Problem 12 - Algebra and Trigonometry

Question

Test for symmetry with respect to a. the polar axis. b. the line \(\theta=\frac{\pi}{2}\) c. the pole. $$ r^{2}=16 \sin 2 \theta $$

Step-by-Step Solution

Answer

The given equation \( r^{2}=16 \sin 2\theta \) is not symmetric about the polar axis. It is symmetric with respect to the line \( \theta = \frac{\pi}{2} \) and the pole.

Step 1: Test for Symmetry with respect to the Polar Axis

To test for symmetry with respect to the polar axis, replace \( \theta \) with \( -\theta \) in the equation. If the equation remains unchanged, then it's symmetric with respect to the polar axis. The equation becomes \( r^{2}=16 \sin 2(-\theta) = r^{2}=-16\sin 2\theta\). The equation differs from the original one, so it is not symmetric with respect to the polar axis.

Step 2: Test for Symmetry with respect to the Line \( \theta = \frac{\pi}{2} \)

To test for symmetry with respect to the line \( \theta = \frac{\pi}{2} \), replace \( \theta \) with \( \pi - \theta \) in the equation. If the equation remains unchanged, then it's symmetric with respect to \( \theta = \frac{\pi}{2} \). The equation becomes \( r^{2}=16 \sin 2(\pi - \theta) = r^{2}=16 \sin (2\pi - 2\theta)\). Simplifying the sine function gives the original equation \( r^{2}=16 \sin 2\theta\). Hence, the graph is symmetric with respect to \( \theta = \frac{\pi}{2} \).

Step 3: Test for Symmetry with respect to the Pole

To test for symmetry with respect to the pole, replace \( r \) with \( -r \) in the equation. If the equation remains unchanged, then it's symmetric with respect to the pole. However, in this exercise, replacing \( r \) with \( -r \) doesn't apply because there are no \( r \) terms on their own in the equation, just \( r^{2} \), and squaring \( r \) or \( -r \) yields the same result. So, the equation is already symmetric with respect to the pole by definition.

Key Concepts

Polar Coordinates

Polar coordinates offer a different way to describe points on a plane. Instead of using x and y as in Cartesian coordinates, polar coordinates use a radius \(r\) and an angle \(\theta\) measured from the positive x-axis.
Understanding polar coordinates can help visualize more complex equations, especially those involving circles and spirals.
For example:

• \(r\) describes how far the point is from the origin, or the pole.
• \(\theta\) measures the direction of the point from the positive x-axis.

This system is particularly helpful in solving problems related to symmetry, as certain transformations in polar coordinates are simpler than their Cartesian counterparts.

Sine Function

The sine function plays a critical role in trigonometry and directly impacts polar coordinates. When working with equations like \(r^2 = 16 \sin 2\theta\), it's important to understand how the sine function affects the graph.
Sine has specific properties:

• Its values range from -1 to 1.
• Sine is periodic, repeating every \(2\pi\).
• Sine is an odd function, meaning \(\sin(-\theta) = -\sin \theta\).

These characteristics influence the symmetry in polar graphs and are essential when performing symmetry tests, as certain transformations may or may not affect the equation.

Symmetry Testing

Symmetry testing helps identify whether a graph remains unchanged under specific transformations. In polar coordinates, there are three common symmetry tests:

• Polar Axis Symmetry: Replace \(\theta\) with \(-\theta\). If the equation remains the same, the graph is symmetric about the polar axis.
• \(\theta = \frac{\pi}{2}\) Symmetry: Replace \(\theta\) with \(\pi - \theta\). If unchanged, the graph is symmetric with respect to \(\theta = \frac{\pi}{2}\).
• Pole Symmetry: Replace \(r\) with \(-r\). If the equation remains unchanged, the graph is symmetric with respect to the pole.

By performing these transformations, we can determine symmetry properties, which simplifies understanding and solving equations in polar coordinates.

Mathematical Equations

Mathematical equations like \(r^2 = 16 \sin 2\theta\) require understanding both their form and transformations for correct analysis. In polar coordinates, equations describe curves or loci of points.
Understanding the form of equations helps in:

• Identifying symmetry and behavior of the graph.
• Predicting how transformations like reflection or rotation affect the equation.
• Simplifying complex equations for easier manipulation.

The given equation showcases how trigonometric functions integrate with algebraic expressions, resulting in curves that are visually appealing and mathematically significant.