Problem 34 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The polar equation \(r \sin \theta = 2\) is symmetric about the y-axis but not about the x-axis or the origin. The graph of this equation is a horizontal line passing through the point (0,2).

Step 1: Test for Symmetry About the X-axis

Let's first test if the equation is symmetric with respect to the x-axis or not. To do that, replace \(\theta\) with \(-\theta\) in the given polar equation. Thus, we obtain \(r \sin (-\theta) = r(-\sin\theta)\). This is not equivalent to the original equation \(r \sin \theta = 2\), therefore the graph is not symmetric with respect to the x-axis.

Step 2: Test for Symmetry About the Y-axis

Now let's test the symmetry with respect to the y-axis. Make the substitution \(r \rightarrow -r\) and \(\theta \rightarrow -\theta\) in the given equation. This will result in \(-r \sin(-\theta) = -r \sin\theta\), which agrees with the original equation \(r \sin\theta = 2\). Thus, the graph is symmetric with respect to the y-axis.

Step 3: Test for Symmetry About the Origin

In this step, we will test the symmetry with respect to the origin. Make the substitution \(r \rightarrow -r\) in the original equation which results in \(-r \sin\theta \), which is not equivalent to the original equation, \(r \sin\theta = 2\). Therefore, the graph is not symmetric with respect to the origin.

Step 4: Graphing the Polar Equation

Now that we know the graph is symmetric about the y-axis, we plot the polar equation. We can convert the polar equation to Cartesian coordinates by using the property \(r\sin\theta = y\). Therefore, \(y = 2\) is the Cartesian equivalent of our given polar equation. It is a horizontal line that passes through the point (0,2).