Problem 28 - Precalculus with Calculus Previews

Question

$$ r^{2}=4 \cos 2 \theta $$

Step-by-Step Solution

Answer

The equation represents a lemniscate with two loops.

Step 1: Identify the Equation Type

The given equation \( r^2 = 4 \cos 2\theta \) is in polar form and represents a conic section. Whenever \( r^2 = a \cdot \cos 2\theta \) or \( r^2 = a \cdot \sin 2\theta \), it indicates a conic section with a specific orientation depending on the trigonometric function used.

Step 2: Classify the Conic Section

Since the equation is in the form of \( r^2 = a \cdot \cos 2\theta \) with \( a = 4 \), this type of equation generally represents a conic section called a 'lemniscate'. A lemniscate has two loops, symmetrical about the polar axis (horizontal axis in polar coordinates).

Step 3: Determine the Graph's Shape

For \( r^2 = a \cdot \cos 2\theta \), the graph is a lemniscate if \( a > 0 \). Here, \( a = 4 \) which is positive, meaning the graph has two loops intersecting at the origin, with the major axis along the polar axis.

Key Concepts

Conic Sections

Conic sections, often a fascinating topic of study in mathematics, are curves obtained from intersecting a plane with a double-napped cone. Based on the angle and position of the intersection, this can form different shapes. The primary types of conic sections are:

• Circles
• Ellipses
• Parabolas
• Hyperbolas

These shapes have unique properties and equations associated with them.
In polar coordinates, conic sections can take on interesting forms due to their dependence on the angle \( \theta \). The equation \( r^2 = a \cdot \cos 2\theta \) is significant, as it represents a special conic known as a lemniscate, which we'll explore further in the next section. Recognizing such equations can help you quickly identify the type of conic section and its general geometry.

Lemniscate

A lemniscate is a distinctive type of conic section resembling the infinity symbol \( \infty \). It is known for its characteristic figure-eight shape or two symmetric loops that intersect at the origin. The general form in polar coordinates is:

• \( r^2 = a \cdot \cos 2\theta \)
• \( r^2 = a \cdot \sin 2\theta \)

For the given equation \( r^2 = 4 \cdot \cos 2\theta \), since \( a = 4 \) is positive, the conic is a lemniscate. This equation results in a graph with the following properties:

• Two loops
• Symmetrical about the polar axis
• Intersecting at the pole (origin)

Knowing these characteristics helps visualize how the graph's major axis aligns with the polar axis, giving it its distinctive shape.

Graphing Polar Equations

Graphing polar equations requires understanding the relationship between \( r \) and \( \theta \), which together determine the position of each point on the graph. Here's a simplified process: 1. Convert Polar to Cartesian Coordinates:

• Use the transformations \( x = r \cos\theta \) and \( y = r \sin\theta \) to switch between coordinate systems if needed.

2. Determine Key Features:

• Identify symmetry, orientation, and where the equation intersects, such as the pole.

3. Plot Points:

• Calculate \( r \) for several values of \( \theta \).
• Plot these points in the polar coordinate plane.

For the example equation \( r^2 = 4 \cos 2\theta \), you recognize it as a lemniscate. The graph formation is automatic, with orientation and symmetry easily found by evaluating where \( \cos 2\theta \) equals positive or negative values, revealing the loop structure typical of a lemniscate.