Question

Equations of the form \(r^{2}=a \sin 2 \theta\) and \(r^{2}=a \cos 2 \theta\) describe lemniscates (see Example 7 ). Graph the following lemniscates. $$r^{2}=4 \sin 2 \theta$$

Short answer

Question: Sketch the graph of the lemniscate given by the polar equation \(r^2 = 4 \sin 2\theta\). Answer: To sketch the graph of the lemniscate given by the polar equation \(r^2 = 4 \sin 2\theta\), you need to understand the equation, calculate the values of \(r\) for different angles \(\theta\), plot the points in polar coordinates, and connect these points to form a smooth curve that repeats itself.

Step-by-step solution

Understanding the equation

Begin by considering the given polar equation: \(r^2 = 4 \sin 2\theta\). The purpose of this step is to identify the range of values for \(\theta\) that will produce the graph of this lemniscate, as well as to determine regions where the graph will cross the polar axis.

Calculate the values of \(r\) for different \(\theta\)

Start by setting \(\theta = 0\). We will then increment the value of \(\theta\) and compute the corresponding \(r\) values for the full range of angles from \(0\) to \(2\pi\). This process will help us understand the behavior of the lemniscate. At \(\theta = 0\), \(r^2 = 4 \sin(0) = 0\), so \(r = 0\). At \(\theta = \frac{\pi}{4}\), \(r^2 = 4 \sin(\frac{\pi}{2}) = 4\), so \(r = \pm 2\). At \(\theta = \frac{\pi}{2}\), \(r^2 = 4 \sin(\pi) = 0\), so \(r = 0\). At \(\theta = \frac{3\pi}{4}\), \(r^2 = 4 \sin(\frac{3\pi}{2}) = -4\), but \(r^2\) cannot be negative, so there is no solution for this angle. At \(\theta = \pi\), \(r^2 = 4 \sin(2\pi) = 0\), so \(r = 0\). At \(\theta = \frac{5\pi}{4}\), \(r^2 = 4 \sin(\frac{5\pi}{2}) = 4\), so \(r = \pm 2\). At \(\theta = \frac{3\pi}{2}\), \(r^2 = 4 \sin(3\pi) = 0\), so \(r = 0\). At \(\theta = \frac{7\pi}{4}\), \(r^2 = 4 \sin(\frac{7\pi}{2}) = -4\), but \(r^2\) cannot be negative, so there is no solution for this angle.

Plot the points in polar coordinates

Now that we have some points to work with, plot them in polar coordinates. Place these points on the polar grid, and begin connecting them to form a smooth curve.

Sketch the graph of the lemniscate

Continue tracing the curve formed by the points from step 3, completing the process when the curve starts to repeat itself. The resulting figure is the graph of the lemniscate \(r^2 = 4 \sin 2\theta\).

Key concepts

Polar Coordinates

Polar coordinates provide an alternative way to describe the position of a point in a plane. Unlike the Cartesian coordinate system, which uses \( x \) and \( y \) coordinates, polar coordinates describe a point with a distance and an angle. This system is especially useful for dealing with problems involving curves or motion that are circular or spiral in nature.

• The first component, \( r \), represents the radial distance from the origin to the point. It can be thought of as how "far" the point is from the center of the circle.
• The second component, \( \theta \), is the angle measured from the positive x-axis. It also helps to direct us on where the point is located, spinning counterclockwise.

When graphing equations in polar coordinates, we often convert between polar and Cartesian coordinates to get a better feel for the shape and positioning of the curve. The relationship is given by:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)

These conversions are fundamental when translating graphical data from one system to another, which can provide insights into the geometry of problems examined in polar coordinates.

Lemniscate

A lemniscate is a fascinating and beautiful plane curve, resembling the shape of a figure-eight or an infinity symbol (∞). It works wonderfully with polar coordinates because of its symmetrical properties. The general forms of a lemniscate's equation are \( r^2 = a \sin 2\theta \) and \( r^2 = a \cos 2\theta \). These equations help us understand some unique features of lemniscates:

• In these equations, \( a \) is a constant that stretches or compresses the shape. A larger \( a \) stretches the lemniscate.
• The \( \sin 2\theta \) and \( \cos 2\theta \) forms determine the orientation of the loop—horizontal or vertical.

For our given equation, \( r^2 = 4 \sin 2\theta \), the graph will have its lobes aligned along the axes at angles \( \theta = \frac{\pi}{4} + n\pi \) (where n is an integer).
The key trait of a lemniscate is its self-intersecting nature at the origin, stemming from the presence of both \( \pm r \) solutions for every valid \( \theta \). It loops around both sides of the origin, highlighting its mirroring nature.

Graph Sketching

Graph sketching in polar coordinates involves plotting points in a circular manner based on the radial distance \( r \) and angle \( \theta \). When sketching complex shapes like lemniscates, you mindfully track changes in angle and corresponding distances from a central point. Here's how to approach sketching a polar curve like \( r^2 = 4 \sin 2\theta \):

• Determine the angles \( \theta \) where the function produces valid \( r \) values (for \( \sin \), these will be at \( \frac{\pi}{4}, \frac{5\pi}{4} \), etc.).
• Calculate \( r \) for each \( \theta \), plotting \( r = \pm \sqrt{4 \sin 2\theta} \). Note values where \( r^2 \) becomes negative, as they indicate no real number solution for \( r \).
• Place points on a polar grid and connect them smoothly, remembering the symmetry about the origin. Start from \( \theta = 0 \) incrementing gradually, marking when \( r \) returns to zero.

Upon completing these steps, you’ll visualize a polar curve that elegantly loops through the origin, forming the interconnected loops of the lemniscate. Keep your sketches and calculations organized to ensure accuracy, capturing the essence and symmetry of the curve.