Problem 22 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

The graph of \(r^2 = 36 \sin(2\theta)\) is symmetric with respect to the line \(\theta = \pi/ 2\), the polar axis, and the pole.

Step 1: Symmetry about the line \(\theta = \pi/ 2\)

We replace \(\theta\) by \(-\theta\). The equation becomes \(r^2 = 36 \sin(-2\theta) = 36 \sin(2\theta)\), which is the original equation. Hence, it is symmetric about the line \(\theta = \pi/ 2\).

Step 2: Symmetry about the polar axis

To test for symmetry about the polar axis, we replace \(r\) with \(-r\) and \(\theta\) with \(\theta + \pi\), which gives \((-r)^2 = 36 \sin[2(\theta + \pi)]\). Upon simplification, \((-r)^2 = r^2 = 36 \sin(2\theta + 2\pi) = 36 \sin(2\theta)\), we get the original equation. This shows that the graph is symmetric about the polar axis.

Step 3: Symmetry about the pole

We replace \(r\) with \(-r\) and we get \((-r)^2 = r^2 = 36 \sin(2\theta)\). This is the same as our original equation, hence, the graph is symmetric about the pole.

Key Concepts

Symmetry Tests in Polar Coordinates

Understanding symmetry in polar coordinates is essential for analyzing curves and shapes. In polar coordinates, symmetry can be tested with respect to the polar axis, the line \( \theta = \pi / 2 \), and the pole.

The key is to make a few simple substitutions. For the polar axis, you replace \( r \) with \( -r \) and \( \theta \) with \( \theta + \pi \). If the equation remains unchanged, it indicates symmetry around the polar axis. For the line \( \theta = \pi / 2 \), you change \( \theta \) to \( -\theta \). Again, if the equation is unaffected, it signifies symmetry. Lastly, symmetry about the pole only requires changing \( r \) to \( -r \), and no change in the equation indicates symmetry around the pole.

When trying to grasp how these substitutions work, visualize flipping a curve over the axis or line. If the curve fits perfectly over its original position, you have symmetry. This understanding helps students to quickly analyze graphs and predict their behavior under these symmetry operations.

Polar Axis Symmetry

Polar axis symmetry is observed when a polar equation remains unchanged after the radial distance, \( r \), is replaced by \( -r \), and the angle, \( \theta \), by \( \theta + \pi \).

In terms of a graphical interpretation, a curve is symmetric about the polar axis if, for every point on the curve with coordinates \( (r, \theta) \), there is also a point \( (-r, \theta + \pi) \). This essentially means that the curve can be 'folded' along the polar axis (the positive x-axis in Cartesian coordinates), and both halves would overlap perfectly.

As in our original exercise, the equation for symmetry about the polar axis maintained its form after application of the symmetry transformations, indicating that the curve for \( r^2 = 36 \sin 2\theta \) is indeed symmetric about the polar axis.

Pole Symmetry

Pole symmetry pertains to a polar graph's invariance when the radial coordinate \( r \) is replaced by \( -r \). In simpler terms, it reflects whether a polar graph looks the same when observed from the 'opposite' direction from the pole (the origin).

This transformation does not alter the angle, only the direction along the angle's line is reversed. If the graph does not change its equation, as seen in our exercise where replacing \( r \) with \( -r \) yielded the original \( r^2 = 36 \sin 2\theta \), the graph exhibits pole symmetry.

Pole symmetry is particularly relevant when analyzing patterns that revolve around the origin, such as certain flowers or sea shells that exhibit this type of symmetry naturally.

Trigonometric Function Transformation

Trigonometric function transformation in the context of polar coordinates is crucial to understanding how polar equations behave under symmetry operations.

In trigonometric terms, certain identities, such as \( \sin(-\theta) = -\sin(\theta) \) and \( \sin(\theta + 2\pi) = \sin(\theta) \), facilitate these symmetry tests. For example, the above transformations show why the equation \( r^2 = 36 \sin 2\theta \) in our initial problem remains unchanged under symmetry conditions.

It's beneficial to remember common trigonometric identities when working with polar equations, as they can often simplify the process of determining symmetry, as seen with the \( \sin \) function in the given exercise. These transformations allow students to quickly evaluate the equation without the need for plotting the graph, offering a powerful tool in their mathematical toolkit.