Problem 42 - Algebra and Trigonometry

Question

Test for symmetry and then graph each polar equation. $$r=\frac{3 \sin 2 \theta}{\sin ^{3} \theta+\cos ^{3} \theta}$$

Step-by-Step Solution

Answer

The graph of the given polar equation doesn't have any symmetry - neither about x-axis, nor y-axis, nor about origin. The graph of the equation can be sketched by plotting the points obtained from choosing some values for \( \theta \) and calculating corresponding \( r \) values.

Step 1: Test for Symmetry

For polar equations, there are three different symmetries that could be considered; symmetry about x-axis, y-axis, and origin. Here, we'll test for these three. - To test for symmetry about the x-axis, we should replace \( \theta \) with \( -\theta \) in the equation and see if the equation remains the same.- To test for symmetry about the y-axis, replace \( \theta \) with \( \pi-\theta \).- To test for symmetry about the origin, replace \( \theta \) with \( \theta+\pi \) or \( r \) with \( -r \).Substituting for all three cases, it is found that the given equation doesn't maintain the original expression. Therefore it doesn't possess any symmetry.

Step 2: Graphing the Polar Equation

To graph this polar equation, plot some points by plugging into the equation some easy to handle values of \( \theta \) such as \( \theta = 0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4} , \pi, \frac{5\pi}{4} , \frac{3\pi}{2} , \frac{7\pi}{4} , 2\pi \) and by finding corresponding \( r \) values. But before that, the expression for \( r \) needs to be simplified. The denominator \( \sin^3\theta + cos^3\theta \) can be rewritten as \( (\sin\theta + \cos\theta)(1 - \sin\theta\cos\theta) \), which would make computation easier. Then plot these points on a polar coordinate system and sketch the graph by making sure the curve passes through the determined points. This might require graphing technology for a more accurate presentation.