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: Graph the lemniscate given by the polar equation \(r^{2}=4 \sin 2 \theta\) and describe its key features. Answer: The graph of the lemniscate described by the polar equation \(r^{2}=4 \sin 2 \theta\) is a figure-eight shaped curve that is symmetrical about the origin and the polar axis. The lemniscate intersects the polar axis at \((0,0)\), \((\frac{\pi}{2}, 0)\), \((\pi,0)\), and \((\frac{3\pi}{2},0)\). The loop sizes occur at the points \((\frac{\pi}{4}, 2)\) and \((\frac{5\pi}{4}, 2)\).

Step-by-step solution

Identify the equation type and properties

Our given equation is \(r^{2}=4 \sin 2 \theta\), which is of the form \(r^{2}=a \sin 2 \theta\), where \(a=4\). This type of equation represents a lemniscate. A lemniscate is a figure-eight shaped curve, and it is symmetrical about the origin and with respect to the polar axis. In this case, the orientation of the lemniscate depends on the trigonometric function in the equation. Since our equation uses \(\sin\), the lemniscate is expected to be symmetrical with respect to the polar axis.

Find the points for the graph

Now we will find the coordinates of the key points on the lemniscate so that we can graph it accurately. We need the locations of the points where the lemniscate intersects the polar axis and the loop sizes. For our equation, we can use the properties of lemniscates to find these points. For the polar axis intersections: We set \(r^{2}=0\). This means \(\sin 2 \theta = 0\), which implies \(2 \theta = n\pi\) for any integer \(n\). So, we find the first few polar angles: \(\theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}\). For the loop sizes: We set \(\sin 2 \theta = 1\) (maximum value). So, we have \(2 \theta = \frac{\pi}{2} + 2n\pi\) Then, we find the first few polar angles: \(\theta = \frac{\pi}{4}, \frac{5\pi}{4}\). With these angles, we can find the corresponding radii by plugging the values of \(\theta\) into our equation: \(r^{2}=4 \sin 2 \theta\).

Choose an appropriate graphing tool

In order to graph the lemniscate, we could use graphing software that has support for polar coordinates, such as Desmos or GeoGebra. Alternatively, we could graph it by hand, using polar graph paper and a protractor for precision.

Plot the points and complete the graph

Now, we will plot the points we found in Step 2 onto the polar axis and complete the graph of the lemniscate. 1. Plot the polar axis intersections: \((0,0)\), \((\frac{\pi}{2}, 0)\), \((\pi,0)\), \((\frac{3\pi}{2},0)\). 2. Plot the points for the loop sizes: \((\frac{\pi}{4}, 2)\), \((\frac{5\pi}{4}, 2)\). 3. Sketch the lemniscate by connecting the plotted points, remembering its symmetrical properties. Once the points are plotted and the lemniscate is sketched, our graph for the equation \(r^{2}=4\sin 2 \theta\) should be complete.

Key concepts

Polar Coordinates

Understanding polar coordinates is crucial for graphing equations that result in figures like lemniscates. Within the polar coordinate system, every point on a plane is defined by two values: the radial distance from the origin, denoted as r, and an angle θ from the reference direction (usually the positive x-axis). Unlike Cartesian coordinates, which use a grid based on perpendicular lines, polar coordinates are based on concentric circles and radial lines emanating from a central point.

In the context of graphing a lemniscate, the value of r can change depending on the angle θ. For instance, when r equals 0, the lemniscate intersects the polar axis. Recognizing these unique points is essential to graphing the shape accurately. A lemniscate, with its figure-eight form, will have various points of intersection and maximum distances from the origin (or 'loop sizes') that are determined by specific angles. Navigating these nuances requires a solid grasp of polar coordinates.

Polar Equations

Polar equations express the relationship between the radius r and the angle θ. A lemniscate, a unique curve resembling the number eight or an infinity symbol, can be described by polar equations like r^2 = a sin 2θ or r^2 = a cos 2θ, where a is a positive real number that affects the size of the lemniscate.

In the example equation r^2 = 4 sin 2θ, understanding the specific traits of a lemniscate's polar equation is essential. When graphing, the symmetry about the origin (polar axis) simplifies the process, as it is only necessary to calculate a quarter of the lemniscate's points -- the rest follows due to symmetry. Moreover, key points, such as loop sizes and axis intersections, are found by setting sin 2θ to its minimum and maximum values. Recognizing how these equations shape the curve helps in quickly graphing accurately, whether it is through technology or by hand on polar graph paper.

Trigonometric Functions

In the realm of mathematics, trigonometric functions like sine and cosine are fundamental in describing the properties of angles and their relationships to a circle. In polar equations that define lemniscates, we often encounter these functions, specifically in forms such as sin 2θ or cos 2θ. These compound trigonometric functions exhibit periodic behavior and determine the points where the shape expands and contracts.

Having a firm grasp of how these functions behave is imperative when graphing a lemniscate. For instance, in the equation r^2 = 4 sin 2θ, the maximum and minimum values of the sine function dictate the size and position of the loops. By knowing that sine reaches its peak at π/2 and its lowest point at 3π/2, we can find critical points that construct the lemniscate. This demonstration of the sine function's influence in polar equations illustrates why a deep understanding of trigonometric functions is essential for students tackling such graphing exercises.