Problem 31 - Precalculus

Question

(a) Use one of the polar symmetry tests to show that the graph of \(r=\cos ^{2} \theta-2 \cos \theta\) is symmetric about the \(x\) -axis. (b) Graph the equation given in part (a) and note that the curve is indeed symmetric about the \(x\) -axis.

Step-by-Step Solution

Answer

The graph of \( r = \cos^2(\theta) - 2 \cos(\theta) \) is symmetric about the x-axis, confirmed both analytically and visually through graphing.

Step 1: Identify the Symmetry Test

In this problem, to test for symmetry about the x-axis in polar coordinates, we use the test that involves replacing \( \theta \) with \( -\theta \). We need to check if \( r(\theta) = r(-\theta) \) holds true for the given polar equation.

Step 2: Determine the Polar Equation

The polar equation given is \( r = \cos^2(\theta) - 2 \cos(\theta) \). We need to rewrite any terms using trigonometric identities if necessary, and prepare to substitute \( \theta \) with \( -\theta \).

Step 3: Substitute for Symmetry Test

Substitute \( \theta \) with \( -\theta \) in the equation: \( r(-\theta) = \cos^2(-\theta) - 2 \cos(-\theta) \). Since the cosine function is even, \( \cos(-\theta) = \cos(\theta) \). So, \( r(-\theta) = \cos^2(\theta) - 2 \cos(\theta) \).

Step 4: Verify the Symmetry Test Result

After substitution, compare \( r(-\theta) = \cos^2(\theta) - 2 \cos(\theta) \) with the original equation \( r(\theta) = \cos^2(\theta) - 2 \cos(\theta) \). Since they are the same, the equation is symmetric about the x-axis.

Step 5: Graph the Equation

To graph the equation \( r = \cos^2(\theta) - 2 \cos(\theta) \), use a polar graphing tool and plot the values for \( \theta \) from 0 to \( 2\pi \). Observe the symmetry visually about the x-axis.

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools for simplifying polar equations. These identities involve various relationships between trigonometric functions like sine, cosine, and tangent, among others. For example, the cosine identity states that \( \cos(-\theta) = \cos(\theta) \). This particular identity shows that the cosine function is even, meaning it reflects symmetry across the y-axis. In our original problem, this identity is crucial because it helps simplify the analysis and shows that the polar equation is symmetric about the x-axis when using the symmetry test.Using such identities allows us to rewrite or manipulate terms in a polar equation, thereby easing comparisons and substitutions. By fully understanding and applying these identities, you can tackle a wide range of problems and tests in polar coordinates, much like the one in this exercise.

Polar Symmetry

Polar symmetry involves examining whether a polar graph is symmetric about the line, such as the x-axis, y-axis, or the origin. To determine such symmetry, several tests can be applied based on the coordinates or angle transformations.

• **Symmetry about the x-axis:** Replace \( \theta \) with \( -\theta \). If \( r(\theta) = r(-\theta) \), the graph is symmetric about the x-axis.
• **Symmetry about the y-axis:** Replace \( r \) with \( -r \). If the equation remains unchanged, there is symmetry about the y-axis.
• **Symmetry about the origin:** Replace \( \theta \) with \( \theta + \pi \) in the equation, or equivalently replace \( r \) with \( -r \).

In the given problem, we use the symmetry test for the x-axis, replacing \( \theta \) with \( -\theta \). Confirming that \( r(\theta) = r(-\theta) \), we verify that the equation is indeed symmetric about the x-axis. Understanding these tests helps ensure accurate analysis of polar graphs.

Graphing Polar Equations

Graphing polar equations might seem daunting, but it becomes manageable with practice and the right tools. The key is to understand how angles and radii cooperate to color the graph.When converting polar equations to a graph, we typically compute values of \( r \) for various angles \( \theta \) within the range of \( 0 \) to \( 2\pi \). Using these values, plot points on a polar coordinate plane where each point is determined by \( (r, \theta) \).A polar graphing tool can expedite this process by automatically computing points and visually displaying the result. For equations with identified symmetry, predictions about the graph's appearance can be made more confident, as in our exercise where the graph was symmetric across the x-axis.Understanding the basics of graphing with symmetry tests not only accelerates the plotting process but also deepens comprehension of polar coordinates and their applications.