To determine how many times the polar curve $r = 2 + \sin(\theta)$ passes through the line $x = 1$ in Cartesian coordinates, let’s proceed step-by-step.

Step 1: Convert the Polar Equation to Cartesian Form

The polar equation is given by: $r = 2 + \sin(\theta)$ In polar coordinates:

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

So, we can express $x$ as: $x = r \cos(\theta) = (2 + \sin(\theta)) \cos(\theta)$

Step 2: Set $x = 1$

To find when the curve passes through $x = 1$, we set: $1 = (2 + \sin(\theta)) \cos(\theta)$ This equation gives us the condition on $\theta$ that we need to solve.

Step 3: Solve for $\sin(\theta)$ in Terms of $\cos(\theta)$

Rearrange the equation to solve for $\sin(\theta)$: $\cos(\theta) = \frac{1}{2 + \sin(\theta)}$ This is a trigonometric equation, and solving it directly requires analyzing possible values of $\theta$ that satisfy this relationship.

Step 4: Check If This Problem is Solvable

Solving the equation explicitly is possible but challenging due to the nonlinear nature. However, we can analyze the behavior of $r$ as a function of $\theta$ and determine the possible solutions without solving it entirely.

Step 5: Analyzing $r = 2 + \sin(\theta)$

Since $\sin(\theta)$ oscillates between -1 and 1, $r = 2 + \sin(\theta)$ oscillates between 1 and 3. Therefore:

• The curve reaches $r = 1$ at its minimum when $\sin(\theta) = -1$.
• The maximum $r = 3$ occurs when $\sin(\theta) = 1$.

This suggests that the curve can reach $x = 1$ multiple times as $\theta$ varies, potentially passing through $x = 1$ twice in each cycle.

Step 6: Final Answer

The problem is solvable in the sense that we can determine the number of intersections by analyzing the range of $r$ and the periodic nature of $\sin(\theta)$, even though solving the exact points would be complex without computational tools. The curve $r = 2 + \sin(\theta)$ will pass through $x = 1$ twice per cycle of $\theta$ over $[0, 2\pi]$ due to the oscillatory nature of the trigonometric terms.

Summary

The curve $r = 2 + \sin(\theta)$ passes through $x = 1$ twice as $\theta$ varies from $0$ to $2\pi$.

Would you like more detailed calculations, or do you have any questions on this?

Related Questions

1. How would you solve for the exact values of $\theta$ where the curve intersects $x = 1$?
2. How can you verify the number of intersections using a graph of $r = 2 + \sin(\theta)$?
3. How would the answer change if the equation were $r = 1 + \sin(\theta)$ instead?
4. Can you derive the Cartesian form of the curve $r = 2 + \sin(\theta)$ entirely?
5. How does the range of $r$ affect the intersection points with various lines $x = a$?

Tip: For complex trigonometric-polar problems, graphing the curve often provides valuable insight into the behavior and possible intersections.