Solve the equation for all solutions in the interval [0,2π).
\[
\cos \left(\frac{x}{2}\right)=\frac{\sqrt{3}}{2}
\]
Express answer in radians in exact form.
If there is more than one solution, enter the solutions as a comma-separated list.
\[
x=
\]

We are given the equation 

\[
\cos \left(\frac{x}{2}\right) = \frac{\sqrt{3}}{2}
\]

and need to find all solutions for \(x\) in the interval \([0, 2\pi)\).

### Step 1: Identify the angles
First, recall that \(\cos(\theta) = \frac{\sqrt{3}}{2}\) at specific standard angles. These angles are:

\[
\theta = \frac{\pi}{6}, \quad \theta = \frac{11\pi}{6}
\]

### Step 2: Set up the equation
Since \(\frac{x}{2} = \theta\), we have:

\[
\frac{x}{2} = \frac{\pi}{6}, \quad \frac{x}{2} = \frac{11\pi}{6}
\]

### Step 3: Solve for \(x\)
Multiply both sides of each equation by 2 to solve for \(x\):

1. For \(\frac{x}{2} = \frac{\pi}{6}\):
   \[
   x = 2 \times \frac{\pi}{6} = \frac{\pi}{3}
   \]

2. For \(\frac{x}{2} = \frac{11\pi}{6}\):
   \[
   x = 2 \times \frac{11\pi}{6} = \frac{11\pi}{3}
   \]

### Step 4: Adjust for the interval
Next, check if these solutions lie within the interval \([0, 2\pi)\).

- \(\frac{\pi}{3}\) is within the interval.
- \(\frac{11\pi}{3}\) needs adjustment since it's larger than \(2\pi\).

Now, reduce \(\frac{11\pi}{3}\) modulo \(2\pi\):

\[
\frac{11\pi}{3} - 2\pi = \frac{11\pi}{3} - \frac{6\pi}{3} = \frac{5\pi}{3}
\]

### Step 5: List all solutions
Thus, the solutions in the interval \([0, 2\pi)\) are:

\[
x = \frac{\pi}{3}, \frac{5\pi}{3}
\]

These are the solutions.

Would you like further details or have any questions?

Here are some related questions:

1. What are the general solutions for \(\cos\left(\frac{x}{2}\right) = \frac{\sqrt{3}}{2}\) for all \(x\)?
2. How do you solve similar equations involving sine functions?
3. What is the significance of reducing angles modulo \(2\pi\)?
4. Can this method be applied to equations involving tangent or cotangent?
5. What are the steps to solve equations with other trigonometric identities?

**Tip:** Always check the interval specified in the problem to ensure all solutions are within the correct range.

Learn how to solve the equation cos(x/2) = sqrt(3)/2 for all solutions in the interval [0, 2π) in exact radians. This problem involves trigonometric identities and equations, suitable for advanced high school students.

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