In Exercises 35–44, test for symmetry and then graph each polar equation. r = 2 + 3 sin 2θ

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1. **Step 1:** Test for symmetry with respect to the polar axis (x-axis). Substitute $\theta$ with $-\theta$ in the equation $r=2+3\sin 2\theta$. This gives $r=2+3\sin (-2\theta )$. Since $\sin (-2\theta )=-\sin (2\theta )$, the equation becomes $r=2-3\sin 2\theta$. This is not equivalent to the original equation, so there is no symmetry with respect to the polar axis.
2. **Step 2:** Test for symmetry with respect to the line $\theta =\frac{\pi }{2}$ (y-axis). Substitute $\theta$ with $\pi -\theta$ in the equation $r=2+3\sin 2\theta$. This gives $r=2+3\sin (2(\pi -\theta ))=2+3\sin (2\pi -2\theta )$. Since $\sin (2\pi -2\theta )=-\sin (2\theta )$, the equation becomes $r=2-3\sin 2\theta$. This is not equivalent to the original equation, so there is no symmetry with respect to the line $\theta =\frac{\pi }{2}$.
3. **Step 3:** Test for symmetry with respect to the pole (origin). Substitute $r$ with $-r$ and $\theta$ with $\theta +\pi$ in the equation $r=2+3\sin 2\theta$. This gives $-r=2+3\sin (2(\theta +\pi ))=2+3\sin (2\theta +2\pi )$. Since $\sin (2\theta +2\pi )=\sin (2\theta )$, the equation becomes $-r=2+3\sin 2\theta$. This is not equivalent to the original equation, so there is no symmetry with respect to the pole.
4. **Step 4:** Identify the type of graph. The equation $r=2+3\sin 2\theta$ is a polar equation that represents a rose curve. The general form of a rose curve is $r=a+b\sin n\theta$ or $r=a+b\cos n\theta$, where $n$ determines the number of petals. Since $n=2$, the graph will have $2n=4$ petals.
5. **Step 5:** Graph the equation. To graph $r=2+3\sin 2\theta$, plot points for various values of $\theta$ from $0$ to $2\pi$. Calculate $r$ for each $\theta$ and plot the corresponding points in polar coordinates. Connect the points smoothly to form the rose curve with 4 petals.

Key Concepts

Polar Coordinates

Polar coordinates represent points in a plane using a distance from a reference point (the pole) and an angle from a reference direction. In polar equations, 'r' denotes the radius (distance from the origin), and 'θ' represents the angle. Understanding how to convert between polar and Cartesian coordinates is essential for graphing polar equations.

Symmetry in Polar Graphs

Symmetry in polar graphs can be tested by analyzing the equation for specific transformations. A polar graph is symmetric about the polar axis if replacing 'θ' with '-θ' yields the same equation. It is symmetric about the line θ = π/2 if replacing 'r' with '-r' results in the same equation. Recognizing these symmetries helps in sketching the graph accurately.

Graphing Polar Equations

Graphing polar equations involves plotting points based on the values of 'r' for various angles 'θ'. The shape of the graph can vary significantly depending on the equation's form. For the given equation, r = 2 + 3 sin 2θ, understanding how the sine function affects the radius at different angles is crucial for accurately representing the graph's features.