Problem 26 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The given polar equation \( r=2 \sin 2 \theta \) is symmetrical about the y-axis. The graph of the polar equation forms the shape of a four-leaf clover, with each leaf lying in a quadrant.

Step 1: Test for Symmetry

Test for symmetry with respect to the three axes: x-axis, y-axis, and the origin. Starting with the x-axis, check the equation \( r=2 \sin 2 (-\theta) \). Simplifying, we will get \( r=2 \sin -2 \theta \), which simplifies further to \( -r=2 \sin 2 \theta \), this is not the same as the initial equation. For the y-axis, check the equation \( r=2 \sin(2\pi - 2\theta) \). Simplifying further leads to \( r=2 \sin 2 \theta \), which is the same as the initial equation, so it is symmetrical along the y-axis. To check for symmetry around the origin, replace r with -r, to get \( -r=2 \sin 2\theta \), which is not equal to the initial equation so it is not symmetrical about the origin.

Step 2: Determine the intervals and plot points on the graph

For the polar equation \( r=2 \sin 2 \theta \), we are working in a range of \( 0 \leq \theta < 2\pi \). Due to the symmetry, we only need to plot the interval \( 0 \leq \theta < \pi \) and reflect these points about the y-axis. The function period is \( \frac{\pi}{2}\) so take several values of \( \theta \) in the interval \( 0 \leq \theta < \pi \) to plot these points.

Step 3: Plot the graph

For each selected value of \( \theta \), find the respective r using the equation \( r=2 \sin 2 \theta \). After finding these coordinates plot these points onto the graph and include the reflected points. The graph provides the visual solution of the given polar equation, showing symmetry about the y-axis.