Problem 46 - A Graphical Approach to Precalculus with Limits

Question

Graph each polar equation for \(\theta\) in \(\left[0^{\circ}, 360^{\circ}\right)\). In Exercises \(39-48\), identify the rype of polar graph. $$r^{2}=4 \sin 2 \theta$$

Step-by-Step Solution

Answer

The polar graph is a lemniscate, characterized by a symmetrical, double-looped curve.

Step 1: Understand the Polar Equation

The given polar equation is \( r^2 = 4 \sin 2\theta \). This equation can also be written as \( r = \pm \sqrt{4 \sin 2\theta} \). Notice how the equation involves \( \sin 2\theta \), which is key to identifying the type of polar graph.

Step 2: Identify the Type of Polar Graph

The presence of \( \sin 2\theta \) in the equation suggests that this polar graph is a type of rose curve. Specifically, equations of the form \( r^2 = a \sin n\theta \) or \( r^2 = a \cos n\theta \) typically result in lemniscates.

Step 3: Transform the Equation

To transform \( r^2 = 4 \sin 2\theta \) into a recognizable form, let \( r = \pm 2 \sqrt{\sin 2\theta} \). This form confirms the lemniscate characteristic, which has a double-loop figure.

Step 4: Graph the Polar Equation

Graphing the equation involves plotting points for various values of \( \theta \) from \( 0^{\circ} \) to \( 360^{\circ} \). Note the symmetry and consistency characteristic of lemniscates: if \( \theta = 45^{\circ} \), \( 90^{\circ} \), or other key angles where \( \sin 2\theta \) reaches 1 or -1.

Step 5: Confirm the Polar Graph Type

The final graph shows the characteristic shape of a lemniscate. This confirms the typical appearance expected from \( r^2 = 4 \sin 2\theta \): a symmetrical, double-looped curve centered at the origin.

Key Concepts

Lemniscates

Lemniscates are interesting and beautiful mathematical figures. They resemble a figure-eight or an infinity symbol, and are a special type of curve known as a rose curve. In polar coordinates, lemniscates are graphed using equations typically of the form \(r^2 = a \sin n\theta\) or \(r^2 = a \cos n\theta\). These equations help generate the characteristic double-loop shape of a lemniscate.

When you plot a lemniscate, it reflects across both the horizontal and vertical axes. This symmetry is due to the trigonometric functions involved, like sine or cosine, which determine the position and orientation of each loop. In the case of \(r^2 = 4 \sin 2\theta\), as seen in the exercise, the equation uses sine, which influences the loops' appearance.

• Each loop is centered on the origin, or the pole in the polar coordinate system.
• The value \(a\) in the equation affects the size of the loops.
• The \(n\) in \(n\theta\) determines how many times the loops are repeated around the center.

Graphing a lemniscate can be a fun challenge, where exploring different angles of \(\theta\) reveals how the curve takes shape. Recognizing these curves as lemniscates helps in anticipating the characteristics of the graph.

Polar Coordinates

Polar coordinates are a way of representing points in a plane using angles and distances. Unlike the Cartesian coordinate system, which uses \(x\) and \(y\)-coordinates, polar coordinates use \(r\) (the radial distance from the origin) and \(\theta\) (the angle from the positive x-axis).

This system is particularly helpful for graphing curves that exhibit rotational symmetry, such as circles and spirals. In a polar coordinate system:

• Each point is defined by its distance from the origin, \(r\), and its angle, \(\theta\).
• \(\theta\) is usually measured in degrees or radians, extending from the positive x-axis.
• The positive direction of \(r\) suggests outwards movement from the pole, while negative values indicate backward movement.

Polar equations, like the one given \(r^2 = 4 \sin 2\theta\), can be transformed to highlight key characteristics of the curve. By evaluating \(r\) at different angles \(\theta\), you gain insights on the symmetry and structure of the graph. This method is ideal for graphing equations that naturally revolve around a central point.

Trigonometric Graphing

Trigonometric graphing involves using sine, cosine, and other trigonometric functions to graph equations. These functions are based on angles and cyclic behaviors, which makes them perfect for polar graphs that are circular or repetitive in nature.

In the equation \(r^2 = 4 \sin 2\theta\), the use of \(\sin 2\theta\) highlights how fundamental trigonometric functions are in shaping the graph. Here’s what makes trigonometric graphing unique:

• Sine and cosine can create periodic graphs that are symmetrical and can repeat patterns efficiently.
• By changing the parameters in the trigonometric functions, you can adjust the amplitude and frequency of cycles.
• This modulation in cycles dictates the loops and turns of rose curves and lemniscates.

When graphing trigonometric functions, considering key angles where these functions reach maximum and minimum values (like \(0\), \(90^\circ\), \(180^\circ\), etc.) is crucial to getting an accurate depiction.In the exercise, understanding these function values and their influence on \(r\) helps shape the graph. Trigonometric graphing allows for a structured analysis of how angles and distances interact to form complex, beautiful shapes like the lemniscates.