Problem 15 - Precalculus

Question

Test for symmetry with respect to the line \(\theta=\pi 2,\) the polar axis, and the pole. $$r=\frac{2}{1+\sin \theta}$$

Step-by-Step Solution

Answer

The polar function \(r = \frac{2}{1 + \sin \theta}\) is not symmetric with respect to the line \(\theta = \frac{\pi}{2}\), the polar axis, or the pole.

Step 1: Test for Symmetry with Respect to the Line \(\theta = \frac{\pi}{2}\)

To test for symmetry about the line \(\theta = \frac{\pi}{2}\), substitute \(-\theta\) for \(\theta\) in the polar equation and simplify. If the new equation is the same as the original equation, the plot is symmetric with respect to the line \(\theta = \frac{\pi}{2}\).\nSo, \[r = \frac{2}{1+\sin(-\theta)} = \frac{2}{1 - \sin(\theta)}\] which is not the same as the original equation. So, the plot is not symmetric with respect to the line \(\theta = \frac{\pi}{2}\)

Step 2: Test for Symmetry with Respect to the Polar Axis

To test for symmetry with respect to the polar axis, replace \(\theta\) by \(-\theta\) in the polar equation and simplify. If we get back the original equation, then the graph of the equation is symmetrical about the polar axis.\nSo, \[r = \frac{2}{1 + \sin(-\theta)} = \frac{2}{1 - \sin(\theta)}\] which does not reduce to the original equation. Therefore, the function is not symmetrical with respect to the polar axis.

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 polar equation and simplify.However, we can see directly that replacing \(r\) with \(-r\) doesn't make any sense as \(r\) equals to a fraction and replacing it with \(-r\) would mean that we replace the fraction with its negative. Hence, the function is not symmetric with respect to the pole.

Key Concepts

Symmetry in Polar Graphs

Symmetry in polar graphs is an important concept for understanding the behavior of curves plotted in polar coordinates. Polar coordinates use a distance from the origin and an angle to define a point in a plane.
The symmetries we typically look for include:

• Symmetry with respect to the line \(\theta = \frac{\pi}{2}\)
• Symmetry with respect to the polar axis (the horizontal axis)
• Symmetry with respect to the pole (the origin)

Testing for these symmetries involves substituting values and observing if an equation remains unchanged. For example, substituting \(-\theta\) for \(\theta\) and checking the equality tests symmetry about the polar axis. If the equation transforms to its original form, the graph is symmetric with respect to that axis. In our problem:

• We substituted \(-\theta\) in \(r = \frac{2}{1+\sin(\theta)}\), getting \(\frac{2}{1-\sin(\theta)}\). This is not the same as the original equation, indicating no symmetry with the polar axis.

This process helps identify symmetry in plot graphs, making it easier to sketch and analyze them.

Rational Functions

Rational functions play a critical role in various areas of mathematics and are defined as the ratio of two polynomials.
In the exercise, we used the rational function \(r = \frac{2}{1+\sin(\theta)}\) .
This particular function is a ratio of a constant and a polynomial involving a trigonometric function.
The behavior of rational functions can significantly change depending on the numerator and the denominator:

• Zeros: Occur when the numerator of the function equals zero.
• Undefined Points: Occur when the denominator equals zero, as division by zero is undefined.
• Horizontal Asymptotes: Appear when the degree of the polynomial in the denominator is equal to or greater than the numerator.

In polar coordinates, these attributes of rational functions can create unique plots that are not necessarily straightforward to understand solely through Cartesian methods. Understanding rational functions helps us describe and predict the shapes and limits within polar plots.

Sin Functions

Sin functions are foundational to trigonometry and play a significant part in describing wave-like shapes and periodic behaviors.
When applied in polar coordinates, they help define points on a graph using \(r\) as the radius and \(\theta\) as the angle. In the exercise, the given function is \(r = \frac{2}{1+\sin(\theta)}\), employing a trigonometric function.Sin functions have unique features in polar graphs:

• Periodicity: The function repeats its values over specific intervals, typically every \(2\pi\).
• Symmetry: Sin functions often create symmetrical shapes, especially around the polar axis.
• Maximum and Minimum: The sinusoidal nature causes alternate crests and troughs, representing maximum and minimum values.

Analyzing these characteristics helps us understand how a function may oscillate or change direction, creating distinct loops or petals in polar graphs. This insight is crucial for predicting how polar equations will behave when graphed.