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Chapter 6: Problem 23

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

Short Answer

Expert verified
The polar equation \(r=2-3\sin\theta\) is symmetric about the y-axis. Plotting the values of \(r\) for different \(\theta\) values results in a graph that illustrates this symmetry.

Step by step solution

01

Test for Symmetry

In polar equations, there are three types of symmetries: symmetry about the x-axis, y-axis, and origin. To test for symmetry about the x-axis, replace \(\theta\) with \(-\theta\). If the equation remains unchanged, it's symmetric about the x-axis. For the y-axis, replace \(\theta\) with \(\pi-\theta\), and for origin, replace \(\theta\) with \(\theta+\pi\). Here, let's replace \(\theta\) with \(-\theta\), \(\pi-\theta\), and \(\theta+\pi\) in the equation \(r=2-3\sin\theta\) and observe if it remains unchanged.
02

Analyze Symmetry

After performing the replacements in Step 1, these are the results: - For x-axis symmetry, \(r = 2 - 3\sin(-\theta) = 2 + 3\sin\theta\), not matching the original equation. - For y-axis symmetry, \(r = 2 - 3\sin(\pi-\theta) = 2 - 3\sin\theta\), matching the original. - For origin symmetry, \(r = 2 - 3\sin(\theta+\pi) = 2 + 3\sin\theta\), not matching the original. Hence, the given polar equation is symmetric about the y-axis.
03

Create Graph

Now, to graph the polar equation \(r = 2 - 3\sin\theta\), plot a series of points for various values of \(\theta\) and then join these points in a smooth curve. Since the equation is symmetric about the y-axis, you can plot for \(\theta\) ranging from 0 to \(\pi\) and then mirror the result about the y-axis to complete the graph. The choice of \(\theta\) values also depends on the nature of the graph, for a usual graph \(\theta\) from 0 to \(2\pi\) can be chosen.

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Key Concepts

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

Graphing Polar Equations
Understanding how to graph polar equations involves several steps. At the core of graphing is the practice of marking points on the polar coordinate system and joining these points to reveal a continuous curve. To graph the equation
\( r = 2 - 3\sin\theta \), we begin by choosing a range of values for \( \theta \) and calculating the corresponding \( r \) values. Since the graph is symmetrical about the y-axis, as determined in the solution, we need to calculate points only for \( \theta \) values from 0 to \( \pi \), and then reflect those points across the y-axis.

Visualizing the graph, we can plot these points on the polar grid: start at the pole (the origin in the polar system) and move outwards the distance \( r \) in the direction of the angle \( \theta \) from the polar axis (positive x-axis). After plotting several points, we connect them with a smooth curve. This resulting curve indicates the shape of the graph for the given polar equation.
Testing for Symmetry in Polar Coordinates
Identifying symmetry in polar equations can save time when graphing and provide insight into the nature of the graph. As explained in the solution, symmetry tests in polar coordinates are performed by substituting the angle \( \theta \) with \( -\theta \) for x-axis symmetry, \( \pi-\theta \) for y-axis symmetry, and \( \theta+\pi \) for origin symmetry.

For the exercise equation, when we test for y-axis symmetry by substituting \( \pi-\theta \), we find the equation remains the same, indicating that any point \( (r,\theta) \) has a corresponding point \( (r,\pi-\theta) \) that also lies on the graph. This is a powerful property because it allows us to graph only half of the equation, and mirror the rest over the y-axis, simplifying the entire graphing process.
Polar Coordinate System
The polar coordinate system is an alternative to the Cartesian coordinate system most students are familiar with. Instead of using x and y coordinates, the polar system uses the distance from a central point (the pole), denoted as \( r \) and the angle \( \theta \) from a reference direction (usually the positive x-axis).

The polar coordinate system can describe the same points in the plane as the Cartesian system but does so with a focus on angles and distances, making it particularly suited for problems that have circular or spiral symmetry. This system is also essential when dealing with complex numbers, electromagnetism, and navigation, among other fields.

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Most popular questions from this chapter

Convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. $$r=12 \cos \theta$$

Show that each statement is true by converting the given polar equation to a rectangular equation. Show that the graph of \(r=a \cos \theta\) is a circle with center at \(\left(\frac{a}{2}, 0\right)\) and radius \(\frac{a}{2}\)

In converting \(r=5\) from a polar equation to a rectangular equation, describe what should be done to both sides of the equation and why this should be done.

Graph \(y=2 \sin \frac{1}{2} x\). Then use the graph to obtain the graph of \(y=2 \csc \frac{1}{2} x . \)

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. $$\mathbf{u} \cdot \mathbf{v}=\mathbf{v} \cdot \mathbf{u}$$
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