Let's carefully revisit and solve the problem step-by-step to ensure accuracy.

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

The point $P$ crosses the x-axis at the origin and three more times. This happens when $y(t) = 0$.

Step 1: Set $y(t)$ to 0

$\sin(t) + \sin\left(\frac{1}{2} t\right) = 0$

Step 2: Solve for $t$

Using 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 $t$:

• $t = 0$
• $t = \frac{4}{3} \pi$
• $t = \frac{8}{3} \pi$
• $t = -\frac{4}{3} \pi$
• $t = -\frac{8}{3} \pi$

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 are outside the given range.

Thus, the valid values of $t$ are: $t_1 = \frac{4}{3} \pi$ $t_2 = -\frac{4}{3} \pi$ $t_3 = \frac{8}{3} \pi$

Let's list these values clearly.

Final Answer

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

Upon further inspection, let's revalidate all possible $t$ values within the given range. The possible solutions are:

For $\sin\left(\frac{3}{4} t\right) = 0$: $t = \frac{4}{3} n \pi$ Where $n = -2, -1, 0, 1, 2$ to fit within the range $-2\pi \leq t \leq 2\pi$.

This yields:

• $t = -\frac{8}{3} \pi$
• $t = -\frac{4}{3} \pi$
• $t = 0$
• $t = \frac{4}{3} \pi$
• $t = \frac{8}{3} \pi$

For $\cos\left(\frac{1}{4} t\right) = 0$: $t = (4n + 2) \pi$ Where $n = -1, 0, 1$ to fit within the range $-2\pi \leq t \leq 2\pi$.

This yields:

• $t = -2\pi$
• $t = 2\pi$

Combining these valid $t$ values, and excluding $t = 0$ as required:

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