Problem 31 - Algebra and Trigonometry

Question

15–36 Sketch the graph of the polar equation. $$r^{2}=\cos 2 \theta \quad \text {(lemniscate)}$$

Step-by-Step Solution

Answer

Graph is a symmetric figure-eight lemniscate based on the given polar equation.

Step 1: Identify the Type of Curve

The given polar equation \(r^2 = \cos 2\theta\) represents a lemniscate, which is a type of figure-eight shape in polar coordinates.

Step 2: Locate Symmetries

This equation is symmetric about the polar axis, the line \(\theta = \frac{\pi}{4}\), and the origin because the cosine function is even and periodic with period \(\pi\). No shifts or transformations change these symmetries in the polar coordinates.

Step 3: Find Key Points

To sketch the lemniscate, let's find when the radius \(r\) is zero: Set \(\cos 2\theta = 0\). Hence, \(2\theta = \frac{\pi}{2} + k\pi\), where \(k\) is an integer. Therefore, \(\theta = \left(\frac{\pi}{4} + \frac{k\pi}{2}\right)\).

Step 4: Evaluate \(r\) at Specific Angles

Evaluate \(r\) at different angles to find points on the graph. For example: \(r = 1\) when \(\cos 2\theta = 1\) which means \(2\theta = 0, \pi,\) etc., giving \(\theta = 0, \frac{\pi}{2}\).

Step 5: Sketch the Graph

Plot the curve based on the calculated points and symmetries. The lemniscate will touch the origin, extend outwards symmetrically around the polar axis and the line \(\theta = \frac{\pi}{4}\), and return symmetrically to the other side of the origin.

Key Concepts

Lemniscate

The term "lemniscate" refers to a specific type of curve, recognized for its unique figure-eight or infinity-like shape in polar coordinates. This curve can be described by equations often involving trigonometric functions such as sine or cosine. In the case of a lemniscate, a classic example is the equation \( r^2 = \cos 2\theta \). The beauty of a lemniscate is in its symmetry, which allows it to be perfectly balanced around certain axes.

Understanding how these curves behave is important for sketching them accurately. A lemniscate "loops" through the origin and mirrors itself, making these crossings both a key property and a focal point for plotting. The symmetry and simplicity of the shape make it essential in studying polar curves, linking mathematical beauty with geometric patterns.

Polar Equations

Polar equations are mathematical expressions where each point on a plane is determined by an angle and a distance from a central point, typically called the pole. Unlike Cartesian coordinates that rely on \(x\) and \(y\) axes, polar coordinates use \(r\) (radius) and \(\theta\) (angle).

One advantage of polar coordinates is their ability to represent circular or rotational features simply. For example, the lemniscate equation, \(r^2 = \cos 2\theta \), elegantly describes the curve without needing complex algebraic terms. In polar form, adjustments such as scaling or rotating become more intuitive because you're dealing directly with radial distances and angular directions.

• \(r^2 = \cos 2\theta\) implies symmetry and involves trigonometric functions.
• Evaluating polar equations often involves calculating \(r\) for various values of \(\theta\).
• Expressing a curve in polar form can simplify analysis of rotational or symmetric properties.

By grasping polar equations, students can explore mathematical concepts related to angles and radii, providing deeper insight into various geometric configurations.

Graph Symmetries

Symmetry in graphs is all about balance and mirroring, which plays a crucial role in visualizing polar equations like the lemniscate. In the context of the polar equation \(r^2 = \cos 2\theta\), the symmetry can be seen in multiple forms:

The equation features symmetry about the origin, the polar axis (which is the horizontal line through the pole), and interestingly, along the line \(\theta = \frac{\pi}{4}\). This means that for any point \((r, \theta)\) on the curve, there exists a mirrored point that maintains this balanced configuration.

• The polar axis symmetry implies that if \((r, \theta)\) is on the graph, \((r, -\theta)\) is also on the graph.
• Origin symmetry means that \((-r, \theta)\) is a point when \((r, \theta)\) is a point on the graph.
• \(\theta = \frac{\pi}{4}\) symmetry results in additional pattern consistency, allowing the curve to reflect with respect to this line.

Recognizing these symmetries aids in accurately sketching the curve and understanding its structure, emphasizing the harmonious and predictable nature of mathematical graphs.