Problem 17 - Precalculus

Question

Test for symmetry with respect to the line \(\theta=\pi 2,\) the polar axis, and the pole. $$r^{2}=36 \cos 2 \theta$$

Step-by-Step Solution

Answer

The polar coordinate equation \(r^{2}=36 \cos 2 \theta\) is symmetric with respect to the line \(\theta = \frac{\pi}{2}\) and the polar axis, but not symmetric with respect to the pole.

Step 1: Test symmetry with respect to \(\theta = \frac{\pi}{2}\)

Replace \(\theta\) with \(-\theta\) in the given equation, yielding \(r^2 = 36 \cos 2(-\theta)\). That simplifies to \(r^2 = 36 \cos 2\theta\), which is the same as the original equation. Thus, the graph is symmetric about the line \(\theta = \frac{\pi}{2}\).

Step 2: Test symmetry with respect to the polar axis

Replace \(r\) with \(-r\) in the equation to yield \((-r)^2 = 36 \cos 2\theta\). This simplifies to \(r^2 = 36 \cos 2\theta\), which is again the same as the initial equation. It shows that the graph is symmetric with respect to the polar axis.

Step 3: Test symmetry with respect to the pole

In the original equation, replace \(\theta\) with \(\theta + \pi\). We get \(r^{2} = 36 \cos 2(\theta + \pi)\), which simplifies to \(r^{2} = 36 \cos (4\theta + 2\pi)\), meaning the equation has changed and isn't the same as the original one. The graph isn't symmetric about the pole.

Key Concepts

Polar Graph Symmetry

Understanding symmetry in polar graphs is crucial for visualizing and analyzing complex equations. Imagine folding a polar graph along a line of symmetry; both sides match exactly. To test for symmetry in polar coordinates, mathematicians have developed specific methods centered around key axes and lines.

For the line \(\theta = \frac{\pi}{2}\), if replacing \(\theta\) with \( -\theta\) in the equation results in an equivalent equation, this indicates symmetry about that line. In our exercise, by subsituting \(\theta\) with \( -\theta\) in \(r^{2} = 36 \cos 2 \theta\), the equation remains unchanged, thus proving the presence of this type of symmetry. This tells us that the graph will mirror across the line \(\theta = \frac{\pi}{2}\), providing a useful clue about the graph's overall shape and structure.

Cosine Function in Polar Equations

The cosine function in polar coordinates provides a direct path to understanding radial symmetry. Specifically, cosine's even property, where \(\cos(-\theta) = \cos(\theta)\), plays a major role in determining symmetry.

In the given expression \(r^{2}=36 \cos 2 \theta\), the presence of the cosine function allows us to apply this even property. During Step 1 of the solution process, we used this property to test for symmetry across \(\theta = \frac{\pi}{2}\) and found the graph to be symmetric. This example demonstrates the importance of recognizing the properties of trigonometric functions, such as cosine, which can simplify the process of identifying symmetries in polar equations.

Polar Axis Symmetry

Polar axis symmetry involves flipping the graph across the polar axis, essentially reversing the direction of radial lines while leaving the angle \(\theta\) unchanged. To test for this, the value of \(r\) is replaced with \( -r\) in the equation.

In our exercise's Step 2, when \(r\) was replaced with \( -r\), the equation \(r^{2} = 36 \cos 2\theta\) remained unchanged, confirming that the graph is symmetric with respect to the polar axis. This means that for every point with coordinates \(r,\theta\), there exists a counterpart at \( -r,\theta\) within the graph. This concept helps in predicting the behavior of the graph without needing to plot all points and assists in understanding the underlying geometry of polar equations.