Sketch a well labeled curve for the following polar equation. r^2=16sin(2theta) 

To sketch the curve given by \( r^2 = 16 \sin(2\theta) \), let's analyze the equation and identify key features of this polar graph.

### Step 1: Identify the Type of Curve
The equation \( r^2 = 16 \sin(2\theta) \) is a type of *lemniscate*, which is characterized by a figure-eight or infinity-like shape. We can confirm this by comparing it with the standard form of a lemniscate, which often takes the form \( r^2 = a^2 \sin(2\theta) \). Here, \( a^2 = 16 \), so \( a = 4 \).

### Step 2: Determine Key Points
1. **Intercepts**: 
   The curve will intersect the origin when \( r = 0 \). This happens when \( \sin(2\theta) = 0 \). Solving for \( \theta \), we get:
   \[
   2\theta = n\pi \implies \theta = \frac{n\pi}{2}, \quad n \in \mathbb{Z}.
   \]
   Thus, the curve passes through the origin at \( \theta = 0, \pi, \frac{\pi}{2}, \frac{3\pi}{2}, \ldots \).

2. **Maximum Radius**: 
   The maximum value of \( r^2 \) is 16 (when \( \sin(2\theta) = 1 \)). This corresponds to:
   \[
   r = \pm 4.
   \]

3. **Symmetry**: 
   The curve is symmetric about both the polar axis (the line \( \theta = 0 \)) and the line \( \theta = \frac{\pi}{2} \), due to the nature of the \( \sin(2\theta) \) term.

### Step 3: Sketch the Curve
1. The curve extends outward to a maximum radius of \( r = 4 \).
2. Since it is a lemniscate, it will resemble a figure-eight pattern.
3. The points where \( r = 0 \) will correspond to the axes at \( \theta = 0 \), \( \pi \), \( \frac{\pi}{2} \), and \( \frac{3\pi}{2} \).

### Final Sketch

In the sketch:
- **Label the points** where the curve reaches its maximum radius at \( r = 4 \).
- **Mark the origin** where the curve intersects at \( r = 0 \).
- **Draw a smooth figure-eight shape** centered at the origin.

### Verbal Description of the Sketch
1. The curve consists of two loops, one lying along the horizontal axis and the other along the vertical axis.
2. Each loop touches the maximum radius of \( 4 \) at its farthest points.
3. The center of the figure-eight shape is the origin, and the loops intersect there.

Would you like a graphical plot of this curve, or do you want to proceed with this detailed explanation?

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**Relative Questions**: 
1. What other polar equations can form a lemniscate shape?
2. How does the value of \( a \) in \( r^2 = a^2 \sin(2\theta) \) affect the size of the lemniscate?
3. How would the curve change if we had \( \cos(2\theta) \) instead of \( \sin(2\theta) \)?
4. What are the intercepts of the curve when expressed in Cartesian coordinates?
5. How would the sketch change if the coefficient of \( \sin(2\theta) \) were negative?

**Tip**: When sketching polar equations, identifying key points and checking for symmetry simplifies the plotting process significantly.

Sketch a well-labeled curve for the following polar equation: r^2 = 16sin(2θ)

Learn how to sketch a polar curve defined by the equation r^2 = 16sin(2θ). This problem explains the characteristics of lemniscates, key points like intercepts, symmetry, and maximum radius, with a step-by-step approach. Ideal for high school students studying advanced trigonometry and polar coordinates.

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