To determine how many times the polar equation $r = 2 + \sin(\theta)$ passes through the Cartesian coordinate $x = 1$, we start by converting the polar equation to Cartesian coordinates.

The relationships between polar and Cartesian coordinates are: $x = r \cos(\theta)$ $y = r \sin(\theta)$

Substituting $r$ into the $x$ equation: $x = (2 + \sin(\theta)) \cos(\theta)$ We want to find $\theta$ such that $x = 1$: $1 = (2 + \sin(\theta)) \cos(\theta)$ This can be rewritten as: $2 \cos(\theta) + \sin(\theta) \cos(\theta) = 1$ Rearranging gives: $\sin(\theta) \cos(\theta) = 1 - 2 \cos(\theta)$

Using the identity $\sin(\theta) = \sqrt{1 - \cos^2(\theta)}$ can be helpful. However, it might be simpler to analyze the equation for intersections by substituting values.

Let's isolate $\sin(\theta)$: $\sin(\theta) = \frac{1 - 2 \cos(\theta)}{\cos(\theta)}$

This leads to: $\sin(\theta) = \tan(\theta) - 2$

This equation will give us the possible values of $\theta$.

Finding the Solutions

Next, we can explore the range of $\theta$. The range of $r$ from the polar equation $r = 2 + \sin(\theta)$ is from $1$ to $3$ as $\sin(\theta)$ varies from $-1$ to $1$.

Checking Points:

1. For $\theta = 0$: $r = 2 + \sin(0) = 2$ $x = 2$.

2. For $\theta = \frac{\pi}{2}$: $r = 2 + \sin\left(\frac{\pi}{2}\right) = 3$ $x = 0$.

3. For $\theta = \pi$: $r = 2 + \sin(\pi) = 2$ $x = -2$.

4. For $\theta = \frac{3\pi}{2}$: $r = 2 + \sin\left(\frac{3\pi}{2}\right) = 1$ $x = -1$.

Given that the curve oscillates with $\sin(\theta)$, we can analyze the graph of $r$ and $x$ through a full rotation from $0$ to $2\pi$ to find out how many times it crosses $x = 1$.

Crossings through $x = 1$

To confirm the crossings, we note that as $\theta$ goes from $0$ to $\pi/2$, the value of $x$ decreases from $2$ to $0$, and then from $\pi/2$ to $\pi$, it increases back from $0$ to $-2$.

This indicates:

1. A crossing occurs as $x$ decreases from $2$ to $0$.
2. A crossing occurs again from $0$ to $-2$.

By symmetry, there will be two crossings in $[0, \pi]$ and two more in $[\pi, 2\pi]$ due to the periodic nature of $\sin(\theta)$.

Conclusion

Therefore, the polar equation $r = 2 + \sin(\theta)$ passes through $x = 1$ a total of 4 times in Cartesian coordinates.

If you have any questions or need further details, feel free to ask! Here are some follow-up questions:

1. How does the shape of the graph change with different values of the coefficient in front of $\sin(\theta)$?
2. What would be the intersections for $r = 1 + \cos(\theta)$?
3. How do polar coordinates relate to complex numbers?
4. Can you explain how to graph polar equations?
5. What are other forms of polar equations and their intersections with Cartesian lines?

Tip: When analyzing polar equations, always consider converting them to Cartesian form for easier visualization and intersection analysis.