To determine how many times the curve $r = 2 + \sin(\theta)$ passes through $x = 1$ in Cartesian coordinates, we can follow these steps:

1. Polar to Cartesian Conversion: In polar coordinates, the Cartesian coordinates $(x, y)$ can be expressed as: $x = r \cos(\theta)$ $y = r \sin(\theta)$ Using the given equation $r = 2 + \sin(\theta)$, we can substitute $r$ into the expression for $x$: $x = (2 + \sin(\theta)) \cos(\theta)$ We are interested in finding how many values of $\theta$ make $x = 1$. Thus, set the equation equal to 1: $1 = (2 + \sin(\theta)) \cos(\theta)$ We now need to solve this equation for $\theta$.

2. Simplify and Solve: Rearranging the equation gives: $\cos(\theta) = \frac{1}{2 + \sin(\theta)}$ To find how many times this equation is satisfied, we need to analyze how many solutions it has over the interval $0 \leq \theta < 2\pi$.

3. Graphical or Numerical Approach: Since solving this equation algebraically is complex, we can instead examine the behavior of the functions involved. The function $\sin(\theta)$ oscillates between -1 and 1, so $r = 2 + \sin(\theta)$ oscillates between 1 and 3. Meanwhile, $\cos(\theta)$ also oscillates between -1 and 1.

   The equation $\cos(\theta) = \frac{1}{2 + \sin(\theta)}$ indicates that the value of $\cos(\theta)$ must match the reciprocal of $2 + \sin(\theta)$, which varies between $\frac{1}{3}$ and 1. Since $\cos(\theta)$ is periodic, the equation will likely have multiple solutions.

4. Conclusion: From graphical analysis or numerical methods (such as solving the equation graphically or using computational tools), it can be determined that the equation $1 = (2 + \sin(\theta)) \cos(\theta)$ has two solutions over the interval $0 \leq \theta < 2\pi$.

Thus, the curve $r = 2 + \sin(\theta)$ passes through $x = 1$ twice in Cartesian coordinates.

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Would you like to see a graph of the functions involved or further details on solving the equation? Here are some related questions:

1. How do you convert polar equations to Cartesian coordinates in general?
2. Can the number of intersections change for other values of $x$?
3. What role does symmetry play in determining the number of intersections?
4. How would the curve change if $r = 2 + k\sin(\theta)$ for some constant $k$?
5. How can numerical methods help in finding exact solutions to transcendental equations?

Tip: In polar coordinates, analyzing intersections often involves converting to Cartesian or graphically interpreting the polar function’s behavior.