Problem 31 - Algebra and Trigonometry

Question

Test for symmetry and then graph each polar equation. $$r=1-3 \sin \theta$$

Step-by-Step Solution

Answer

The given polar equation \(r = 1 - 3 \sin \theta\) shows no symmetry about the x-axis, y-axis or origin after testing. To graph this equation, plot several points using different \(\theta\) values and then connect these points to form the polar curve.

Step 1: Test for Symmetry

First, test the polar equation for three types of symmetries. 1. Symmetry about x-axis: Replace \( \theta\) by \(-\theta\) . If the original equation remains unchanged, then it has x-axis symmetry. 2. Symmetry about y-axis: Replace \( r\) by \(-r\) . If the original equation remains unchanged, then it has y-axis symmetry. 3. Symmetry about the origin: Replace \( \theta\) by \(\theta + \pi\) and \( r\) by \(-r\) . If the original equation remains unchanged, then it has symmetry about the origin.

Step 2: Plot the Polar Equation

To plot the given equation in a polar graph, it's necessary to select several values of \(\theta\) and calculate corresponding \(r\) values using our given equation. Make a theta-r table and plot these points on a polar grid.

Step 3: Draw the Graph

After determining the points from the table in step 2, plot those points on a polar coordinate plane. Connect these points smoothly to form the complete polar curve representing the given equation.