Review the response and rate each piece. Only mark steps incorrect if they contain a mathematical error. The model may add preamble or summary steps which seem superfluous, but do not mark them incorrect unless they contain a math error.

Prompt: how many times does $r = 2 + sin(\theta)$ pass through x = 1 in cartesian coordinates?

Step 1

To gain insight into this problem, we can plot the function $f(\theta) = 2\cos(\theta) + \frac{1}{2}\sin(2\theta) - 1$ and observe how many times it crosses the $\theta$-axis, which corresponds to the instances where $x = 1$.

Step 2

Alternatively, we can analyze the problem geometrically by plotting the polar equation $r = 2 + \sin(\theta)$ and observing how many times it intersects the vertical line $x = 1$.

Step 3

By plotting the polar equation, we can see that it forms a cardioid shape with the cusp pointing to the right. The line $x = 1$ intersects this cardioid shape twice.

Step 4

Therefore, based on the geometric analysis, the polar equation $r = 2 + \sin(\theta)$ passes through $x = 1$ twice.

Which step is wrong in getting the answer