Problem 16 - Precalculus Mathematics for Calculus

Question

Test the polar equation for symmetry with respect to the polar axis, the pole, and the line \(\theta=\pi / 2\) $$r^{2}=9 \sin \theta$$

Step-by-Step Solution

Answer

Symmetric about the line \(\theta = \pi/2\) only.

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

To test for symmetry with respect to the polar axis, check if replacing \((r, \theta)\) with \((r, -\theta)\) yields the original equation. Substitute \(\theta = -\theta\) into the given equation: \[ r^2 = 9 \sin(-\theta) \]Knowing \(\sin(-\theta) = -\sin\theta\), the equation becomes: \[ r^2 = -9 \sin\theta \]This is not identical to the original equation \(r^2 = 9 \sin \theta\), hence the equation does not have symmetry about the polar axis.

Step 2: Test for Symmetry with respect to the Pole

To test for pole symmetry, check if replacing \((r, \theta)\) with \((-r, \theta)\) or \((r, \theta + \pi)\) yields the original equation. Substitute \((r, \theta + \pi)\) into the equation: Since \(r^2\) will remain positive, we consider \(\sin(\theta + \pi) = -\sin \theta \).Thus, the equation becomes: \[ r^2 = 9(-\sin \theta) \]This simplifies to: \[ r^2 = -9 \sin \theta \]This is not the same as the original equation, so there is no symmetry with respect to the pole.

Step 3: Test for Symmetry with respect to the Line \(\theta = \pi/2\)

For symmetry about the line \(\theta = \pi/2\), substitute \(\theta = \pi - \theta\):\[ r^2 = 9 \sin(\pi - \theta) \]Using the identity \(\sin(\pi - \theta) = \sin \theta\), the equation reduces to:\[ r^2 = 9 \sin \theta \]This matches the original equation, so the equation is symmetric about the line \(\theta = \pi/2\).

Key Concepts

Symmetry in Polar Equations

In polar equations, symmetry is an important property to examine because it reveals insightful characteristics about the graph of an equation. Testing for symmetry helps to determine if the graph looks the same when manipulated in specific ways. There are three common symmetries to test:

• Symmetry with respect to the polar axis.
• Symmetry with respect to the pole.
• Symmetry about the line \(\theta = \pi/2\).

Let's break these down a bit.
The **polar axis symmetry** means if we reflect the equation over the horizontal axis, the graph remains unchanged. By substituting \(\theta\) with \(-\theta\), we assess whether the equation remains the same. If it does, it is symmetrical about the polar axis.
The **pole symmetry** checks if swapping positive and negative \(r\) values, or shifting the angle by \(\pi\), leaves the equation unchanged.
Finally, **symmetry about the line \(\theta = \pi/2\)** involves reflecting over the vertical line where \(\theta = \pi/2\). Substituting \(\theta\) with \(\pi - \theta\) helps test for this type of symmetry.

Understanding Polar Coordinates

Polar coordinates are a two-dimensional coordinate system, which represent points on a plane. Unlike Cartesian coordinates that use \((x, y)\) pairs, polar coordinates use a radius and an angle, \((r, \theta)\).

• \(r\) represents the distance from the origin (or pole) to the point.
• \(\theta\) is the angle from the polar axis, usually measured in radians.

This system is especially useful for equations and graphs involving circles and spirals. Testing an equation like \(r^{2}=9\sin\theta\) uses these polar coordinates to analyze the position and behavior of points on the graph.
Working with polar coordinates requires understanding how the radius and angle relate to each other and how changes to either affect the graph's representation.

Trigonometric Identities in Polar Graphs

Trigonometric identities are essential tools when working with polar equations. These identities allow you to simplify and manipulate trigonometric expressions to aid in analyzing symmetry and solving equations. Common identities used include:

• \(\sin(-\theta) = -\sin\theta\)
• \(\sin(\pi - \theta) = \sin\theta\)
• \(\sin(\theta + \pi) = -\sin\theta\)

These identities play a crucial role when testing for symmetry. For example, when assessing pole symmetry, the identity \(\sin(\theta + \pi) = -\sin\theta\) helps determine if an equation is symmetric around the pole. By using identities like these, you can simplify the process of checking symmetry in polar graphs.