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The problem asks to plot the polar equation \( r^2 = 25 \sin(2\theta) \). Here's a step-by-step process to interpret and graph the equation:

### Step 1: Analyze the Equation
The equation is given as:
\[
r^2 = 25 \sin(2\theta)
\]

To make it easier for graphing purposes, we can solve for \( r \):
\[
r = \pm \sqrt{25 \sin(2\theta)}
\]
\[
r = \pm 5 \sqrt{\sin(2\theta)}
\]

### Step 2: Key Points
The behavior of \( \sin(2\theta) \) repeats every \( \pi \) radians because of the periodicity of the sine function. The value of \( r \) will depend on the value of \( \theta \):
- For \( \sin(2\theta) > 0 \), \( r \) is real.
- For \( \sin(2\theta) \leq 0 \), \( r \) is imaginary, meaning there's no real solution.

### Step 3: Symmetry
Since the equation involves \( \sin(2\theta) \), which is symmetric with respect to rotation by \( \pi \), the graph will have symmetry about the origin, with lobes corresponding to intervals where \( \sin(2\theta) \) is positive.

### Step 4: Graph Shape
This type of polar equation generates a **lemniscate** (figure-eight-shaped curve). For the given equation:
- The curve will have two lobes.
- The maximum radius will be \( 5 \), because \( r = 5 \) when \( \sin(2\theta) = 1 \).

### Step 5: Plotting on Polar Graph
Using the given polar grid:
- At \( \theta = 0 \), \( r = 0 \) (since \( \sin(2\theta) = 0 \)).
- As \( \theta \) increases to \( \pi/4 \), \( r \) increases, reaching \( r = 5 \) when \( \theta = \pi/4 \).
- As \( \theta \) continues to \( \pi/2 \), \( r \) goes back to zero (due to \( \sin(2\theta) = 0 \)).

The lobes will be centered at angles \( \pi/4 \) and \( 5\pi/4 \), and the maximum distance will be \( r = 5 \).

Would you like me to plot the curve or go into further details about any of these steps?

Here are 5 related questions you might consider:
1. What is the general form of a lemniscate in polar coordinates?
2. How does the equation \( r^2 = a \sin(2\theta) \) vary when the value of \( a \) changes?
3. What is the role of symmetry in plotting polar curves?
4. How do trigonometric identities simplify graphing polar equations?
5. Can this polar equation be transformed into Cartesian coordinates?

**Tip:** When working with polar graphs, always consider the periodicity of trigonometric functions for symmetry and key points.

Trazar el gráfico de la ecuación polar: r^2 = 25 sin(2θ)

Discover how to graph the polar equation r^2 = 25 sin(2θ). This problem involves key trigonometric concepts and graphing techniques, ideal for students in Grades 11-12. The solution includes symmetry analysis and curve properties, yielding a lemniscate with a maximum radius of 5.

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