Problem 43 - Algebra and Trigonometry

Question

Test for symmetry and then graph each polar equation. $$r=2+3 \sin 2 \theta$$

Step-by-Step Solution

Answer

The given polar coordinate equation \( r = 2 + 3 \sin 2 \theta \) has symmetry about the origin. This symmetry aids in visualizing and graphing the rose curve represented by this equation.

Step 1: Test for Symmetry

Check for three kinds of symmetry in polar coordinates: the symmetry about the x-axis, y-axis, and the origin. This can be done by plugging \(-\theta\) into the equation and seeing if it simplifies to the original equation or with negative r.

Step 2: Identify the Type of Symmetry

For the given equation \(r=2+3 \sin 2 \theta \), replace \(\theta\) with \(-\theta\) to give \(r = 2 + 3 \sin -2\theta = 2 - 3 \sin 2\theta = -(2 - 3 \sin 2\theta) \). This simplifies to \(-r\). This means the graph has symmetry about the origin.

Step 3: Graph the Polar Equation after Recognizing the Symmetry

Create a table of values for \(\theta\) and r, and then graph these points. Remember that only some points have to be graphed due to the identified symmetry. Mirror the plotted points about the origin to get the other part of the graph because of its symmetry. The equation is a type of rose curve, and the full plot can be completed by considering the symmetry.