Problem 17 - Precalculus with Limits

Question

In Exercises 13-18, test for symmetry with respect to \(\theta = \pi/2\), the polar axis, and the pole. \(r^2 = 36\ \cos\ 2\theta\)

Step-by-Step Solution

Answer

The polar equation \(r^2 = 36 \cos 2\theta\) is symmetric with respect to the line \(θ = π/2\) and the polar axis, but not with respect to the pole.

Step 1: Testing Symmetry for θ = π/2

To test for symmetry about the line θ = π/2, replace θ with (-θ) in the given equation and check if the new form coincides with the original one. Doing so gives us: \(r^2 = 36 \cos 2(-\theta)\). Since \( \cos -\theta\) = \( \cos \theta\), the equation simplifies to: \(r^2 = 36 \cos 2\theta\), which is the same as the original equation. Therefore, the graph is symmetric about the line θ = π/2.

Step 2: Testing Symmetry for the Polar Axis

To check for symmetry around the polar axis, we replace r with -r in the equation. The equation now becomes: \((-r)^2 = 36 \cos 2\theta\). Simplification yields: \(r^2 = 36 \cos 2\theta\), which is identical to the original equation. Hence, the graph is also symmetric with respect to the polar axis.

Step 3: Testing Symmetry for the Pole

The final test is for symmetry about the pole. This is done by replacing θ by (θ + π) in the equation. We get: \(r^2 = 36 \cos 2(θ + π)\). However, as \( \cos (θ + π) = - cos θ\), the equation becomes: \(r^2 = -36 \cos 2\theta\). Given that this is not similar to the original equation, the graph is not symmetric with respect to the pole.

Key Concepts

Symmetry in Polar Coordinates

Symmetry is a fascinating concept that is essential in understanding the behavior of graphs in polar coordinates.
In the context of polar coordinates, symmetry can be described as the ability of a graph to look the same after specific transformations. There are three common types of symmetry usually tested: symmetry with respect to the polar axis, θ = π/2 (the vertical line through the origin), and the pole itself.
Each of these types of symmetry provides insight into the inherent properties of the polar equation in question and helps simplify the graphing process.

Understanding the Polar Axis

The polar axis in polar coordinates is the equivalent of the horizontal x-axis in Cartesian coordinates.
It serves as a baseline for various transformations and is crucial for determining symmetry.
When we test for symmetry with respect to the polar axis, we are essentially checking whether the equation remains unchanged when the radius, r, is negated.

• If the graph, under the transformation of replacing r with -r, remains unchanged, then the graph is symmetric about the polar axis.
• This indicates that whatever is above the polar axis is mirrored below it.

For example, in our given equation, this symmetry is confirmed since substituting r with -r still resulted in the same equation.

Exploring the Pole

The pole is the origin point in the polar coordinate system, represented by r = 0.
Testing for symmetry around the pole involves replacing θ with θ + π.
This leads to an interesting transformation, as you are checking whether shifting the entire angle by π will still yield the same graph.

• However, a graph symmetric about the pole will often be unchanged if both r and θ are subjected to this transformation.
• In our specific scenario, the equation changes sign, indicating that it is not symmetric about the pole.

Symmetry around the pole can impact how a graph behaves around the origin and influences both the completeness and orientation of the graph.

Graph Symmetry and Its Importance

Graph symmetry provides crucial insights into the structural behavior of polar equations.
Symmetry helps us predict and visualize parts of the graph without needing to plot every single point.
There are key points to consider when analyzing symmetry in graphs:

• If a graph is symmetric about θ = π/2, as it was in the given problem, certain points and their corresponding negative angles will have identical radii.
• Graph symmetry simplifies graphing because you need to graph only a portion and use symmetry to complete the whole.
• Detection of symmetry can also aid in solving polar equations more efficiently by reducing redundancy.

Understanding these symmetries is vital for delving deeper into more complex functions and applications.