I apologize for any confusion. Let's verify the solution more carefully, ensuring all steps and conditions are correctly followed.

Given:
\[ x(t) = \cos(t) + \cos\left(\frac{1}{2} t\right) \]
\[ y(t) = \sin(t) + \sin\left(\frac{1}{2} t\right) \]

The problem requires finding the values of \( t \) where \( y(t) = 0 \).

### Step-by-Step Solution:

1. **Set \( y(t) = 0 \):**
\[ \sin(t) + \sin\left(\frac{1}{2} t\right) = 0 \]

2. **Use the sum-to-product identities:**
\[ \sin(t) + \sin\left(\frac{1}{2} t\right) = 2 \sin\left(\frac{3}{4} t\right) \cos\left(\frac{1}{4} t\right) = 0 \]

This gives us two conditions:
\[ \sin\left(\frac{3}{4} t\right) = 0 \]
\[ \cos\left(\frac{1}{4} t\right) = 0 \]

3. **Solve \( \sin\left(\frac{3}{4} t\right) = 0 \):**
\[ \frac{3}{4} t = n \pi \]
\[ t = \frac{4}{3} n \pi \]

4. **Solve \( \cos\left(\frac{1}{4} t\right) = 0 \):**
\[ \frac{1}{4} t = \left(n + \frac{1}{2}\right) \pi \]
\[ t = 4\left(n + \frac{1}{2}\right) \pi \]

5. **Identify the range \( -2\pi \leq t \leq 2\pi \):**
For \( t = \frac{4}{3} n \pi \):
- \( n = -1, 0, 1 \)
- Possible \( t \) values: \( -\frac{4}{3} \pi, 0, \frac{4}{3} \pi \)

For \( t = 4\left(n + \frac{1}{2}\right) \pi \):
- \( n = -1, 0 \)
- Possible \( t \) values: \( -2\pi, 2\pi \)

### Verification:
We need values of \( t \) where the point \( P \) passes through the x-axis three more times besides the origin. We discard \( t = 0 \) and check other values within the given range.

### Possible values of \( t \) from \(\sin\left(\frac{3}{4} t\right) = 0 \):
\[ t = -\frac{4}{3} \pi \]
\[ t = \frac{4}{3} \pi \]

### Possible values of \( t \) from \(\cos\left(\frac{1}{4} t\right) = 0 \):
\[ t = -2\pi \]
\[ t = 2\pi \]

Given the correct interval:
\[ t \in [-2\pi, 2\pi] \]

### Filtering:
- Discard \( t = 0 \) as it’s the origin.
- \( t = \pm 2\pi \) are valid but should be checked in context.

### Conclusion:
The valid values where \( y(t) = 0 \) and point \( P \) passes through the x-axis (not origin):

1. \( t_1 = -\frac{4}{3} \pi \)
2. \( t_2 = \frac{4}{3} \pi \)
3. \( t_3 = 2\pi \)

### Final Answer:
\[ [t_1, t_2, t_3] = \left[ -\frac{4}{3} \pi, \frac{4}{3} \pi, 2\pi \right] \]

If you have any further questions or need more details, feel free to ask! Here are some questions you might be interested in:

1. How can I verify the solutions for \( t \)?
2. What is the geometric interpretation of this parametric equation?
3. How do I apply these parametric equations in a different context?
4. Can you provide a graph for better visualization?
5. Are there alternative methods to solve this problem?

**Tip:** Always double-check boundary conditions and values to ensure they fit within the given range for parametric equations.

Learn how to verify solutions for parametric equations involving trigonometric functions step-by-step. This problem covers trigonometry concepts, including sum-to-product identities. Suitable for undergraduate students studying advanced trigonometry.

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