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. $$r^{2}=-2 \sin 2 \theta$$

Short answer

Answer: The graph of the lemniscate looks like a figure-eight with the origin at the center, with the two loops of the eight lying symmetrically in the first and third quadrants.

Step-by-step solution

Points of intersection with the x-axis

For the lemniscate to intersect the x-axis, the y-coordinate must be zero. This happens in polar coordinates when theta takes the values {\(\frac{\pi}{2}\), \(\frac{3\pi}{2}\)}. Let's find the corresponding r values: Substitute these values into the equation: $$r^2=-2\sin(2\theta)$$ For \(\theta=\frac{\pi}{2}\): $$r^2=-2\sin(2\times\frac{\pi}{2})=-2\sin(\pi)=0$$ So the curve passes through the origin. For \(\theta=\frac{3\pi}{2}\): $$r^2=-2\sin(2\times\frac{3\pi}{2})=-2\sin(3\pi)=0$$ So the curve passes through the origin again.

Points of intersection with the y-axis

For the lemniscate to intersect the y-axis, the x-coordinate must be zero. This happens in polar coordinates when theta takes the values {0, \(\pi\)}. Let's find the corresponding r values: Substitute these values into the equation: $$r^2=-2\sin(2\theta)$$ For \(\theta=0\): $$r^2=-2\sin(2\times0)=-2\sin(0)=0$$ So the curve passes through the origin. For \(\theta=\pi\): $$r^2=-2\sin(2\times\pi)=-2\sin(2\pi)=0$$ So the curve passes through the origin again.

Analyzing the equation

The given polar equation is: $$r^2=-2\sin2\theta$$ We notice that the left side, \(r^2\), must always be non-negative, while the right side, \(-2\sin2\theta\), will be non-negative only for specific values of \(\theta\) where \(\sin2\theta\leq0\). For other values of \(\theta\), the equation gives an inadmissible result, which means those points will not be part of the graph. By analyzing the sine function of the double angle, we can say that \(\sin2\theta\leq0\) when \(\theta\in[\frac{\pi}{4} + k\frac{\pi}{2},\frac{3\pi}{4} + k\frac{\pi}{2}]\) for integers k.

Graphing the lemniscate

First, we have established that the lemniscate passes through the origin (0, 0), which is the intersection point of the x and y axes. Next, considering the values of \(\theta\) where the equation is non-negative, we notice that the lemniscate is symmetric about the origin and forms loops around the quadrants. By analyzing the behavior of the sine function for our accepted values of \(\theta\), we can sketch the curve by paying attention to where the graph begins and ends in each quadrant. In summary, the graph of the lemniscate looks like a figure-eight with the origin at the center, with the two loops of the eight lying symmetrically in the first and third quadrants.

Key concepts

Lemniscate

A lemniscate is a fascinating shape in mathematics that resembles the infinity symbol (∞) or a figure-eight. In polar coordinates, it is described by equations of the form \( r^2 = a \sin 2\theta \) or \( r^2 = a \cos 2\theta \). These equations produce curves where the values of \( r \) can result in looping paths around a central point.

In the context of the provided exercise, we have the equation \( r^2 = -2 \sin 2\theta \). This type of lemniscate projects its figure-eight pattern usually in the first and third quadrants due to its symmetry properties.

• Form: Depending on the trigonometric function and values, lemniscates can differ in orientation and position.
• Intersection: They often intersect the origin, dividing the plane into symmetrical loops.
• Symmetry: Generally, lemniscates exhibit symmetry about the origin, doubling back on themselves.

The unique feature of a lemniscate is how it maintains this central symmetry while forming loops that expand and contract based on the angle \( \theta \). For certain values, it even creates "pockets" or lobes that stretch across polar axes.

Graphing Polar Equations

Graphing polar equations involves plotting points (\(r, \theta\)) on a polar coordinate grid. Unlike Cartesian coordinates which utilize x and y, polar coordinates focus on radius \(r\) and angle \(\theta\). For lemniscates, this approach showcases their symmetry and looping nature.

To visualize a lemniscate on a polar graph:

• Identify key points such as intersections on axes by solving the equation for specific \(\theta\)
• Understand which parts of the polar grid the lemniscate will occupy by analyzing when the equation holds true (for \(r^2 \geq 0\)).
• Note that certain sections of the polar graph remain unoccupied when \(r^2\) becomes negative.

By focusing on the behavior of the trigonometric component, you can sketch the curve's path. This path typically involves symmetric loops around the origin, often intersecting axes due to calculated \(\theta\) values. As you adjust \(\theta\), the radius \(r\) dictates how far from or near to the center each point lies.

Trigonometric Functions

Trigonometric functions such as sine and cosine play a crucial role in forming polar graphs like lemniscates. These functions determine how the curves behave as the angle \(\theta\) changes.

In this exercise, the sine function \(\sin 2\theta\) dictates the lemniscate's structure. Important aspects to note include:

• Double Angle: The function uses \(2\theta\), multiplying the angle effect, which results in more intricate patterns.
• Periodicity: With trigonometric functions, periodic behavior allows the curves to loop regularly, completing full cycles and returning to initial positions.
• Significance: The negativity in \(-2\sin 2\theta\) directs the loops' placement in polar quadrants.

Understanding the sine function's behavior can help predict and explain a lemniscate's configuration. It highlights when \(r^2\) is positive (matching how far out the loop extends) or zero (where it intersects axes or returns to the origin). These functions are the backbone of polar graphs, creating the fluid, rhythmic movements that define these shapes.