Problem 8 - Calculus

Question

Sketch the curve and check for \(x, y,\) and \(r\) symmetry. \(r^{2}=4 \sin 2 \theta \quad\) (lemniscate)

Step-by-Step Solution

Answer

The curve is symmetric about the origin.

Step 1: Identify the Type of Curve

The given equation is in polar form: \( r^2 = 4 \sin 2\theta \). This is the equation of a lemniscate, which is a type of curve that resembles the infinity symbol (∞). Lemniscates often have symmetry properties that we need to check.

Step 2: Check for Symmetry about the x-axis

For symmetry about the x-axis in polar coordinates, we replace \( \theta \) with \(-\theta \). The equation becomes \( r^2 = 4 \sin(-2\theta) = -4 \sin 2\theta \). This does not result in the original equation, so the curve is not symmetric about the x-axis.

Step 3: Check for Symmetry about the y-axis

For symmetry about the y-axis, we replace \( \theta \) with \( \pi - \theta \). Thus, the equation becomes \( r^2 = 4 \sin(2(\pi - \theta)) = 4 \sin(2\pi - 2\theta) = 4 \sin(-2\theta) = -4 \sin 2\theta \). Since this does not return the original equation, there is no y-axis symmetry.

Step 4: Check for Symmetry about the Origin

For symmetry about the origin, replace \( r \) with \(-r \). Thus, if \( r^2 = 4 \sin 2\theta \), then \((-r)^2 = r^2 = 4 \sin 2\theta \), which matches the original equation. This indicates that the curve is symmetric about the origin.

Step 5: Sketch the Curve

To sketch the curve, recognize that as \(\theta\) varies, \(r^2 = 4\sin 2\theta\) traces a symmetric figure around the origin. The lemniscate's shape is like an infinity sign (∞), with loops emerging at \(\theta = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}\) when the sine value reaches its maximum or minimum.

Key Concepts

Lemniscate

A lemniscate is a fascinating curve in mathematics, known for its distinctive shape resembling the infinity symbol (∞). It is often used to illustrate concepts in polar coordinates due to its unique properties.
Polar coordinates are a two-dimensional coordinate system, where each point on a plane is determined by an angle and a distance. In the case of a lemniscate, the equation involves the sine function, which influences its symmetry and form.
The specific lemniscate under study, given by the equation \( r^2 = 4\sin 2\theta \), is a classic example. As \( \theta \) changes, the product \( 2\theta \) changes the sine value, and consequently, the radius squared \( r^2 \) dictates how the curve is traced. This tracing creates two symmetrical loops extending from the origin, which is a characteristic feature of lemniscates.

Curve Symmetry

Symmetry in curves is an essential concept, particularly in polar coordinates, where the axis and origin symmetries offer insights into the nature of the curve.
When analyzing symmetry, we examine how the curve reacts to specific transformations:

• X-axis symmetry: Replace \( \theta \) with \( -\theta \). For the lemniscate \( r^2 = 4\sin 2\theta \), this substitution yields \( r^2 = -4\sin 2\theta \), which is not the original equation, indicating no x-axis symmetry.


• Y-axis symmetry: Replace \( \theta \) with \( \pi - \theta \). The equation transforms to \( r^2 = -4\sin 2\theta \), again deviating from the original, confirming no y-axis symmetry.


• Origin symmetry: Replace \( r \) with \( -r \). Here, the equation remains \( r^2 = 4\sin 2\theta \), demonstrating that the lemniscate maintains symmetry about the origin. The curve thus spins around the midpoint, ensuring that both loops are mirror images of each other.

Understanding symmetry aids in sketching the curve and predicting its behavior based on its mathematical properties.

Polar Equation Analysis

Analyzing polar equations is a crucial skill when dealing with curves like lemniscates. It involves understanding how equations in polar form translate into geometric shapes on a plane.
The equation \( r^2 = 4\sin 2\theta \) is in polar coordinates, meaning the relationship between the radius \( r \) and angle \( \theta \) defines the curve. Here, the sine function's role is pivotal as it oscillates between -1 and 1, dictating how the curve's shape unfolds.
For values of \( \theta \) where the sine function reaches its peaks, such as \( \theta = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4} \), the radius \( r \) is maximized, creating the loops of the lemniscate. Additionally, the factor of 2 inside the sine function doubles the rate at which this oscillation occurs compared to a standard sine wave, impacting the curve's form.
Polar equation analysis helps identify these critical points and understand how \( r \) and \( \theta \) cooperate to produce the elegant, infinite loops of a lemniscate.