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calculate this math question step by step, thanks.

Let's solve this step-by-step.

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

We need to find the values of \( t \) where the point \( P \) crosses the x-axis. This happens when \( y(t) = 0 \).

### Step 1: Set the \( y(t) \) equation to 0
\[ \sin(t) + \sin\left(\frac{1}{2} t\right) = 0 \]

### Step 2: Solve for \( t \)
We can use the identity for the sum of sines:
\[ \sin(A) + \sin(B) = 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \]

Let's apply this to our equation:
\[ \sin(t) + \sin\left(\frac{1}{2} t\right) = 2 \sin\left(\frac{t + \frac{1}{2} t}{2}\right) \cos\left(\frac{t - \frac{1}{2} t}{2}\right) \]
\[ = 2 \sin\left(\frac{3}{4} t\right) \cos\left(\frac{1}{4} t\right) = 0 \]

This product equals zero if either factor is zero:
1. \( \sin\left(\frac{3}{4} t\right) = 0 \)
2. \( \cos\left(\frac{1}{4} t\right) = 0 \)

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

Where \( n \) is any integer. We need to find values of \( t \) within the given range \( -2\pi \leq t \leq 2\pi \).

Possible values of \( t \):
- \( t = 0 \)
- \( t = \frac{4}{3} \pi \)
- \( t = \frac{8}{3} \pi \)
- \( t = -\frac{4}{3} \pi \)
- \( t = -\frac{8}{3} \pi \)

Only the values within the given range are valid:
\[ t = 0, \frac{4}{3} \pi, -\frac{4}{3} \pi \]

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

Where \( n \) is any integer. We need to find values of \( t \) within the given range \( -2\pi \leq t \leq 2\pi \).

Possible values of \( t \):
- \( t = 2\pi \)
- \( t = -2\pi \)

### Step 5: Combine and filter the solutions
We discard \( t = 0 \) since we are looking for values where \( P \) crosses the x-axis besides the origin. We also discard \( \pm 2\pi \) as they fall outside the range.

Thus, the valid values of \( t \) are:
\[ t_1 = \frac{4}{3} \pi \]
\[ t_2 = -\frac{4}{3} \pi \]

### Final Answer
\[ [t_1, t_2, t_3] = \left[ \frac{4}{3} \pi, -\frac{4}{3} \pi, \frac{8}{3} \pi \right] \]

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:** When solving trigonometric equations, always consider using identities to simplify the expressions before solving them.

Learn how to solve parametric equations involving trigonometric identities step-by-step. This problem covers parametric equations and trigonometric identities, focusing on finding specific values of t where a point crosses the x-axis. Suitable for advanced high school students.

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