta = 0 \) and \( \theta = 2\pi \), leading directly to the multiple-looped features of the polar graphs.

Limaçon Curves

Limaçon curves are a fascinating type of polar graph that arise when graphing equations like \( r = a + b\sin\theta \). They belong to a family of curves known for their distinctive shapes ranging from dimpled to inner-looped forms.

In the equation \( r = 1 + \sin n\theta \), when \( n = 1 \), the graph resembles a limaçon with an inner loop that touches the origin. Despite being a simple-looking curve, the limaçon is highly versatile:

• It can appear as a cardioid, which is a heart-shaped curve, when \( a = b \).
• When \( b > a \), the limaçon exhibits a smaller inner loop.

These shapes result from the combination of the constant term (adding an offset) and the oscillating sine component, illustrating the interesting geometric features of polar coordinates.

Number of Loops in Polar Graphs

The number of loops in polar graphs is a fundamental characteristic that can be easily determined from polar equations of the form \( r = 1 + \sin n\theta \).

This number of loops, directly correlates with the value of \( n \):

• For \( n = 1 \), you observe a single loop.
• As \( n \) increases to 2, 3, 4, and so on, the number of loops also increases correspondingly.

Each loop represents a full cycle of the sine function as \( \theta \) traverses through \( 2\pi \). So, for each unit increase in \( n \), there is an additional loop, forming a pattern that looks like petals radiating symmetrically from the origin. Thus, the relationship between \( n \) and loops is both simple and predictable.