Problem 56 - Algebra and Trigonometry

Question

Identify and graph each polar equation. $$ r^{2}=\sin (2 \theta) $$

Step-by-Step Solution

Answer

The equation graphs as a lemniscate (figure-eight shape) centered at the origin.

Step 1: Understanding the Polar Equation

The given polar equation is \[ r^2 = \sin(2\theta) \]. In this equation, \( r \) represents the radius and \( \theta \) represents the angle in polar coordinates. We need to first understand what the equation represents before graphing it.

Step 2: Determine the Nature of the Equation

The equation \( r^2 = \sin(2\theta) \) indicates a relationship between the radius and the doubled angle. Since \( r \) squared is equal to \( \sin(2\theta) \), the graph will include points where \( r \) can be both positive and negative values.

Step 3: Find Key Points and Symmetry

To graph the equation, identify key angles and their corresponding radius values. Note that \( \sin(2\theta) \) is periodic with a period of \( \pi \). Evaluate the equation at key points: \( \theta = 0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \pi \). For example, if \( \theta = \frac{\pi}{4} \), \( r^2 = \sin\left(2 \cdot \frac{\pi}{4}\right) = \sin(\frac{\pi}{2}) = 1 \), so \( r = \pm 1 \).

Step 4: Graph the Equation

Plot the radius values obtained in the previous step at the respective angles. Since the equation suggests a lemniscate (figure-eight shape), reflect these points in all four quadrants. Radius values that are positive will be directed outward from the origin, while negative values will be inward.

Step 5: Draw the Curve

Connect the points smoothly to form the lemniscate. The resulting shape should look like a figure-eight centered at the origin, symmetric with respect to the origin and the line \( \theta = \frac{\pi}{4} \).

Key Concepts

Polar Equations

A polar equation is a mathematical expression where the variables are in terms of polar coordinates, namely radius \( r \) and angle \( \theta \). Unlike Cartesian coordinates, which use (x, y) pairs to represent points on a plane, polar coordinates employ a distance from a fixed point (the origin) and an angle from a fixed direction (usually the positive x-axis). For example, the equation \( r^2 = \sin(2\theta) \) involves the radius squared and the sine of twice the angle. This relationship makes it different from Cartesian equations and gives it unique graphing properties. Polar equations often describe curves more naturally than Cartesian forms, allowing for elegant representations of complex shapes.

Graphing Polar Equations

Graphing polar equations involves plotting points based on their radius and angle. Each point is plotted by moving a distance \( r \) from the origin at an angle \( \theta \). It's crucial to determine key points that help visualize the curve. For example, let's graph the given polar equation \( r^2 = \sin(2\theta) \). Key points are calculated at specific angles like 0, \( \frac{\pi}{4} \), \[ \pi \], \( \frac{3\pi}{4} \), and \ \pi \. Once the radius for these angles is calculated, each point (r, \( \theta \)) is plotted on polar graph paper or a coordinate plane. The resultant graphing process forms a unique shape, characteristic of the equation. In this case, it forms a lemniscate, resembling a figure-eight.

Lemniscate

A lemniscate is a special type of curve shaped like a figure-eight or an infinity symbol (∞). It is characterized by its symmetrical nature and unique geometric properties. The given polar equation \( r^2 = \sin(2\theta) \) plots into a lemniscate. This curve has significant features:

• It is centered around the origin.
• It is symmetric about the origin.
• It intersects itself at the origin.

The lemniscate has points where \( r \) takes equal positive and negative values at different angles. For example, when \( \theta = \frac{\pi}{4} \), \( r \) can be both 1 and -1. These reflective properties make the lemniscate a popular subject of study in polar coordinates.

Periodic Functions

Periodic functions repeat their values in regular intervals. In the context of polar coordinates, \( \sin(2\theta) \) is a periodic function with a period of \( \pi \), meaning \( \sin(2(\theta + \pi)) = \sin(2\theta) \). This periodicity influences the graph of the polar equation \( r^2 = \sin(2\theta) \), creating repeating sections of the curve. Understanding the periodic nature helps in identifying and plotting key points across the entire range of \( \theta \). Periodic functions like sine and cosine are staples in polar equations, often defining circles, spirals, and other intricate shapes based on their repeating patterns. They also facilitate symmetry in polar graphs, making graphing more predictable and comprehensible.

Symmetry in Polar Graphs

Symmetry plays a crucial role in polar graphs, simplifying the graphing process and aiding in visualization. For the polar equation \( r^2 = \sin(2\theta) \), symmetry can be observed:

• About the origin (central symmetry).
• About the line \( \theta = \frac{\pi}{4} \).

To determine symmetry, you can test if \( (r,\theta) \) and \( (-r,\pi + \theta) \) or other such pairs satisfy the equation. Symmetry indicates that once a segment of the graph is plotted, the rest can be mirrored accordingly. This helps in creating accurate and complete graphs more efficiently, promoting a deeper understanding of the structure and form of polar equations.