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

Short answer

Question: Sketch the lemniscate given by the polar equation \(r^2 = -8\cos(2\theta)\) and describe its behavior in the first and second quadrants. Answer: The lemniscate has a point of intersection with the polar axis at \((2, 2)\) in rectangular coordinates. In the first quadrant, the lemniscate doesn't exist for \(\theta\) in the interval \((0, \pi/4)\) and approaches the point \((2, 2)\) as \(\theta\) increases from \(\pi/4\) to \(\pi/2\). The graph in the second quadrant is a reflection of the graph in the first quadrant across the polar axis.

Step-by-step solution

Find the points of intersection with the polar axis

Plug in \(\theta=0\) and \(\theta=\pi/2\) into the equation \(r^2=-8\cos(2\theta)\) to find the intersection points with the polar axis. For \(\theta=0\): $$r^2=-8\cos(2(0))=-8\cos(0)=-8(1)=-8$$ Since \(r^2\) cannot be negative, there are no intersection points with the polar axis at \(\theta=0\). For \(\theta=\pi/2\): $$r^2=-8\cos(2(\pi/2))=-8\cos(\pi)=-8(-1)=8$$ So, there is an intersection point at \((\sqrt{8},\pi/2)\), which corresponds to the point \((2,2)\) in rectangular coordinates.

Analyze behavior in the first and second quadrants

Observe that the graph is symmetric with respect to the polar axis because \(r^2\) is an even function of \(\theta\). Therefore, we only need to consider the behavior in the first quadrant, and we can reflect the resulting graph across the polar axis to obtain the graph in the second quadrant. In the first quadrant, start with \(\theta=0\) and increase \(\theta\) towards \(\pi/4\), and then from \(\pi/4\) to \(\pi/2\). As \(\theta\) increases from \(0\) to \(\pi/4\), we find that \(r^2\) becomes more negative, which means that \(r\) is imaginary, and thus the lemniscate doesn't exist in this interval. As \(\theta\) increases from \(\pi/4\) to \(\pi/2\), \(r^2\) becomes less negative, approaching zero as \(\theta\) approaches \(\pi/2\), and therefore, the lemniscate approaches the point \((2,2)\) in rectangular coordinates.

Sketch the graph

Start by plotting the point of intersection \((2,2)\) with the polar axis in rectangular coordinates. Sketch a curve in the first quadrant that approaches \((2,2)\) as \(\theta\) increases from \(\pi/4\) to \(\pi/2\), and doesn't exist for \(\theta\) in the interval \((0,\pi/4)\). Reflect the curve across the polar axis to obtain the graph in the second quadrant. The resulting graph represents the lemniscate described by the polar equation \(r^2=-8\cos(2\theta)\).

Key concepts

Lemniscate

The lemniscate is a fascinating mathematical shape that resembles a figure-eight. In polar coordinates, lemniscates are usually described by the equations \(r^{2}=a \sin 2 \theta\) or \(r^{2}=a \cos 2 \theta\). These formulas define the curves as the angle \(\theta\) changes, with the parameter \(a\) affecting the size of the lemniscate.

In our specific example, the equation is \(r^{2}=-8 \cos 2 \theta\). Here, the negative coefficient indicates that the curve is mirrored or flipped compared to a standard lemniscate where the coefficient is positive. Understanding how the equation's terms affect the graph is crucial for sketching the lemniscate accurately.

When plotted, the lemniscate will appear symmetric and balanced around the polar axis, displaying two loops. The visualization of such a curve strengthens the understanding of polar equations and the intriguing beauty of mathematical graphs.

Trigonometric Graphing

Creating graphs from trigonometric functions like sine and cosine in polar coordinates can be both challenging and fun. The graphing of the equation \(r^2 = -8 \cos 2\theta\) involved utilizing symmetry and identifying crucial points derived from zero crossings.

In this case, by plugging different values of \(\theta\) such as \(\theta = 0\) and \(\theta = \pi/2\), you can determine essential intersection points. These values help locate where the graph enters or exits certain axes. For the lemniscate, much like in cartesian graphing, symmetry can be employed to simplify the process by reducing the area and angles that need to be calculated independently.

Another critical aspect of trigonometric graphing in polar coordinates involves recognizing the periods over which the trigonometric components create patterns. Since the function involves \(2\theta\), it effectively doubles the number of cycles in your usual trigonometric function, providing insightful experimentation with angles in terms of graph shape and direction.

Polar Axis Symmetry

Symmetry, particularly polar axis symmetry, plays a critical role in simplifying the graphing of certain polar equations like the lemniscate. Polar axis symmetry implies that for every point \((r, \theta)\) on the graph, there exists a corresponding point \((-r, -\theta)\), effectively mirroring a section across the polar axis.

In our example of \(r^2 = -8 \cos 2\theta\), leveraging symmetry allows you to only calculate and sketch one-half of the lemniscate, then reflect this to complete the other half. By understanding this principle, you can significantly reduce the complexity involved in plotting such intricate graphs.

Symmetry ensures that time spent on complex calculations is minimized, allowing for a more efficient approach to visualizing the mathematical constructs and creating aesthetically pleasing graphs. It highlights how mathematics can offer elegant solutions through clever techniques and properties of the functions involved.