Problem 20 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The graph of the polar equation \(r = 2 - \sin \theta\) has no symmetry.

Step 1: Test for symmetry

Test for symmetry by replacing \(\theta\) in the equation \(r = 2-\sin \theta'\) with \(-\theta\), \(\pi - \theta\) and \(\theta+\pi\). Once \(\theta\) is replaced, if the original equation is obtained, then the graph is symmetric according to the replaced variable's axis.

Step 2: Check for x-axis symmetry

Replace \(\theta\) in the equation \(r = 2 - \sin \theta\) with \(-\theta\). The equation \(r = 2 - \sin (-\theta) = 2 + \sin \theta\) is not the same as the original equation. So the graph is not symmetric about the x-axis.

Step 3: Check for y-axis symmetry

Replace \(\theta\) in the equation \(r = 2 - \sin \theta\) with \(\pi - \theta\). This gives the equation \(r = 2 - \sin (\pi - \theta) = 2 + \sin \theta\), which does not match the original equation. So the graph is not symmetric about the y-axis.

Step 4: Check for origin symmetry

Replace \(\theta\) in the equation \(r = 2 - \sin \theta\) with \(\theta + \pi\) to get \(r = 2 - \sin (\theta + \pi)\). This simplifies to \(r = 2 + \sin \theta\), which doesn't match the original equation, thus the graph is not symmetric about the origin.

Step 5: Plot the graph

Now graph the equation \(r = 2 - \sin \theta\). It is customary to select values for \(\theta\) in the interval [0, 2\pi] and find the corresponding values of r.