Problem 24 - 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). \(r=4 \cos 2 \theta\) (four-leaved rose)

Step-by-Step Solution

Answer

The graph is a four-leaved rose symmetric about the x-axis, y-axis, and origin.

Step 1: Understanding the Polar Equation

The polar equation given is \(r=4 \cos 2\theta\). This represents a four-leaved rose, where the equation is of the form \(r=a \cos n\theta\). Here, \(a=4\) and \(n=2\). This leads to a rose with \(2n=4\) petals.

Step 2: Analyze the Leaf Pattern

Since \(n=2\), a classic property of rose curves states if \(n\) is even, there will be \(2n\) petals in the rose. Thus, for \(n=2\), the graph will depict four petals or leaves.

Step 3: Check for Symmetry

To verify symmetry, observe the function with transformations:- **Symmetric about the x-axis**: Replace \(\theta\) with \(-\theta\). The equation becomes \(r = 4 \cos 2(-\theta) = 4 \cos 2\theta\), so it is symmetric about the x-axis.- **Symmetric about the y-axis**: Substitute \(r\) with \(-r\) gives the same equation \(r = 4 \cos 2\theta\) because cosine is even.- **Symmetric about the origin**: Try \(\theta + \pi\) which remains unchanged \(4 \cos(2\theta + 2\pi)=4\cos 2\theta\).

Step 4: Sketch the Rose Curve

Draw the rose curve by plotting points at various angles \(\theta\). Compute \(r\) for angles like \(\theta=0\), \(\pi/4\), \(\pi/2\), and \(\pi\) to find distinct point positions and plot smoothly through these points to form symmetric leaves.

Key Concepts

Polar Coordinates

Polar coordinates offer a distinct method of representing points on a plane. Unlike the usual Cartesian coordinates, which use \(x\) and \(y\) to specify a location, polar coordinates rely on the distance from the origin, called the radius \(r\), and the angle \(\theta\) measured from the positive x-axis. This system is particularly useful in scenarios involving angles and distances.
\[r(\theta)\]
By understanding polar coordinates,

• we can represent curves and shapes more efficiently when angle and radius play a central role, such as in circular patterns.
• Conversion between Cartesian and polar coordinates involves the use of trigonometric relations, where \(x = r \cos \theta\) and \(y = r \sin \theta\).

Polar coordinates can simplify equation representation in situations where symmetry or rotation is involved. To fully appreciate these advantages, it's important to become comfortable with thinking about angles and radii as fundamental descriptors of a point's location.

Symmetry in Polar Graphs

Symmetry plays a crucial role in analyzing polar graphs, making it easier to predict their behavior and shape without plotting every point.
Considering symmetry:

• X-axis Symmetry: Replace \(\theta\) with \(-\theta\). If the resulting equation is similar to the original, the curve is symmetric about the x-axis.
• Y-axis Symmetry: Substitute \(r\) with \(-r\). If unchanged, the curve is symmetric about the y-axis. This test checks if reflection across the y-axis leads to the same point.
• Origin Symmetry: Use \(\theta + \pi\) or \(r + \pi\). Consistency shows the curve is symmetric about the origin, indicating rotational symmetry.

Understanding these types of symmetry helps in sketching graphs more efficiently. Symmetry can mean less work due to repetition across different axes or quadrants.

Rose Curve

The rose curve is a fascinating example of polar graphs, often characterized by its petal-like pattern. The general form \(r = a \cos n\theta\) or \(r = a \sin n\theta\) determines its shape, with the parameter \(n\) driving its petal count.
When \(n\) is even, the curve produces \(2n\) petals, while odd \(n\) results in \(n\) petals.
Key properties of rose curves include:

• The amplitude \(a\) controls the overall size of the petals. Larger \(a\) means larger petals.
• The function \(n\theta\) determines angular frequency, influencing how many petals are present.
• Rose curves often exhibit multiple types of symmetry, making them easier to sketch and analyze by exploiting these repetitive patterns.

These curves are not just theoretical but have practical applications in fields like engineering and design, where symmetrical patterns are desired.

Mathematical Visualization

Visualizing mathematical concepts like polar equations involves more than just sketching lines and curves. It requires an understanding of how equations translate into distinct shapes and patterns on a graph. By using polar graphing, we can better understand:

• Sine and Cosine Functions: They are fundamental in defining radial distance as a function of angle, crucial in forming patterns like the rose curve.
• Transformation Techniques: Using symmetry and periodicity helps reduce complexity when analyzing repetitive or symmetrical structures.
• Graphical Interpretation: Through visual tools, abstract equations become tangible, aiding comprehension of theoretical concepts.

For educators and learners alike, mathematical visualization is an indispensable tool in simplifying complex ideas and turning them into intuitive and understandable visuals. It fosters a deeper comprehension of geometric configurations and their real-world applications.