Problem 22 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems \(1-32\), sketch the graph of the given polar equation and verify its symmetry (see Examples 1-3). \(r=3 \sin 3 \theta\) (three-leaved rose)

Step-by-Step Solution

Answer

The graph is a three-leaved rose symmetric about the y-axis.

Step 1: Understanding Polar Equations

The given polar equation is \(r = 3 \sin 3\theta\). In this equation, \(r\) represents the radius, and \(\theta\) represents the angle. The equation is in the form \(r = a \sin(n\theta)\), which is known for producing rose curves.

Step 2: Recognizing Symmetry

A rose curve of the form \(r = a \sin(n\theta)\) is symmetric with respect to the y-axis (polar axis) when \(n\) is odd. For the equation \(r = 3 \sin 3 \theta\), \(n = 3\), which is odd, so we can verify symmetry with respect to the y-axis.

Step 3: Sketching the Graph

To sketch this graph, consider various values of \(\theta\). Notice that as \(\theta\) varies from 0 to \(2\pi\), the sine function completes three positive lobes of the rose curve. At \(\theta = 0\), \(r = 0\); at \(\theta = \frac{\pi}{6}\), \(r = 3\); and at \(\theta = \frac{\pi}{2}\), \(r = 0\) again. This pattern continues for full symmetry in the rotation.

Step 4: Verifying Symmetry (Again)

To further verify symmetry over different axes, change \(\theta\) to \(-\theta\) in the equation. The resulting equation \(r = 3 \sin(-3\theta)\) simplifies to \(r = -3\sin(3\theta)\), demonstrating symmetry about the y-axis as explained in Step 2. The curve is symmetric with respect to the y-axis.

Key Concepts

Symmetry in Polar Graphs

In polar coordinate systems, symmetry is a key attribute that can simplify the analysis and sketching of graphs. Symmetry can be about different axes or the origin, each providing unique insights into the graph's structure. For polar graphs, two main types of symmetry often occur:

• Symmetry about the Polar Axis (x-axis): This occurs when replacing \( \theta \) with \( -\theta \) in the polar equation yields an identical equation. It suggests that the graph is mirrored across the polar axis.
• Symmetry about the Line \(\theta = \frac{\pi}{2}\) (y-axis): For certain equations, substituting \( \theta \) with \( \frac{\pi}{2} - \theta \) or testing with the substitution \( r = -r \) can verify symmetry about this line.
• Origin Symmetry: Also known as rotational symmetry, this occurs when replacing both \( \theta \) with \( \theta + \pi \) and \( r \) with \( -r \) yields the original equation.

In the case of the rose curves, symmetry is crucial for ensuring that the pattern of petals repeats uniformly. Particularly for the equation \( r = 3 \sin 3\theta \), symmetry about the y-axis—established as the polar axis—informs us about how the petals will arrange themselves around the center.

Rose Curves

Rose curves are named for their petal-like shapes appearing in polar graphs. They are an interesting mathematical phenomenon, defined by equations of the form \( r = a \sin(n\theta) \) or \( r = a \cos(n\theta) \). These curves vary notably based on the integer \( n \):

• If \( n \) is odd, the rose will have \( n \) petals.
• If \( n \) is even, the rose will have \( 2n \) petals.

For instance, with the equation \( r = 3 \sin 3\theta \), since \( n = 3 \) (an odd number), the curve will feature three symmetrical petals. Importantly, the value \( a \) affects the petal's length but does not change the petal count. Thus, different values of \( a \) will create larger or smaller roses with the same number of petals. This consistency in appearance with varied amplitude and rotational symmetry around the origin makes rose curves a striking example of polar graph symmetry and beauty.

Graph Sketching in Polar Coordinates

Graph sketching in polar coordinates might be a bit different from the Cartesian system, yet it offers unique visualization options for certain mathematical equations. When sketching polar graphs, follow these simple steps:

• Understand the equation's form: Recognize whether the equation belongs to standard polar graph types like circles, spirals, or rose curves.
• Identify symmetry attributes: As highlighted, checking for symmetry can greatly simplify the drawing process by reducing the amount of manual plotting needed.
• Choose key angles: Start by selecting values such as \( \theta = 0, \frac{\pi}{6}, \frac{\pi}{3} \), and so on, to find corresponding \( r \) values, which help determine the graph's path.
• Plot the graph: Using the points calculated, begin at the origin and plot |\( r \)| along the polar axis direction determined by \( \theta \). Reflect on symmetry and repeat due to the periodic nature.

Polar graphs offer an elaborate and artistic perspective on equations, with the ability to demonstrate complex periodicity and rotational symmetry visually. This is particularly apparent in sketching rose curves, where the petals distinctly articulate these characteristics.