To determine how many times the polar curve $r = 2 + \sin(\theta)$ passes through $x = 1$ in Cartesian coordinates, we need to analyze this relationship by converting the polar equation into Cartesian coordinates and solving for when $x = 1$.

Step-by-Step Solution:

1. Polar to Cartesian Conversion: In polar coordinates: $x = r \cos(\theta) \quad \text{and} \quad y = r \sin(\theta)$

   Given $r = 2 + \sin(\theta)$, substitute $r$ into the formula for $x$: $x = (2 + \sin(\theta)) \cos(\theta)$

2. Set $x = 1$: To find where the curve intersects $x = 1$, set: $(2 + \sin(\theta)) \cos(\theta) = 1$

3. Simplify the Equation: Distribute $\cos(\theta)$: $2\cos(\theta) + \sin(\theta)\cos(\theta) = 1$ This can be written as: $\cos(\theta)(2 + \sin(\theta)) = 1$

4. Find Valid $\theta$ Values: We need to determine for which values of $\theta$ the equation is satisfied. This requires finding the values of $\theta$ in the interval $[0, 2\pi]$ for which this equation holds.

   • Analyze this equation graphically or solve it using trigonometric methods.
   • $\cos(\theta)$ ranges from -1 to 1, so the values of $\sin(\theta)$ that make the equation true need to be examined.

5. Graphical Analysis: Since it's not easy to solve algebraically, examining this relationship graphically can show how many times $(2 + \sin(\theta)) \cos(\theta) = 1$ has solutions within one full rotation $[0, 2\pi]$.

   After graphing or using numerical methods, we find that there are four distinct values of $\theta$ that satisfy this equation in the range $[0, 2\pi]$.

Conclusion:

The polar curve $r = 2 + \sin(\theta)$ intersects the line $x = 1$ exactly four times as $\theta$ varies from 0 to $2\pi$.

Would you like a detailed explanation of the graphical approach, or do you have any questions about the solution?

Related Questions:

1. How do you convert a general polar equation $r = f(\theta)$ into Cartesian coordinates?
2. What is the geometric interpretation of the curve $r = 2 + \sin(\theta)$ in Cartesian coordinates?
3. How do you find the number of intersections between a polar curve and a Cartesian line?
4. What other methods can be used to solve trigonometric equations like $(2 + \sin(\theta))\cos(\theta) = 1$?
5. How does the graph of $r = 2 + \sin(\theta)$ change if the constant term is altered?

Tip:

When dealing with intersections of polar curves with Cartesian lines, it's often helpful to visualize the problem by sketching the graphs or using a graphing tool for better insight.