Problem 11 - Algebra and Trigonometry

Question

7-14 Test the polar equation for symmetry with respect to the polar axis, the pole, and the line \(\theta=\pi / 2 .\) $$r=\frac{4}{3-2 \sin \theta}$$

Step-by-Step Solution

Answer

The polar equation has no symmetry with the polar axis, the pole, or the line \(\theta = \pi/2\).

Step 1: Test for Symmetry with the Polar Axis

The polar axis relates to symmetry across the horizontal axis. To test this, replace \( \theta \) with \( -\theta \) in the polar equation \( r = \frac{4}{3 - 2 \sin \theta} \): \[ r' = \frac{4}{3 - 2 \sin(-\theta)} = \frac{4}{3 + 2 \sin \theta} \].Since \( r' eq r \), the equation is not symmetric with respect to the polar axis.

Step 2: Test for Symmetry with the Pole

Symmetry with respect to the pole implies that the plot of \( r \) for \( \theta \) should be the same as for \( -r \) and \( -\theta \). Replace \( r \) with \( -r \) and \( \theta \) with \( \pi + \theta \):\[ r' = \frac{4}{3 - 2 \sin(\pi + \theta)} = \frac{4}{3 + 2 \sin \theta} \].Thus, \( r' eq -r \), so the equation is not symmetric with respect to the pole.

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

This symmetry is vertical. Substitute \( \theta \) with \( \pi - \theta \) in the polar equation \( r = \frac{4}{3 - 2 \sin \theta} \):\[ r'' = \frac{4}{3 - 2 \sin(\pi - \theta)} = \frac{4}{3 - 2(-\sin \theta)} = \frac{4}{3 + 2 \sin \theta} \].Since \( r'' eq r \), there is no symmetry with respect to the line \( \theta = \pi/2 \).

Key Concepts

Polar Axis

The polar axis is an important concept in polar coordinates, analogous to the x-axis in Cartesian coordinates. To check a polar equation for symmetry with respect to the polar axis (the horizontal axis), you can substitute \( \theta \) with \( -\theta \) in the equation and see if the modified equation matches the original equation.

This type of symmetry means that the shape or graph looks the same when reflected across the polar axis. This is like how water reflects a tree perfectly across a calm, straight pond.

• If the modified equation equals the original equation, then the shape is symmetric with respect to the polar axis.
• In the case of the equation \( r = \frac{4}{3 - 2 \sin \theta} \), substituting \( -\theta \) results in \( r' = \frac{4}{3 + 2 \sin \theta} \), which is not equal to the original. Therefore, the equation is not symmetric with respect to the polar axis.

This lack of matching indicates that the graph of the equation does not mirror itself over the horizontal line through the pole.

Pole Symmetry

Pole symmetry, also known as origin symmetry in the Cartesian sense, involves examining whether a polar equation is invariant when both \( r \) and \( \theta \) are replaced with \( -r \) and \( -\theta \).

This kind of symmetry means that the graph can be rotated around the origin without changing its appearance. Imagine spinning a coin on a table; its appearance doesn’t change even as it moves around.

• To test for pole symmetry, modify the equation by substituting \( r \) with \( -r \) and \( \theta \) with \( \pi + \theta \).
• For the equation \( r = \frac{4}{3 - 2 \sin \theta} \), this results in \( r' = \frac{4}{3 + 2 \sin \theta} \), which is not equal to \( -r \). Thus, the graph is not symmetrically about the pole.

Understanding whether a graph has pole symmetry can help design systems that need rotational consistency, such as fans or propellers.

Vertical Symmetry

Vertical symmetry in polar coordinates refers to a graph being symmetric with respect to the vertical line \( \theta = \pi/2 \), which is equivalent to reflection over the y-axis in Cartesian coordinates.

To check for this kind of symmetry, substitute \( \theta \) with \( \pi - \theta \) in the polar equation and see if the new equation matches the original.

• This type of symmetry is like flipping a piece of paper from left to right; the folds should line up perfectly if symmetrical.
• In the case of our equation \( r = \frac{4}{3 - 2 \sin \theta} \), substituting \( \theta \) with \( \pi - \theta \) gives \( r'' = \frac{4}{3 + 2 \sin \theta} \), which differs from the original equation. Therefore, it does not exhibit vertical symmetry.

Understanding vertical symmetry helps in designing objects with reflections such as in architecture or art installations.