Problem 15 - Precalculus with Limits

Question

In Exercises 13-18, test for symmetry with respect to \(\theta = \pi/2\), the polar axis, and the pole. \(r = \dfrac{2}{1\ +\ \sin\ \theta}\)

Step-by-Step Solution

Answer

The given polar equation is not symmetric about the line \(\theta = \pi/2\) nor about the pole, but it is symmetric about the polar axis.

Step 1: Check for Symmetry about \(\theta = \pi/2\)

Replace \(\theta\) with \(-\theta\) in the given equation \(r = \dfrac{2}{1 + \sin\theta}\). This gives us: \[r = \dfrac{2}{1 + \sin(-\theta)}\] Since \(\sin(-\theta) = -\sin(\theta)\), we can rewrite the equation as: \[r = \dfrac{2}{1 - \sin\theta}\] This is not the same as the original equation, so the equation is not symmetric about \(\theta = \pi/2\).

Step 2: Check for Symmetry about the Polar Axis

Replace \(\theta\) with \(\pi - \theta\) in the original equation. This gives us: \[r = \dfrac{2}{1 + \sin(\pi - \theta)}\] Since \(\sin(\pi - \theta) = \sin\theta\), we can rewrite the equation as: \[r = \dfrac{2}{1 + \sin\theta}\] This is the same as the original equation, so the polar equation is symmetric about the polar axis.

Step 3: Check for Symmetry about the Pole

Replace \(r\) with \(-r\) in the original equation. This would give us: \[-r = \dfrac{2}{1 + \sin\theta}\] Clearly, this is not the same as the original equation, so the equation is not symmetric about the pole.

Key Concepts

Symmetry Tests in Polar Equations

Understanding the symmetry of polar equations is a foundational aspect of working with polar coordinates. Symmetry in polar equations allows us to anticipate the graph and identify attributes of the curve, which aids greatly in graphing complex equations more efficiently.

For any given polar equation, there are three classic symmetry tests:

• The test for symmetry about the line \(\theta = \frac{\pi}{2}\) which is a vertical line through the pole.
• The test for symmetry about the polar axis, which is the positive x-axis in Cartesian coordinates.
• The test for symmetry about the pole (origin), which would mean the curve looks the same when rotated 180 degrees.

To perform these tests, certain substitutions are made into the polar equation and the results are compared to the original equation. If the form remains unchanged, the graph possesses that particular symmetry. Let's take an example of a polar equation \(r = \frac{2}{1 + \sin\theta}\).

For symmetry about \(\theta = \frac{\pi}{2}\), \(\theta\) is replaced with \(\-\theta\), for symmetry about the polar axis \(\theta\) is replaced with \(\pi-\theta\), and for symmetry about the pole, \(r\) is replaced with \(\-r\). The outcomes of these substitutions determine the symmetry of the equation, which ultimately guides the graphing of the polar curve.

Polar Coordinates

Polar coordinates offer a different perspective from the traditional Cartesian system for defining the location of points. Instead of using x and y coordinates, polar coordinates specify a point's position in terms of its distance from a reference point, called the pole, and an angle from a reference direction, known as the polar axis.

In polar coordinates, a point P is represented by a pair \( (r, \theta) \). Here, \(r\) denotes the radial distance from the pole, and \(\theta\) is the angle measured in radians, typically from the positive x-axis. This system is particularly useful when dealing with problems involving circular or spiral paths, or when the relationship between variables is simpler to express in polar form.

Polar coordinates are defined as follows:

• \(r\): The length of the line segment from the pole to the point P.
• \(\theta\): The angle measured counter-clockwise from the polar axis to the line segment joining the pole with the point P.

Understanding polar coordinates is essential for solving polar equations as it lays the foundation for analyzing curves in a polar context.

Sin Function Symmetry

The sine function holds interesting symmetry properties, which are leveraged when analyzing polar equations. Since \(\sin\) is an odd function, it satisfies the property that \(\sin(-\theta) = -\sin(\theta)\), which reflects across the origin in Cartesian coordinates. This means that for any angle \(\theta\), its sine will be the negative of the sine of the angle with the opposite sign.

Moreover, the sine function has rotational symmetry of period \(\pi\). This implies that \(\sin(\theta) = \sin(\pi - \theta)\), holding true for any angle \(\theta\). In the context of polar equations, these properties of the sine function allow us to determine the symmetry of the curve with respect to the polar axis by substituting \(\theta\) with \(\pi - \theta\) in the given polar equation, as seen in our example equation \(r = \frac{2}{1 + \sin\theta}\).

Recognizing the sine function's symmetry properties can often simplify the process of plotting the curve of a polar equation and facilitate understanding the underlying geometry of the graph.