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Chapter 6: Problem 43

Test for symmetry and then graph each polar equation. $$r=2+3 \sin 2 \theta$$

Short Answer

Expert verified
The polar equation \(r = 2 + 3\sin2\theta\) is neither symmetric about the x-axis, y-axis, nor the origin. The graph of this equation can be drawn by plotting various data points (r, \theta) and joining them.

Step by step solution

01

Test for Symmetry

For Symmetry about x-axis, replace \((r, \theta)\) by \((r, -\theta)\) or \((r, \theta + \pi)\) in the equation.For Symmetry about y-axis, replace \((r, \theta)\) by \((-r, \theta)\) or \((-r, \theta + \pi)\) in the equation.For Symmetry about the origin, replace \((r, \theta)\) by \((-r, -\theta)\) or \((-r, \theta + \pi)\) in the equation. The substitutions will be done one after the another in the given equation \(r = 2 + 3\sin2\theta\). If the equation remains invariant, then it's symmetric about that axis or origin.
02

Testing for Symmetry in the equation

Substitute \((r, \theta)\) by \((r, -\theta)\), \((r, \theta + \pi)\), \((-r, \theta)\), \((-r, \theta + \pi)\), \((-r, -\theta)\), and \((-r, \theta + \pi)\) one by one in the equation \(r = 2 + 3\sin2\theta\). It is found that the equation does not remain unchanged. Thus, the graph is not symmetric about the x-axis, y-axis, or the origin.
03

Plotting the Graph

First, for various \(\theta\) values, calculate the corresponding \(r\) values using the equation \(r = 2 + 3\sin2\theta\).Plot the polar coordinates \((r, \theta)\) on the polar graph. Once all the data points are plotted, join them to get the graph of the polar equation \(r = 2 + 3\sin2\theta\).

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Key Concepts

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

Symmetry Testing
Symmetry testing is an essential process when dealing with polar equations. It's a technique used to determine if a graph will be mirrored across certain axes or the origin. To test for symmetry around the x-axis, replace
  • \((r, \theta)\) with \((r, -\theta)\)
  • \((r, \theta + \pi)\)
For the y-axis, use:
  • \((-r, \theta)\)
  • \((-r, \theta + \pi)\)
Finally, for origin symmetry, substitute:
  • \((-r, -\theta)\)
  • \((-r, \theta + \pi)\)
In the given equation \(r = 2 + 3\sin2\theta\), applying these tests shows no invariance. This means the graph will not display symmetry about the x-axis, y-axis, or origin. Symmetry testing provides insight into the nature of the graph before plotting, simplifying the graphing process by anticipating its shape.
r and theta coordinates
Polar coordinates use two values, \(r\) and \(\theta\), to describe a point's position in the plane. Unlike Cartesian coordinates, which rely on horizontal and vertical distances, polar coordinates use these two measures:
  • \(r\): the radial distance from the origin to the point.
  • \(\theta\): the angle formed with the positive x-axis.
When working with the equation \(r = 2 + 3\sin2\theta\), different \(\theta\) values will lead to corresponding \(r\) values, creating a unique set of points. Understanding these coordinates and how they plot is fundamental, especially since the transition between \(r\) and \(\theta\) allows graphs to represent complex and interesting shapes that differ from traditional Cartesian graphs.
Graphing Polar Functions
Graphing polar functions involves finding and plotting the set of points generated by polar equations like \(r = 2 + 3\sin2\theta\). Here鈥檚 how you do it step by step:
  • Calculate \(r\) for Various \(\theta\): By substituting different values of \(\theta\) into the polar equation, you can calculate \(r\). Remember, the angle \(\theta\) is often taken in radians.
  • Plot the Coordinates: Once \(r\) values are determined, plot each \((r, \theta)\) coordinate on a polar grid.
  • Connect the Points: After plotting several points, join them to visualize the graph. For complex graphs, this step might require some smoothing.
If the function \(r = 2 + 3\sin2\theta\) is plotted, it will display interesting patterns and shapes typical for sinusoidal polar equations, yet it lacks symmetry as identified through symmetry testing. This can produce intricate and visually appealing curves.

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Most popular questions from this chapter

Show that each statement is true by converting the given polar equation to a rectangular equation. Show that the graph of \(r=a \csc \theta\) is a horizontal line \(a\) units above the \(x\) -axis if \(a>0\) and \(|a|\) units below the \(x\) -axis if \(a<0\)

Polar coordinates of a point are given. Find the rectangular coordinates of each point. $$\left(4,90^{\circ}\right)$$

Explain how to convert a point from polar to rectangular coordinates. Provide an example with your explanation.

Polar coordinates of a point are given. Find the rectangular coordinates of each point. $$(7.4,2.5)$$

Use the vectors \(\mathbf{u}=a_{1} \mathbf{i}+b_{1} \mathbf{j}, \quad \mathbf{v}=a_{2} \mathbf{i}+b_{2} \mathbf{j}, \quad\) and \(\quad \mathbf{w}=a_{3} \mathbf{i}+b_{3} \mathbf{j}\) to prove the given property. $$\mathbf{u} \cdot(\mathbf{v}+\mathbf{w})=\mathbf{u} \cdot \mathbf{v}+\mathbf{u} \cdot \mathbf{w}$$
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