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 graph of the polar equation given by the lemniscate: \(r^2 = -8\cos(2\theta)\) and provide insights about its properties. Answer: The given lemniscate \(r^2 = -8\cos(2\theta)\) has a shape that repeats itself every \(180^\circ\). \(r^2\) is positive when \(\cos(2\theta)\) is negative, i.e., between \(90^\circ\) and \(270^\circ\) for \(2\theta\). The zeros occur at \(\theta = 45^\circ\) and \(\theta = 135^\circ\). The graph is symmetrical about the polar axis and crosses itself at the origin. The size of the lemniscate is controlled by the coefficient of the cosine function.

Step-by-step solution

Understand the lemniscate

Recall that the lemniscate formula we have is given by \(r^2 = a \cos(2\theta)\), where \(a\) is the constant that controls the size of the figure. In our case, the lemniscate is defined as \(r^2 = -8\cos(2\theta)\). Since the cosine is a periodic function with a period of \(360^\circ\), the lemniscate's shape will repeat itself every \(180^\circ\).

Analyze when \(r^2\) is positive

To plot the curve, we first need to find the values of \(\theta\) when the equation \(r^2 = -8\cos(2\theta)\) results in a positive value (since \(r^2\) can't be negative). The cosine function's range is between \(-1\) and \(1\). Therefore, \(r^2\) will be positive when \(\cos(2\theta)\) is negative, corresponding to \(2\theta\) values between \(90^\circ\) and \(270^\circ\) or between \(450^\circ\) and \(630^\circ\). However, we can limit our analysis to a range of \(0\) to \(360^\circ\) for \(\theta\), as the patterns repeat themselves.

Calculate the zeros

To graph the lemniscate, we need to find the zeros of the equation \(r^2 = -8\cos(2\theta)\). The zeros occur when \(r=0\), which implies \(\cos(2\theta) = 0\). This happens when \(2\theta = 90^\circ, 270^\circ, 450^\circ, \dots\). Hence, for the range of \(0^\circ\) to \(360^\circ\) of \(\theta\), we have two zeros, one at \(\theta = 45^\circ\) and another at \(\theta = 135^\circ\).

Plot the lemniscate

Now that we have analyzed the given lemniscate's main properties, we can sketch its graph in a polar coordinate system. First, draw the polar axis and label the angle \(\theta\) starting from the polar axis. Next, plot the curve, keeping in mind the range of \(r^2\) positive values and the zeros found earlier. The lemniscate will be symmetrical about the polar axis and cross itself at the origin. Ensure to draw the loops of the lemniscate where \(r^2\) is positive. In conclusion, after following the above steps, we will have a graph that represents the lemniscate given by \(r^2 = -8\cos(2\theta)\). It's important to notice that the shape and behavior of the lemniscate will depend on the coefficient of the cosine function and the periodical repetition of its structure.

Key concepts

Polar Coordinates

Polar coordinates provide a unique way to represent points on a plane using a distance from a reference point and an angle from a reference direction. Unlike Cartesian coordinates, which use an X and Y axis, polar coordinates use:

• \(r\): the radial distance from the origin, or pole.
• \(\theta\): the angle measured counterclockwise from the positive x-axis (the polar axis).

This system is especially useful for graphing complex curves such as the lemniscate. The values of \(r\) can be both positive and negative, indicating direction from the pole. An important concept is that angles are periodic, generally rotating every \(360^\circ\) or \(2\pi\) radians, allowing for repetitive patterns and shapes like circles and lemniscates.

Trigonometric Functions

Trigonometric functions such as sine and cosine are fundamental in polar coordinates. They relate the angle \(\theta\) to ratios of sides in a right triangle and are periodic functions.
For example, the cosine function, used in the given lemniscate equation \(r^2 = a \cos(2\theta)\), has a range between -1 and 1 and repeats every \(360^\circ\). The function's periodic nature means it can model cyclic phenomena perfectly.
In the equation \(r^2 = -8\cos(2\theta)\), the presence of cosine indicates symmetry along the x-axis. When \(\cos(2\theta)\) is negative, \(r^2\) becomes positive because it is multiplied by a negative number, showing when the lemniscate exists.

Graphing Polar Equations

Graphing polar equations involves plotting points by calculating their \((r, \theta)\) values. For the lemniscate, \(r^2 = -8\cos(2\theta)\), several steps are required.
First, establish when \(r^2\) is positive — when \(\cos(2\theta)\) is negative in this case, specifically between certain angles. These values dictate where the curve will exist. Then, determine zeros where \(r = 0\) by solving \(\cos(2\theta) = 0\), indicating points where curves intersect the pole.
Finally, sketch the curves and ensure to follow the symmetry and zero patterns. This graph will reveal a looped figure, symmetric about the origin, known as the lemniscate, recognizable for its figure-eight shape. Understanding these trigonometric influences helps in visually mapping the equation onto a polar plane.