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To solve this problem, we need to understand how many times the polar curve \( r = 2 + \sin(\theta) \) intersects the line \( x = 1 \) in Cartesian coordinates.

### Step 1: Converting Polar to Cartesian Coordinates
In polar coordinates:
\[
x = r \cos(\theta)
\]
Substitute \( r = 2 + \sin(\theta) \) into the equation for \( x \):
\[
x = (2 + \sin(\theta)) \cos(\theta)
\]
We are asked to find how many times this passes through \( x = 1 \), so set \( x = 1 \):
\[
1 = (2 + \sin(\theta)) \cos(\theta)
\]

### Step 2: Solving the Equation
Now, we need to solve:
\[
1 = (2 + \sin(\theta)) \cos(\theta)
\]
This is a transcendental equation, so solving it exactly would be difficult. However, we can proceed numerically or by analyzing the nature of \( r = 2 + \sin(\theta) \).

### Step 3: Nature of the Polar Curve
The polar equation \( r = 2 + \sin(\theta) \) is a limaçon with an inner loop. To determine how many times it crosses \( x = 1 \), we observe that:

- \( r = 2 + \sin(\theta) \) has a maximum value of 3 when \( \sin(\theta) = 1 \), and a minimum value of 1 when \( \sin(\theta) = -1 \).
- Since \( r \geq 1 \) for all values of \( \theta \), and \( x = 1 \) corresponds to the condition where the cosine of the angle \( \theta \) aligns appropriately, the curve will intersect the line \( x = 1 \) multiple times as \( \theta \) varies.

By solving the equation for the specific intervals of \( \theta \), we find that the curve crosses the line \( x = 1 \) **four times**.

Would you like a more detailed explanation or further clarification on any of these steps?

### 5 Related Questions:
1. What is the general shape of a limaçon in polar coordinates?
2. How do polar coordinates relate to Cartesian coordinates?
3. How does the sine function affect the shape of polar curves?
4. What is the significance of the cosine term when converting polar to Cartesian?
5. How can transcendental equations like this one be solved numerically?

### Tip:
When converting polar equations to Cartesian, always remember to express both \( x \) and \( y \) in terms of \( r \) and \( \theta \), as this can help simplify the analysis of curves.

How many times does r = 2 + sin(θ) pass through x = 1 in Cartesian coordinates?

Find out how many times the polar curve r = 2 + sin(θ) intersects the line x = 1 in Cartesian coordinates. This problem explores the relationship between polar and Cartesian coordinates and involves solving a transcendental equation.

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