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

The given equation, \(r^2 = 4\sin 2\theta\), is a lemniscate. To graph it, we calculate and plot the values of \(r\) for various angles of \(\theta\). A few of the points we find include (0, 0), \(\left(\dfrac{\pi}{4}, 2\right)\), \(\left(\dfrac{\pi}{4}, -2\right)\), and \(\left(\dfrac{\pi}{2}, 0\right)\). After plotting the points and connecting them with a smooth curve, we obtain a figure-eight-shaped graph.

Step-by-step solution

Verify the given equation is in the form of a lemniscate

The given equation is \(r^2 = 4\sin 2\theta\). Comparing it with the general form of a lemniscate, \(r^2 = a\sin 2\theta\) or \(r^2 = a\cos 2\theta\), we can see it is in the correct form, with \(a = 4\).

Find the values of \(r\) at different angles of \(\theta\)

To graph the lemniscate, we need to find the values of \(r\) for several points on the curve. We can choose values of \(\theta\) in increments of \(\dfrac{\pi}{4}\) and calculate the corresponding values of \(r\). Remember that since the equation is given in polar coordinates, we have to convert our values of \(\theta\) from degrees to radians if needed. For instance, we can start by choosing \(\theta = 0, \dfrac{\pi}{4}, \dfrac{\pi}{2},\ldots\) and then find the corresponding values of \(r\) using the equation: - When \(\theta = 0\): \(r^2 = 4\sin 2(0) = 4\sin 0 = 0\), so \(r = 0\). - When \(\theta = \dfrac{\pi}{4}\): \(r^2 = 4\sin 2\left(\dfrac{\pi}{4}\right) = 4\sin \dfrac{\pi}{2} = 4\), so \(r = \pm 2\). - When \(\theta = \dfrac{\pi}{2}\): \(r^2 = 4\sin 2\left(\dfrac{\pi}{2}\right) = 4\sin \pi = 0\), so \(r = 0\). - When \(\theta = \dfrac{3\pi}{4}\): \(r^2 = 4\sin 2\left(\dfrac{3\pi}{4}\right) = 4\sin \dfrac{3\pi}{2} = -4\), so \(r^2\) cannot be negative, no point for this value of \(\theta\). Do this for other values of \(\theta\) up to \(2\pi\) to have a sufficient number of points to plot.

Plot the points on a polar coordinate plane

Using the points we calculated in step 2, plot them on a polar coordinate plane. For example, with \(\theta = 0\) and \(r = 0\), our point is at the origin. With \(\theta = \dfrac{\pi}{4}\) and \(r = \pm 2\), we have two points, one at a distance of 2 from the origin in the direction of \(\dfrac{\pi}{4}\) and another at a distance of 2 in the direction of \(\dfrac{5\pi}{4}\). Continue plotting the calculated points in this manner. Connect the points with a smooth curve to form the figure-eight-shaped lemniscate. The graph of the given equation, \(r^2 = 4\sin 2\theta\), is a lemniscate.

Key concepts

Lemniscate

A lemniscate is a special type of curve resembling a sideways figure-eight. It is distinguished in polar coordinates through specific equations that define its shape. Consider equations like \(r^2 = a \sin 2\theta\) or \(r^2 = a \cos 2\theta\). These indicate we are dealing with a lemniscate, with variations in the curve's orientation and dimensions.
Understanding the variable \(a\) is crucial. It signifies the size of the lemniscate. For instance, in the equation \(r^2=4 \sin 2\theta\), \(a=4\), suggesting that the lemniscate will expand wider, with loops spread over a farther distance from the origin.
Knowing the behavior of sine and cosine functions helps predict the lemniscate's orientation. A sine-based lemniscate, like the one we've been examining, tends to rotate along diagonal axes, whereas cosine-based lemniscates align more horizontally or vertically. Recognizing these patterns allows easier graphing and comprehension of polar coordinates.

Graphing Polar Equations

To successfully graph polar equations, such as a lemniscate, it's helpful to follow a systematic approach. Here are some key steps:

• Identify the form of the polar equation, ensuring it matches one of the standard lemniscate equations \(r^2 = a \sin 2\theta\) or \(r^2 = a \cos 2\theta\).
• Choose a range of \(\theta\) values to plot. Opt for increments that will capture important points of the graph, such as \(\theta = 0\), \(\frac{\pi}{4}\), \(\frac{\pi}{2}\), up to \(2\pi\).
• Calculate the corresponding \(r\) values using the given equation. Note that \(r^2\) cannot be negative, which affects the feasibility of certain points.

Using our example, for \(\theta = \frac{\pi}{4}\), the equation gives \(r^2=4\), hence \(r = \pm 2\). Similarly evaluate different \(\theta\) values to get various points.
When plotting, translate each \(r, \theta\) pair onto the polar coordinate system, marking points accordingly. Then gently connect them with a curve, revealing the lemniscate's shape. Polar graphing effectively showcases symmetries and behaviors distinct to polar forms.

Trigonometric Identities

Trigonometric identities play a crucial role in simplifying and precisely interpreting polar equations, like those representing lemniscates. They allow transformations in equations by revealing relationships between angles and their trigonometric functions.
A vital identity, \(\sin 2\theta = 2 \sin\theta \cos\theta\), helps in rewriting and solving equations. Understanding how sin and cos interact can simplify real-world calculations, especially when converting polar equations to better understand or visualize graphs.
For lemniscate equations, such as \(r^2 = 4 \sin 2\theta\), be conscious of how trigonometric identities indicate symmetry and cyclic behavior. They suggest points where curves cross or overlap, guiding plotting and interpretation. Mastery of trigonometric identities enhances skills in analyzing and graphing polar coordinates and patterns efficiently.