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

solve this problem carefully. also is this prompt solveable?

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

### Step 1: Convert the Polar Equation to Cartesian Form
The polar equation is given by:
\[
r = 2 + \sin(\theta)
\]
In polar coordinates:
- \( x = r \cos(\theta) \)
- \( y = r \sin(\theta) \)

So, we can express \( x \) as:
\[
x = r \cos(\theta) = (2 + \sin(\theta)) \cos(\theta)
\]

### Step 2: Set \( x = 1 \)
To find when the curve passes through \( x = 1 \), we set:
\[
1 = (2 + \sin(\theta)) \cos(\theta)
\]
This equation gives us the condition on \( \theta \) that we need to solve.

### Step 3: Solve for \( \sin(\theta) \) in Terms of \( \cos(\theta) \)
Rearrange the equation to solve for \( \sin(\theta) \):
\[
\cos(\theta) = \frac{1}{2 + \sin(\theta)}
\]
This is a trigonometric equation, and solving it directly requires analyzing possible values of \( \theta \) that satisfy this relationship.

### Step 4: Check If This Problem is Solvable
Solving the equation explicitly is possible but challenging due to the nonlinear nature. However, we can analyze the behavior of \( r \) as a function of \( \theta \) and determine the possible solutions without solving it entirely.

### Step 5: Analyzing \( r = 2 + \sin(\theta) \)
Since \( \sin(\theta) \) oscillates between -1 and 1, \( r = 2 + \sin(\theta) \) oscillates between 1 and 3. Therefore:
- The curve reaches \( r = 1 \) at its minimum when \( \sin(\theta) = -1 \).
- The maximum \( r = 3 \) occurs when \( \sin(\theta) = 1 \).

This suggests that the curve can reach \( x = 1 \) multiple times as \( \theta \) varies, potentially passing through \( x = 1 \) twice in each cycle.

### Step 6: Final Answer
The problem **is solvable** in the sense that we can determine the number of intersections by analyzing the range of \( r \) and the periodic nature of \( \sin(\theta) \), even though solving the exact points would be complex without computational tools. The curve \( r = 2 + \sin(\theta) \) will pass through \( x = 1 \) **twice per cycle of \( \theta \) over \( [0, 2\pi] \)** due to the oscillatory nature of the trigonometric terms.

### Summary
The curve \( r = 2 + \sin(\theta) \) passes through \( x = 1 \) **twice** as \( \theta \) varies from \( 0 \) to \( 2\pi \).

Would you like more detailed calculations, or do you have any questions on this? 

### Related Questions
1. How would you solve for the exact values of \( \theta \) where the curve intersects \( x = 1 \)?
2. How can you verify the number of intersections using a graph of \( r = 2 + \sin(\theta) \)?
3. How would the answer change if the equation were \( r = 1 + \sin(\theta) \) instead?
4. Can you derive the Cartesian form of the curve \( r = 2 + \sin(\theta) \) entirely?
5. How does the range of \( r \) affect the intersection points with various lines \( x = a \)?

**Tip:** For complex trigonometric-polar problems, graphing the curve often provides valuable insight into the behavior and possible intersections.

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

Explore how often the polar curve r = 2 + sin(theta) intersects the line x = 1 in Cartesian coordinates. This problem involves polar-to-Cartesian conversion, trigonometric identities, and analysis of trigonometric behavior, suitable for advanced math students.

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