Problem 94 - Calculus Early Transcendentals

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. \(r^{2}=-8 \cos 2 \theta\)

Step-by-Step Solution

Answer

Question: Sketch the graph of the polar equation \(r^2 = -8 \cos 2\theta\) and provide its description. Answer: The graph of the polar equation \(r^2 = -8 \cos 2\theta\) represents an inverted lemniscate (figure 8 shape) with both halves lying in the second and fourth quadrants. It intersects the axes at \((0, \frac{\pi}{4}), (0, \frac{3\pi}{4}),(0, \frac{5\pi}{4}),\) and \((0, \frac{7\pi}{4})\).

Step 1: Find the points of intersection with the axes

To find the points of intersection with the axes, we will set r = 0 and solve for \(\theta\). $$r^2=-8\cos 2\theta \Rightarrow 0 = -8\cos 2\theta$$ From this equation, we can find the values of \(\theta\) when \(r = 0\). Cosine function becomes 0 when its argument is equal to \(2\theta = (2n+1)\frac{\pi}{2}\), where n is an integer. Solving for \(\theta\) gives: $$\theta = (n+\frac{1}{2})\frac{\pi}{2}$$ Taking n = 0, 1, 2, and 3, we obtain four different values of \(\theta\): \(\frac{\pi}{4}\), \(\frac{3\pi}{4}\), \(\frac{5\pi}{4}\), and \(\frac{7\pi}{4}\). These correspond to the points where the graph intersects the axes.

Step 2: Determine the shape of the lemniscate

As we know that the given equation is of the form \(r^2 = -a \cos 2\theta\), it will represent a lemniscate with an inverted orientation with both halves lying in the second and fourth quadrants. We can also see that the magnitude of the coefficient is 8, so the curve will have a large size.

Step 3: Sketch the graph

Now that we have found the points of intersection with the axes and know the shape of lemniscate, we can sketch the graph. 1. Mark the points corresponding to the axes intersection we found in step 1: \((0, \frac{\pi}{4}), (0, \frac{3\pi}{4}),(0, \frac{5\pi}{4}),\) and \((0, \frac{7\pi}{4})\). 2. Draw the curve of a lemniscate with an inverted orientation with both halves lying in the second and fourth quadrants, making sure it passes through the points we marked. The graph of the lemniscate \(r^{2}=-8 \cos 2 \theta\) is now complete.

Key Concepts

Lemniscate

A lemniscate is a unique and interesting type of curve that resembles the shape of a figure eight. The term "lemniscate" comes from the Latin "lemniscus," meaning ribbon, which is quite fitting given its shape. Lemniscates are often described by equations of the form \(r^2 = a \sin 2\theta\) or \(r^2 = a \cos 2\theta\). These polar equations showcase the inherent symmetry and derive distinct loop shapes:

• The "\(\sin\)" variant houses its loops on the diagonal line, at 45-degree angles to the principal axes.
• Conversely, the "\(\cos\)" variant aligns its loops horizontally or vertically along the axes.

For the specific equation \(r^2 = -8 \cos 2\theta\), it describes a lemniscate with its loops located in the second and fourth quadrants. This negative sign indicates an inverted orientation compared to standard lemniscates. The number "8" in the equation affects the size of the loops. Larger values result in curves that expand further from the origin.

Graphing Polar Equations

When graphing polar equations, like our lemniscate \(r^2 = -8 \cos 2\theta\), it can be helpful to understand the polar coordinate system. In polar coordinates, each point on the plane is determined by an angle \(\theta\) and a radius \(r\). Unlike the Cartesian system, which uses \(x\) and \(y\) values:

• \(\theta\) is the angle from the positive x-axis.
• \(r\) is the distance from the origin to the point.

To graph a polar equation effectively, it's common to calculate specific points by setting \(r = 0\) or solving for \(\theta\). This is crucial to determine where the graph will intersect the polar axes or the lines defined by specific angles. Consider our equation \(r^2 = -8 \cos 2\theta\). By setting \(r = 0\) and solving for \(\theta\), we determine interaction points, helping to plot the general shape and direction of the lemniscate. Once these key points are known, sketching the curve involves connecting them while respecting the symmetrical properties inherent in the polar equation.

Trigonometric Equations

Trigonometric equations often play a key role in polar equations, especially those involving sine and cosine functions. Understanding these functions helps in interpreting how angles and distances relate on a polar graph. In the case of lemniscates:

• The cosine function varies between \(-1\) and \(1\) over its cycle, determining how \(r\) changes as \(\theta\) varies.
• The "2\theta" term indicates that the period of the cosine wave is halved, producing two cycles over \(0\) to \(2\pi\), promoting the lemniscate's distinct loops.
• The amplitude (in our case determined by the absolute value of \(-8\)) influences the size and spread of these loops.

In solving for specific values like \(\theta\) when \(r = 0\), you plug into trigonometric identities or reference angles to find solutions. Notice in this task, using \(\cos 2\theta = 0\) leads to special angles \((n+\frac{1}{2})\frac{\pi}{2}\), where n is an integer, providing the orientation alignment for our lemniscate's axis intersections.