Problem 44 - Precalculus

Question

Graph equation. \(r^{2}=16 \sin (2 \theta)\)

Step-by-Step Solution

Answer

The graph is a 4-petal rose symmetric about \(\theta = \frac{\pi}{4}\) with each petal extending 4 units from the origin.

Step 1: Identify the Type of Polar Equation

The equation given is in the form of a polar equation, specifically, it resembles the form of a limaçon. In polar coordinates, sinusoidal terms like \(\sin(2\theta)\) suggest that the curve repeats symmetrically about certain axes.

Step 2: Rewrite into Standard Polar Form

We observe that \(r^{2}=16\sin(2\theta)\). Let's convert this equation into a more recognizable form. Here, arrange the equation as \(r^{2}=16\sin(2\theta)\), equivalent to \((r) = \sqrt{16\sin(2\theta)}\). Here, remember that \(r\) can be positive or negative.

Step 3: Determine the Symmetry

Since the function involves \(\sin(2\theta)\), we know the graph has symmetry about \(\theta = \frac{\pi}{4}\) and \(\theta = \frac{3\pi}{4}\). The factor of 2 in \(\theta\) indicates that the graph will have multiple petals.

Step 4: Analyze the Graphical Features

We start by considering some easy values for \(\theta\) to find \(r\). For \(\theta = 0\), \(r^2 = 0\), meaning \(r = 0\). For \(\theta = \frac{\pi}{4}\), \(\sin(2\theta) = 1\), thus \(r^2 = 16\) leading to \(r = \pm 4\). These indicate key points on the graph and show it extends 4 units along the angle \(\theta = \frac{\pi}{4}\).

Step 5: Sketch the Graph

Now, plot the points obtained and note that the graph has 4 petals because of the factor 2 in \(\sin(2\theta)\). These petals will be oriented symmetrically about the angle \(\theta = \frac{\pi}{4}\). Each petal extends a distance of 4 units from the origin.

Key Concepts

Limaçon

The term "limaçon" refers to a family of curves that are best described in polar coordinates. A limaçon is characterized by its specific equation form, typically appearing as \( r = a + b \, \sin(\theta) \) or \( r = a + b \, \cos(\theta) \). These curves are intriguing due to their general shape, which can vary from a heart-like structure to a looped figure. Limaçons display a variety of features based on the values of \( a \) and \( b \):

• If \( a = b \), the curve will resemble a cardioid.
• If \( b > a \), the curve will have an inner loop.
• If \( b < a \), the curve will resemble an oval shape, without a loop.

The name "limaçon" originates from a Latin word meaning "snail", aptly describing the curves' coiled shape. These curves are important in mathematics not only for their aesthetic but also due to their occurrence in physics and engineering scenarios.

Symmetry in Polar Graphs

In polar graphs, symmetry plays a crucial role as it helps us quickly understand the overall layout of the graph. Symmetry can occur in several common forms:

• **Symmetry about the polar axis (horizontal axis):** If replacing \( \theta \) with \( -\theta \) results in an equivalent expression, the graph reflects over the horizontal axis.
• **Symmetry about the line \( \theta = \frac{\pi}{2} \) (vertical axis):** If replacing \( r \) with \( -r \) retains the same curve, then the graph is symmetric about the vertical axis.
• **Symmetry about the pole (origin):** When the expression remains the same after replacing both \( r \) with \( -r \) and \( \theta \) with \( \theta + \pi \), symmetry about the origin is present.

For our given equation \( r^2 = 16\sin(2\theta) \), the presence of \( \sin(2\theta) \) metric implies inherent symmetry. The angle multiplication by 2 suggests that the graph repeats symmetrically more frequently, creating a balanced and predictable pattern that aligns around axes such as \( \theta = \frac{\pi}{4} \) and \( \theta = \frac{3\pi}{4} \). This information aids significantly when sketching or interpreting polar graphs.

Sinusoidal Features in Polar Graphs

Sinusoidal features in polar graphs arise from equations involving trigonometric functions like \( \sin \) or \( \cos \). These features imbue the graph with periodic and oscillatory characteristics, akin to wave patterns. The standard form typically appears as \( r = a \cdot \sin(n\theta) \) or \( r = a \cdot \cos(n\theta) \), where \( n \) is an integer that adjusts the number of 'petals' or repeats observed in the graph.Here are a few notable aspects to consider:

• The coefficient \( n \) determines the number of oscillations or petals. Specifically, \( n \) petals in the entire graph if \( n \) is odd, and \( 2n \) petals if \( n \) is even.
• The amplitude \( a \) defines the radial distance from the pole to the petal tip, representing the graph's largest radius.
• The graph has a repetitive symmetry that gifts it a predictable structure, facilitating easier graph interpretations and drawings.

For the equation \( r^2 = 16\sin(2\theta) \), the factor of "2" inside the sine function doubles the typical number of symmetrical divisions, resulting in a graph with 4 petals. Each petal reaches a maximum radial extent based on the equation's amplitude expression, a fundamental feature in enhancing the graph's visual and mathematical understanding.