Problem 41 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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 \(r = yx^2/(x^2 + y^2)\).

Step 3: 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.