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Chapter 6: Problem 20

Test for symmetry and then graph each polar equation. $$r=2-\sin \theta$$

Short Answer

Expert verified
The given polar equation \(r=2-\sin\theta\) has x-axis symmetry. The graph of the polar equation based on plotted polar coordinates forms a circle with a dent towards the origin at \(0,2\)

Step by step solution

01

Test for Symmetry

We need to test the given polar equation \(r = 2 - \sin \theta\) for symmetry about the x-axis, y-axis, and the origin. For x-axis symmetry, we replace \(-\theta\) for \(\theta\) and \(-r\) for \(r\) in the equation. For y-axis, we replace \(\pi - \theta\) for \(\theta\) and \(r\) for \(r\) in the equation. For origin symmetry, we replace \(\pi + \theta\) for \(\theta\) and \(-r\) for \(r\) in the equation. The equation remains unchanged for negative angle, hence it shows symmetry about the x-axis.
02

Create a Table Value for \(\theta\) and \(r\)

Create a table using the values of \(\theta\) ranging from 0 to \(2\pi\) (increment it by \(\pi/2\) for simplicity). Substitute each value of \(\theta\) into the given equation to get the corresponding \(r\) values.
03

Plot the Polar Coordinates

Plot the polar coordinates \((r, \theta)\) obtained from step 2 on the polar graph. It involves plotting the points on the polar grid and connecting these dots smoothly. Since the equation has x-axis symmetry, the plotted points will reflect below the x-axis, giving the graph of the polar equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Symmetry in Polar Equations
Symmetry in polar equations is crucial for understanding and simplifying polar plots. By identifying symmetry, you can often predict the shape of the graph without plotting every single point.
When dealing with polar equations like \( r = 2 - \sin \theta \), you test for three types of symmetry:
  • X-axis Symmetry: Replace \( \theta \) with \( -\theta \) and \( r \) with \( -r \). If the equation remains unchanged, it has x-axis symmetry.
  • Y-axis Symmetry: Substitute \( \theta \) with \( \pi - \theta \). Retain \( r \). If the equation is unchanged, it has y-axis symmetry.
  • Origin Symmetry: Replace \( \theta \) with \( \pi + \theta \) and \( r \) with \( -r \). If the equation remains the same, it has origin symmetry.
In the given example \( r = 2 - \sin \theta \), testing shows it is symmetric about the x-axis. It means that whatever happens above the x-axis will be mirrored below it, making graphing more efficient.
Graphing Polar Equations
Graphing polar equations involves plotting points on a circular grid, using the angle\( \theta \) and radius \( r \). Each point is determined by its distance (r) from the pole, at a specific angle (\( \theta \)) measured from the polar axis (typically the x-axis equivalent).
For the equation \( r=2-\sin \theta \), you first create a range of \( \theta \) between 0 and \( 2\pi \). Calculate each \( r \) value by substituting respective \( \theta \) into the polar equation.
  • Start with a convenient set of angles, such as \( 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2},\) and \( 2\pi \).
  • Substitute each angle into the equation to find corresponding \( r \).
  • Once you have \( r \) and \( \theta \) pairs, it's time to plot them on the polar graph.
Finally, connect these plotted points smoothly while considering symmetry to generate the full graph of the polar equation.
Plotting Polar Coordinates
When plotting polar coordinates, it鈥檚 essential to use a polar grid, which is a circular graph with radii extending outward at various angles. This grid helps translate polar equations into visual curves.
Begin by marking each \( \theta \) value on the circle, starting from the positive x-axis direction. For each angle, move radially outward for the corresponding \( r \) value.
For example, with the given equation \( r = 2 - \sin \theta \), the process involves
  • Identifying \( \theta \) from the incremented values like \( 0, \pi/2, \pi, \) etc.
  • Calculating \( r \) by substituting \( \theta \) back into the equation.
  • Using the calculated \( r \) to plot the point on the graph at the specified angle \( \theta \).
By following these steps and respecting symmetry, you generate a plot that accurately represents the polar equation. This process makes complex functions manageable and their graphs visually approachable.

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Most popular questions from this chapter

a. Does \((4,-1)\) satisfy \(x+2 y=2 ?\) b. Does \((4,-1)\) satisfy \(x-2 y=6 ?\)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When I convert an equation from polar form to rectangular form, the rectangular equation might not define \(y\) as a function of \(x .\)

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Convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. $$r=\cos \theta$$

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