Problem 28 - Algebra and Trigonometry

Question

Test for symmetry and then graph each polar equation. $$r=4 \cos 3 \theta$$

Step-by-Step Solution

Answer

The polar graph of the given equation, r = 4 cos 3θ, is symmetric about the polar axis. It has peaks at r=4, valleys at r=0, and it exhibits a 3-leaf clover shape when drawn.

Step 1: Tests for Symmetry

There are essentially three tests for symmetry pertaining to polar equations. Test for symmetry with respect to the polar axis by replacing \( \theta \) with \( -\theta \) and simplifying, this becomes \( r = 4 \cos (-3\theta) = 4 \cos (3\theta) \) which is the identical to the original equation. Therefore, this graph is symmetric about the polar axis. Then test for symmetry with respect to the pole by replacing \( r \) with \( -r \), and simplify to see if the result is identical to the original equation. Lastly, test for symmetry about the line θ = π/2. Because this is a cosine function and not a sine function, this graph is not symmetric about the line θ = π/2.

Step 2: Create a Table of Values

The next step in sketching the graph is to come up with a table of values for \( r \) based on various values of \( \theta \). Due to the symmetry, it's only necessary to compute values for \( \theta \) from 0 to π and use symmetry to reflect these for the remaining angles.

Step 3: Sketch the Graph

After listing key values from the table, plot these points on the polar grid. Paying special attention to the peaks and regions where \( r = 0 \). Considering the symmetry of the polar axis, reflected points over the polar axis will fill the graph for \( \theta \) from π to 2π. Then, sketch a smooth curve that fits these points.