Problem 30 - Algebra and Trigonometry

Question

Test for symmetry and then graph each polar equation. $$r^{2}=9 \sin 2 \theta$$

Step-by-Step Solution

Answer

The given polar equation \(r^{2}=9 \sin 2 \theta\) is symmetric about the x-axis only. For graphing, select some values of \(\theta\), compute respective \(r\) values and plot the points in the polar coordinate system.

Step 1: Test for Symmetry about the Pole

Substitute \(-\theta\) for \(\theta\), if the equation remains the same then it’s symmetric about the origin (pole). The equation becomes \(r^{2}=9 \sin 2 (-\theta)\) or \(r^{2}=9 \sin -2 \theta\) and since \(\sin -2 \theta\) is \( -\sin 2 \theta)\), the equation is not symmetric around the pole.

Step 2: Test for Symmetry about the x-axis

Substitute \(-r\) for \(r\) and \(\pi-\theta\) for \(\theta\), if the equation remains the same, it is symmetrical about the x-axis. The equation becomes \((-r)^{2}=9 \sin 2 (\pi-\theta)\) or \(r^{2}=9 \sin (2\pi-2\theta)\) and is same as the original equation, indicating that the graph is symmetric about the x-axis.

Step 3: Test for Symmetry about the y-axis

Substitute \(-r\) for \(r\) and \(-\theta\) for \(\theta\), if it reverts back to the original equation, it’s symmetric about the y-axis. The equation becomes \((-r)^2 = 9 \sin 2 (-\theta)\) or \(r^{2}=9 \sin -2 \theta\) which is not the same as the original equation, hence it is not symmetric around the y-axis.

Step 4: Plot Critical Points

Once the symmetries are tested, start to graph the equation by selecting values for \(\theta\), computing corresponding \(r\) values and plot the points in a Polar Graph.