how many times does $r = 2 + sin(\theta)$ pass through x = 1 in cartesian coordinates?

To determine how many times the polar curve $ r = 2 + \sin(\theta) $ passes through $ x = 1 $ in Cartesian coordinates, let’s walk through the problem step-by-step.

### 1. Polar to Cartesian Conversion  
In polar coordinates:
\[
x = r \cos(\theta) \quad \text{and} \quad y = r \sin(\theta)
\]

For the given curve:
\[
r = 2 + \sin(\theta)
\]
Thus, the $x$-coordinate becomes:
\[
x = (2 + \sin(\theta)) \cos(\theta)
\]

### 2. Setting $x = 1$
We are interested in the points where this curve passes through $x = 1$. Therefore:
\[
(2 + \sin(\theta)) \cos(\theta) = 1
\]

### 3. Rearranging the Equation
\[
2 \cos(\theta) + \sin(\theta) \cos(\theta) = 1
\]
This simplifies to:
\[
2 \cos(\theta) + \frac{1}{2} \sin(2\theta) = 1
\]

This is a transcendental equation, and exact solutions are difficult to find algebraically, so we need to focus on **the nature of the curve** to understand how often this equation can hold.

### 4. Nature of the Polar Curve $ r = 2 + \sin(\theta) $
- The curve $ r = 2 + \sin(\theta) $ is a limaçon with an inner loop.
- The value of $ r $ ranges from 1 to 3, since:
  \[
  2 + \sin(\theta) \in [1, 3]
  \]
- This limaçon is symmetric about the $x$-axis, and it loops around the origin.

### 5. Analyzing Solutions for $x = 1$
Because the curve loops around and intersects the vertical line $x = 1$ multiple times, we expect multiple solutions. By symmetry and the behavior of the limaçon, it will cross $x = 1$ **four times**: twice for the outer loop and twice for the inner loop.

---

### Final Answer
The polar curve $ r = 2 + \sin(\theta) $ passes through the line $x = 1$ exactly **four times**.

---

Do you need more details or have any further questions?  
Here are 5 related questions to deepen your understanding:  
1. How do limaçons with different parameters behave in polar coordinates?  
2. What is the range of $y$-coordinates for the curve $ r = 2 + \sin(\theta) $?  
3. How do you convert other polar curves, like circles, to Cartesian coordinates?  
4. How does the symmetry of the curve help simplify finding intersections?  
5. What is the geometric interpretation of transcendental equations like $2 \cos(\theta) + \frac{1}{2} \sin(2\theta) = 1$?  

**Tip:** When dealing with polar curves, symmetry and understanding their overall shape often simplify intersection problems.

Explore how the polar curve r = 2 + sin(θ), a limaçon with an inner loop, intersects the Cartesian line x = 1 four times. This detailed solution covers polar-to-Cartesian conversion, transcendental equations, and the symmetry of polar curves. Ideal for Grades 10-12 students.

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