To determine how many times the polar curve $r = 2 + \sin(\theta)$ passes through the line $x = 1$ in Cartesian coordinates, we need to translate the problem into a coordinate system that allows us to compare both equations.

Step 1: Convert Polar to Cartesian Coordinates

In polar coordinates, we know the following relationships between $r$ and $\theta$, and the Cartesian coordinates $x$ and $y$:

$x = r \cos(\theta)$ $y = r \sin(\theta)$

Substituting $r = 2 + \sin(\theta)$ into the equation for $x$:

$x = (2 + \sin(\theta)) \cos(\theta)$

We are looking for the values of $\theta$ where $x = 1$, so set the equation equal to 1:

$(2 + \sin(\theta)) \cos(\theta) = 1$

Step 2: Solve the Equation for $\theta$

To solve $(2 + \sin(\theta)) \cos(\theta) = 1$, we can attempt to solve it directly or graphically analyze it to determine how many solutions exist within one period of $\theta$.

Let's proceed graphically or numerically. The equation depends on trigonometric functions, which have periodic behavior, so there might be multiple values of $\theta$ within the range $[0, 2\pi]$ for which this equation holds.

Step 3: Numerical or Graphical Solution (Discussion)

Without performing a detailed numerical analysis here, it's known that polar