To sketch the curve for \( r = 2\sin(x) \) in polar coordinates, let's go through the key points.

### Step-by-Step Guide:

1. **Understanding the Polar Equation**:
   - In polar coordinates, the equation \( r = 2\sin(x) \) represents a relationship between the radial distance \( r \) from the origin and the angle \( x \), which is the polar angle (in radians).
   - For each value of \( x \), \( r \) gives the distance from the origin (pole).

2. **Key Features**:
   - The curve repeats every \( 2\pi \), since \( \sin(x) \) has a period of \( 2\pi \).
   - The values of \( r \) vary depending on the angle \( x \). Specifically:
     - When \( x = 0 \), \( r = 0 \).
     - When \( x = \frac{\pi}{2} \), \( r = 2 \).
     - When \( x = \pi \), \( r = 0 \).
     - When \( x = \frac{3\pi}{2} \), \( r = -2 \) (meaning the point is plotted on the opposite side of the origin).
   - Since \( r \) can be negative, the curve reflects across the origin for certain angles.

3. **Shape of the Curve**:
   - The curve is symmetric about the polar axis (horizontal axis), and the resulting shape is known as a **limacon**.
   - It has an inner loop due to the negative values of \( r \).

4. **Plotting Important Points**:
   - At \( x = 0 \), \( r = 0 \).
   - At \( x = \frac{\pi}{2} \), \( r = 2 \), so plot a point 2 units away from the origin.
   - At \( x = \pi \), \( r = 0 \).
   - At \( x = \frac{3\pi}{2} \), \( r = -2 \), meaning the point is 2 units away but on the opposite side of the origin.

5. **Final Shape**:
   - The curve forms a limacon with no dimple and a loop. It starts at the origin, loops outward to a maximum of 2 units, and then reflects to form a symmetrical shape.

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Learn how to sketch the polar curve r = 2sin(x) with a detailed explanation. This trigonometry problem covers polar coordinates, sine function periodicity, and the symmetry properties of polar curves. Ideal for high school students in Grades 10-12.

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