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. $$r^{2}=\cos 2 \theta$$

Short answer

Answer: The lemniscate has an "X" shape with the origin at the center. The zeroes of the function are at \(\theta = \frac{\pi}{2}\) and \(\theta = \frac{3\pi}{2}\). The maximum points are at \(\theta = 0\) and \(\theta = \pi\), and the minimum points are at \(\theta = \frac{\pi}{4}\) and \(\theta = \frac{3\pi}{4}\).

Step-by-step solution

Identify the nature of the lemniscate

The given polar equation \(r^2 = \cos 2 \theta\) is of the form \(r^2 = a \cos 2 \theta\), where \(a=1\). This type of lemniscate has an "X" shape, with the origin (pole) at the center.

Find the zeroes of the function

To find the zeroes of the function, set \(r^2=0\), and solve for \(\theta\): $$0 = \cos 2\theta$$ The cosine function is zero at \((2n + 1)\frac{\pi}{2}\), where \(n\) is an integer. Since we are only looking for zeroes in one period of the function, we can take \(n = 0\) and \(n = 1\). Thus, the zeroes of the function are at \(\theta = \frac{\pi}{2}\) and \(\theta = \frac{3\pi}{2}\).

Find the maximum/minimum values of the function

To find the maximum and minimum values of the function, find \(\theta\) where the cosine function achieves its maximum or minimum. Recall that the maximum value of the cosine function is \(1\), and the minimum value is \(-1\). For a maximum, set \(r^2 = 1\): $$1 = \cos 2\theta$$ Solving for \(\theta\), we get \(\theta = 0\) and \(\theta = \pi\). For a minimum, set \(r^2 = -1\): $$-1 = \cos 2\theta$$ Solving for \(\theta\), we get \(\theta = \frac{\pi}{4}\) and \(\theta = \frac{3\pi}{4}\). Now that we have zeros and max/min values, we can plot these points on the polar coordinate plane.

Plot the important points on the coordinate plane

Plot the following points: 1. Zeroes: at \(\theta = \frac{\pi}{2}\), \(\theta = \frac{3\pi}{2}\). 2. Maxima: at \(\theta = 0\), \(\theta = \pi\). 3. Minima: at \(\theta = \frac{\pi}{4}\), \(\theta = \frac{3\pi}{4}\).

Sketch the graph of the lemniscate

Using the important points plotted on the coordinate plane, sketch the lemniscate curve. Notice that it has an "X" shape, with the origin at the center, and two smooth loops in the first and third quadrants.

Key concepts

Lemniscates

A lemniscate is a unique type of curve that looks like the figure-eight or an "infinity" symbol. In polar coordinates, lemniscates have equations of the form \(r^2 = a \sin 2\theta\) or \(r^2 = a \cos 2\theta\). These equations create a symmetrical curve around the origin in the polar plane.

The interesting part about a lemniscate is that it's not centered at any particular point on a Cartesian plane. Instead, it revolves around the origin, meaning it is centered at the pole in a polar coordinate system. By analyzing the polar equation, we can see certain key features and behaviors:

• When \(a\) is positive, the lemniscate can exhibit two loops.
• For \(\cos 2\theta\), its shape is akin to an "X".

Understanding these features makes graphing lemniscates much easier.

Graphing Polar Equations

Graphing polar equations involves understanding how the radius \(r\) changes with different angles \(\theta\). Since polar coordinates are based on circles, the process of graphing involves rotating around the origin while plotting points radiating outward.

For a lemniscate equation like \(r^2 = \cos 2\theta\), identifying important points involves:

• Finding zeroes: where the curve intersects the origin.
• Locating maxima and minima: points where the radius is longest or shortest.

In practice, this involves calculating \(\theta\) values that give you these critical points. As you plot these points, remember that each provides a guide to the shape's curve. For example, the maximum value points \(\theta = 0\) and \(\theta = \pi\) mark where the loops of a lemniscate expand the furthest.

Trigonometric Functions

Trigonometric functions are deeply connected to polar coordinates, particularly when it comes to graphing equations like lemniscates. In polar equations, these functions define the relationship between the angle \(\theta\) and the radius \(r\).

The cosine function, used in lemniscates such as \(r^2 = \cos 2\theta\), oscillates between -1 and 1.

• At \(\theta = 0\) and \(\theta = \pi\), \(\cos 2\theta\) reaches its peak value of 1, resulting in maximum radius.
• Zeroes occur at \(\theta = \frac{\pi}{2}\) and \(\theta = \frac{3\pi}{2}\), where the radius becomes zero.
• The minimum value of \(-1\) appears where the curve doesn’t form an actual circle, illustrated by points \(\theta = \frac{\pi}{4}\) and \(\theta = \frac{3\pi}{4}\).

By understanding these behaviors, you can predict and plot points accurately on polar graphs. The trigonometric function not only influences the radius but also guides the overall shape and symmetry of the lemniscate. This harmony between angles and distances forms the beauty of graphing such intricate curves.