Problem 10 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The polar equation \(r=2 \cos 2\theta\) is symmetric with respect to the polar axis, the line \(\theta=\frac{\pi}{2}\), and the pole.

Step 1: Symmetry about Polar Axis

To test for symmetry about the polar axis, replace \(\theta\) with \(-\theta\) and simplify the equation. If it stays the same as the original equation, then the graph is symmetric about the polar axis. Here, the modified equation becomes \(r=2 \cos -2\theta\), which further simplifies to \(r=2 \cos 2\theta\). This is the original equation, thus concluding that the equation is symmetric with respect to the polar axis.

Step 2: Symmetry about the line \(\theta=\frac{\pi}{2}\)

To test for symmetry about \(\theta=\frac{\pi}{2}\), replace \(\theta\) with \(-\theta + \pi\) and simplify. Therefore, the modified equation becomes \(r=2 \cos 2(-\theta + \pi)\) = \(r=2 \cos -2\theta + 2\pi\) which equals \(r=2 \cos 2\theta\). This matches the original polar equation, indicating symmetry about \(\theta=\frac{\pi}{2}\).

Step 3: Symmetry about the Pole

Symmetry around the pole is tested by replacing \(r\) with \(-r\). We replace \(r\) with \(-r\) and simplify. Here, the equation does not get impacted and stays to be \(r=2 \cos 2\theta\). The polar equation is the same as the original equation, which implies a symmetry about the pole.

Key Concepts

Polar Coordinates

Polar coordinates, unlike Cartesian coordinates, provide a way to represent points in a plane using a radius and an angle. Instead of using

• x-values and y-values, we work with a distance from the origin, labeled as r, and an angle θ, which is measured from the positive x-axis.

This method is particularly useful for equations and graphs that exhibit circular or rotational properties.
For example, the point (3, \( \frac{\pi}{4} \)) represents a location 3 units away from the origin with an angle of \( 45^\circ \) from the positive x-axis.Polar coordinates provide a unique perspective, simplifying complex systems, especially those involving circles and spirals.

Symmetry Tests

Symmetry tests in polar coordinates help us identify if a polar graph remains unchanged after transformations.

• Polar Axis Symmetry: Replace \( \theta \) with \( -\theta \).
• Line \( \theta = \frac{\pi}{2} \): Replace \( \theta \) with \( -\theta + \pi \).
• Pole Symmetry: Replace \( r \) with \( -r \).

If the equation remains the same after these substitutions, symmetry is present. For example, in the equation \( r=2 \cos 2\theta \), all these tests show symmetry, indicating that the graph is symmetric about the polar axis, the line \( \theta = \frac{\pi}{2} \), and the pole. These tests simplify understanding complicated polar graphs.

Polar Equations

Polar equations describe curves using polar coordinates. These equations often involve trigonometric functions such as sine and cosine.
A common example is \( r = a \cos(n\theta) \), which generates rose curves when \( n \) is an integer.Polar equations offer unique insights:

• They effortlessly represent circular patterns, spirals, and other complex shapes in a more simplified form.
• The equation \( r=2 \cos 2\theta \) is an example of a polar equation. It's known for its symmetrical properties, as explored through symmetry tests.

Understanding polar equations allows us to visualize and analyze patterns not easily managed in Cartesian coordinates.

Trigonometric Identities

Trigonometric identities are essential tools for simplifying and solving polar equations. These identities allow for easier manipulation of trigonometric expressions.

• Cosine and Sine Symmetries: \( \cos(-x) = \cos(x) \) and \( \sin(-x) = -\sin(x) \).
• Periodicity: Trigonometric functions like sine and cosine repeat values over intervals of \( 2\pi \).

In the exercise \( r = 2 \cos 2\theta \), the identity \( \cos(-x) = \cos(x) \) was used to determine symmetry about the polar axis.
This property of even functions simplifies exploring symmetrical properties. Using trigonometric identities effectively can make solving polar equations straightforward and intuitive.