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Chapter 7: Problem 41

Test for symmetry and then graph each polar equation. $$r=\sin \theta \cos ^{2} \theta$$

Short Answer

Expert verified
The polar equation \(r = \sin(\theta) \cos^2(\theta)\) is symmetric about the polar axis. As a Cartesian equation, it becomes \(r = yx^2/(x^2 + y^2)\). The graph of the equation can be drawn by plotting some key points.

Step by step solution

01

Test for Symmetry

First, let's test for possible symmetries. Replace \(\theta\) by \(-\theta\) in the given equation, we get \(r = \sin(-\theta) \cos^2(-\theta)\). But we know that \(\sin(-\theta) = - \sin(\theta)\) and \(\cos(-\theta) = \cos(\theta)\). Hence it yields \(-r = -\sin(\theta) \cos^2(\theta)\) which simplifies to \(r = \sin(\theta) \cos^2(\theta)\). Since this is the same as the original equation, the graph is symmetrical about the polar axis.
02

Convert Polar to Cartesian Coordinates

Now, let's convert the polar equation into cartesian coordinates for easier plotting. The polar equation is given as \(r = r = \sin(\theta) \cos^2(\theta)\). We can convert this into Cartesian coordinates using the identities \(r^2 = x^2 + y^2\) and \(\tan(\theta) = y/x\), where \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\). Hence the Cartesian equation is \(r = yx^2/(x^2 + y^2)\).
03

Graph the equation

To sketch the graph, you will identify key points such as the origin, and intercepts if applicable. Since we don't have a clear 'r' isolated value, we can plot several points using varying values of \(\theta\) from 0 to \(2\pi\). Calculate the corresponding 'r' values and plot these points on polar graph paper. Connect these points smoothly to get the graph.

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Explaining the Concepts What is the graph of a polar equation?
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