Solve the equation for solutions over the interval $[0,2 \pi)$. Write solutions as exact values or to four decimal places, as appropriate.
\[
\sin \frac{x}{3}-\cos \frac{x}{3}=0
\]
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution set is $\{$.
(Simplify your answe. Type an exact answer, using $\pi$ as needed. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.)
B. The solution is the empty set.

Step 1 ：First, we rewrite the equation \(\sin \frac{x}{3}-\cos \frac{x}{3}=0\) as \(\sin \frac{x}{3}=\cos \frac{x}{3}\).

Step 2 ：We know that \(\sin \theta = \cos \theta\) when \(\theta = \frac{\pi}{4} + n\pi\), where n is an integer.

Step 3 ：So, we can write \(\frac{x}{3} = \frac{\pi}{4} + n\pi\).

Step 4 ：Solving for x, we get \(x = 3\left(\frac{\pi}{4} + n\pi\right)\).

Step 5 ：We are looking for solutions in the interval [0, 2\pi), so we substitute n = 0, 1, 2, 3, 4, 5, 6, 7 into the equation and check which values of x fall within this interval.

Step 6 ：For n = 0, we get \(x = 3\left(\frac{\pi}{4}\right) = \frac{3\pi}{4}\), which is in the interval [0, 2\pi).

Step 7 ：For n = 1, we get \(x = 3\left(\frac{\pi}{4} + \pi\right) = \frac{7\pi}{4}\), which is in the interval [0, 2\pi).

Step 8 ：For n = 2, we get \(x = 3\left(\frac{\pi}{4} + 2\pi\right) = \frac{11\pi}{4}\), which is not in the interval [0, 2\pi).

Step 9 ：For n = 3, we get \(x = 3\left(\frac{\pi}{4} + 3\pi\right) = \frac{15\pi}{4}\), which is not in the interval [0, 2\pi).

Step 10 ：For n = 4, we get \(x = 3\left(\frac{\pi}{4} + 4\pi\right) = \frac{19\pi}{4}\), which is not in the interval [0, 2\pi).

Step 11 ：For n = 5, we get \(x = 3\left(\frac{\pi}{4} + 5\pi\right) = \frac{23\pi}{4}\), which is not in the interval [0, 2\pi).

Step 12 ：For n = 6, we get \(x = 3\left(\frac{\pi}{4} + 6\pi\right) = \frac{27\pi}{4}\), which is not in the interval [0, 2\pi).

Step 13 ：For n = 7, we get \(x = 3\left(\frac{\pi}{4} + 7\pi\right) = \frac{31\pi}{4}\), which is not in the interval [0, 2\pi).

Step 14 ：So, the solutions to the equation \(\sin \frac{x}{3}-\cos \frac{x}{3}=0\) in the interval [0, 2\pi) are \(\boxed{\frac{3\pi}{4}, \frac{7\pi}{4}}\).