Problem 7 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The graph is not symmetric with respect to the polar axis, the pole, or the line \(\theta = \pi/2\).

Step 1: Test Symmetry with Respect to the Polar Axis

To check for symmetry with respect to the polar axis, replace \(\theta\) with \(-\theta\) in the equation. The given equation is \(r = 2 - \sin \theta\). Substituting, we get:\[r = 2 - \sin (-\theta) = 2 + \sin \theta\]Since this equation is not equivalent to the original equation \(r = 2 - \sin \theta\), the graph is not symmetric with respect to the polar axis.

Step 2: Test Symmetry with Respect to the Pole

For symmetry with respect to the pole, replace \(r\) with \(-r\). Start with the original equation:\[r = 2 - \sin \theta\]Replace \(r\) with \(-r\):\[-r = 2 - \sin \theta\,\text{or equivalently,}\,r = \sin \theta - 2\]Since this equation is not the same as the original, the polar graph is not symmetric with respect to the pole.

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

For symmetry with respect to the line \(\theta = \pi/2\), replace \(\theta\) with \(\pi - \theta\). The original equation is \(r = 2 - \sin \theta\). Substitute to get:\[r = 2 - \sin(\pi - \theta) = 2 + \sin \theta\]Since this equation is not equivalent to the original equation \(r = 2 - \sin \theta\), the graph is not symmetric with respect to the line \(\theta = \pi/2\).

Key Concepts

Polar Equations

Polar equations are a special way of describing curves on a plane using polar coordinates \((r, \theta)\) rather than the traditional Cartesian coordinates \((x, y)\). Here, \(r\) represents the distance from the origin, also known as the pole, and \(\theta\) is the angle from the positive x-axis, referred to as the polar axis. The equation given, \(r = 2 - \sin \theta\), is an example of a polar equation. Using polar equations can be very useful in many mathematical applications as they provide a more natural description for circular and spiral shapes. Unlike Cartesian coordinates, where curves are described based on x and y positions, polar coordinates offer a more intuitive representation when dealing with rotations, angles, and radii. Keep in mind, interpreting these equations often involves transforming them back into Cartesian coordinates to visualize or analyze graphically.

Symmetry Testing

Symmetry is an important property in graphing as it helps us understand the behavior and shape of the graph. For polar equations like \(r = 2 - \sin \theta\), symmetry can be tested with respect to different lines and points. Each symmetry test gives insight into the equation's pattern and balance, which can simplify graphing and solving related problems.

• Symmetry with respect to the polar axis: To test this, replace \(\theta\) with \(-\theta\) in the equation. If the resulting equation is equivalent to the original, the graph is symmetric about the polar axis (horizontal line).
• Symmetry with respect to the pole: Replace \(r\) with \(-r\). A result that is equivalent means that the graph is symmetrical around the origin.
• Symmetry with respect to the line \(\theta = \pi/2\): Substitute \(\theta\) with \(\pi - \theta\). If this version is the same as the original, the graph is symmetric with respect to the vertical line through the origin.

By conducting these tests on our polar equation, \(r = 2 - \sin \theta\), it was discovered that the graph does not exhibit any of these symmetries.

Mathematical Graphing

Graphing polar equations involves plotting points on a coordinate plane using the radius \(r\) and angle \(\theta\). It's a bit different from Cartesian graphing, but with practice, it becomes second nature. Here's how you can approach graphing polar equations:1. **Identify Key Points**: Start by calculating a set of \((r, \theta)\) pairs. For example, plug in various \(\theta\) values into the equation \(r = 2 - \sin \theta\) to get a range of \(r\) values.2. **Plotting**: Use these values to plot each point in polar coordinates. Remember, \(\theta\) determines the direction from the origin, and \(r\) determines the distance from the origin.3. **Connecting Dots**: Once your points are plotted, connect them smoothly to reveal the curve's shape.This process highlights the charm of polar graphing: it often reveals intricate and beautiful patterns. With the given equation, you might expect an interesting shape that reflects the equation's lack of symmetry. Embrace this method, as polar graphing opens a window into a world of mathematical elegance.