Problem 55 - Algebra and Trigonometry

Question

Identify and graph each polar equation. $$ r^{2}=9 \cos (2 \theta) $$

Step-by-Step Solution

Answer

This polar equation graphs a lemniscate with maximum \( r = 3\).

Step 1: Identify the Type of Polar Equation

The given polar equation is \(r^2 = 9 \cos(2\theta)\). Recognize that this equation involves \(r^2\) and \(\cos(2\theta)\), which is a characteristic of limacons.

Step 2: Relate to Known Polar Forms

Know that the general form for a limacon with a squared radius is \(r^2 = a \cos(2\theta)\), where \(a \) is a constant. Here, \(a = 9\).

Step 3: Find Key Features

Understand the properties of this limacon. The maximum value of \(r^2\) is \(9\), and the corresponding \(r\) value is \( r = \/9 = 3\). The graph will have symmetry about the x-axis.

Step 4: Plotting Points

Calculate values for \(r\) at specific angles. When \(\theta = 0\), \( r^2 = 9 \cos(0) = 9\), so \(r = \/9 = 3\). When \(\theta = \pi/2 \,\), \( r^2 = 9 \cos(\pi) = -9\), so \(r = \pm i3\).

Step 5: Sketch the Graph

Use the values from Step 4 to sketch the graph. Start with the main points \(r = 3\) at \(\theta = 0\) and further points generated symmetrically. The graph will show a lemniscate shape.

Key Concepts

limacons

Limacons are a specific type of polar graph. They are characterized by their unique shapes, which can look like a heart, a dimpled heart, or even pass through the origin like a loop. Limacons can be described by the general form of the equation, either involving a cosine or sine function. For instance, the polar equation with cosine as given, is usually in the form of \[r = a + b\text{cos}(\theta)\] or \[r = b \text{cos}(2\theta)\]. A limacon’s shape depends heavily on the relationship between the constants 'a' and 'b'. When you see squared radius \(r^2\) with cosine or sine of double the angle \(2\theta\), you are usually looking at a special case of a limacon that will end up forming specific and interesting shapes like the lemniscate. Understanding limacons will help you categorize and graph these equations more easily.

polar coordinates graphing

Polar coordinates graphing helps in visualizing equations described using the distance from the origin and an angle. In polar coordinates, any point can be represented as \((r, \theta)\), where \(r\) is the radius (distance from the origin), and \(\theta\) is the angle from the positive x-axis. To graph a polar equation, you calculate the radius for various values of the angle. For example, in the exercise, the given equation \(r^2 = 9 \text{cos} (2\theta)\), you calculate the radius \(r\) using values of \(\theta\). These calculated points \((r, \theta)\) are then plotted on the polar graph, taking care to respect the symmetry properties of the equation (such as active angles like 0, \(\pi/2\), \(\pi\), etc.), aiding in creating an accurate graph. By systematically calculating and plotting points, you develop a visual representation of the polar equation.

trigonometric identities

Trigonometric identities are mathematical truths involving trigonometric functions. They are invaluable for transforming and simplifying equations. For instance, in polar equations, the identity \( \text{cos}(2\theta) = 2 \text{cos}^2(\theta) - 1 \) or \( \text{cos}(2\theta) = 1 - 2 \text{sin}^2(\theta) \) helps solve or transform the given equation. Recognizing these identities is crucial for converting complex polar equations into more familiar forms or for computing trigonometric values easily. Simplifying using identities makes it easier to plot points and understand the graph's behavior. Remember, these identities are tools to manipulate and simplify your equations, leading to a more straightforward graphing process.

lemniscate

The lemniscate is a specific type of polar graph that looks like a figure-eight or an infinity symbol. This shape can appear when dealing with equations involving \(r^2\) and double angles (like \(r^2 = 9 \text{cos} (2\theta)\)). The general forms of lemniscate equations are \(r^2 = a^2 \text{cos}(2 \theta) \) or \( r^2 = a^2 \text{sin}(2 \theta) \). Graphing a lemniscate involves identifying key symmetrical points. For the given exercise, when \( \theta = 0\), \( r = 3 \), and as the angles change, you can compute \( r \) values accordingly. By plotting these points and using symmetry, the distinctive lemniscate shape emerges on the polar coordinate graph. This specific shape is important in various fields of mathematics, most notably in complex analysis and algebraic geometry. Understanding the lemniscate helps in mastering the sketching and properties of such intricate graphs.