To solve the problem of determining how many times the polar curve $r = 2 + \sin(\theta)$ passes through $x = 1$ in Cartesian coordinates, we need to translate this equation into the Cartesian system and then find the values of $\theta$ that satisfy the condition $x = 1$.

Step 1: Express the Polar Equation in Cartesian Coordinates

In polar coordinates, the relationship between $r$ and Cartesian coordinates $(x, y)$ is given by: $x = r \cos(\theta) \quad \text{and} \quad y = r \sin(\theta)$ Given $r = 2 + \sin(\theta)$, we substitute this into the expression for $x$: $x = (2 + \sin(\theta)) \cos(\theta)$ Now we want to find how many times this equation satisfies $x = 1$, so we set: $1 = (2 + \sin(\theta)) \cos(\theta)$ This is the equation we need to solve for $\theta$.

Step 2: Solve the Equation

We now have: $1 = (2 + \sin(\theta)) \cos(\theta)$ Expanding and rearranging the equation, we get: $\cos(\theta) = \frac{1}{2 + \sin(\theta)}$ This equation needs to be solved for $\theta$, and we will need to check how many values of $\theta$ satisfy this equation over the range $\theta \in [0, 2\pi]$.

Step 3: Analyze the Equation

The equation $\cos(\theta) = \frac{1}{2 + \sin(\theta)}$ is transcendental, so we can solve it by analyzing it graphically or numerically. However, we can infer the general behavior by considering:

1. Domain of the Equation: The right-hand side of the equation, $\frac{1}{2 + \sin(\theta)}$, is always positive and lies between $\frac{1}{3}$ (when $\sin(\theta) = 1$) and $\frac{1}{2}$ (when $\sin(\theta) = 0$).
2. Periodicity of Trigonometric Functions: Since both $\sin(\theta)$ and $\cos(\theta)$ are periodic with a period of $2\pi$, the equation could potentially have multiple solutions within a single period.

Step 4: Find Number of Solutions

Numerical or graphical analysis shows that this equation has two distinct solutions within the interval $[0, 2\pi]$. Therefore, the curve $r = 2 + \sin(\theta)$ passes through $x = 1$ twice.

Conclusion

The curve $r = 2 + \sin(\theta)$ passes through $x = 1$ in Cartesian coordinates 2 times.

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Would you like to go over any specific steps in detail or have any further questions?

Here are 5 related questions you could explore:

1. What is the geometric interpretation of polar curves in the Cartesian plane?
2. How do you find the points where a polar curve intersects the x-axis?
3. What are some general methods for solving transcendental equations like this one?
4. How does the behavior of trigonometric functions affect the number of solutions in equations like this?
5. How can graphical or numerical methods be applied to solve trigonometric equations?

Tip: For transcendental equations involving sine and cosine, graphical analysis often helps reveal how many solutions exist across specific intervals.