To determine how many times the polar curve $r = 2 + \sin(\theta)$ passes through $x = 1$ in Cartesian coordinates, let's go through the following steps:

1. Convert to Cartesian coordinates: In polar coordinates, we have: $x = r \cos(\theta) \quad \text{and} \quad y = r \sin(\theta).$ Since $r = 2 + \sin(\theta)$, we can write: $x = (2 + \sin(\theta)) \cos(\theta).$

2. Set $x = 1$: We are interested in when $x = 1$, so we set up the equation: $(2 + \sin(\theta)) \cos(\theta) = 1.$

3. Simplify the equation: Expanding this equation does not yield a simple factorization, but we can analyze it by testing specific values of $\theta$ to see if they satisfy this equation.

4. Test values of $\theta$ in the range $0 \leq \theta < 2\pi$:

   • If we substitute values of $\theta$ where $\sin(\theta)$ reaches extreme points, like $\theta = \frac{\pi}{2}$, $\theta = \frac{3\pi}{2}$, $\theta = 0$, and $\theta = \pi$, we can determine when $x = 1$.
   • Through testing and substitution, you will find that there are two solutions for $\theta$ that satisfy $x = 1$ in the interval $0 \leq \theta < 2\pi$.

Thus, the curve $r = 2 + \sin(\theta)$ passes through $x = 1$ two times as $\theta$ varies from $0$ to $2\pi$.

Would you like more details on each step?

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1. How do we convert polar equations to Cartesian coordinates in general?
2. Why does setting $x = 1$ help us find specific points on a polar curve?
3. What are common strategies for testing trigonometric values when solving polar equations?
4. How can symmetry in polar curves simplify solving such problems?
5. Are there other ways to solve this without substituting specific $\theta$ values?

Tip: When analyzing polar curves, try graphing the function to get a visual sense of how the curve behaves as $\theta$ varies.