Question

identify the graph of r = 2 sin²(3θ).

Explanation:

Step1: Simplify the polar equation

Use the double - angle identity \(\sin^{2}\theta=\frac{1 - \cos(2\theta)}{2}\). For \(r = 2\sin^{2}(3\theta)\), we substitute \(\sin^{2}(3\theta)=\frac{1-\cos(6\theta)}{2}\). Then \(r = 2\times\frac{1 - \cos(6\theta)}{2}=1-\cos(6\theta)\).

Step2: Analyze the number of petals

For a polar curve of the form \(r = a\pm\cos(n\theta)\) or \(r = a\pm\sin(n\theta)\), the number of petals (leaves) is \(n\) when \(n\) is an integer. Here \(n = 6\), so the curve should have 6 petals.

Now let's analyze the given graphs:

• The first graph (top - left) has more than 6 petals (it looks like 12 petals, maybe a different form).
• The second graph (top - right) has 6 petals.
• The third graph (bottom - left) has more than 6 petals (maybe 12 petals).
• The fourth graph (bottom - right) has 3 petals.

Answer:

The graph on the Top - Right (the second graph in the 2x2 grid)