The least positive solution of `cot (pi/(3 sqrt(3)) sin 2x)=sqrt(3)` lies in

Options

A. `(0, pi/6]`
B. `(pi/9, pi/6)`
C. `(pi/12, pi/9]`
D. `(pi/3, pi/2]`

Explanation

To solve the equation \( \cot\left(\frac{\pi}{3\sqrt{3}} \sin 2x\right) = \sqrt{3} \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \cot\left(\frac{\pi}{3\sqrt{3}} \sin 2x\right) = \sqrt{3} \]

### Step 2: Find the angle whose cotangent is \(\sqrt{3}\)
We know that:
\[ \cot\left(\frac{\pi}{3}\right) = \sqrt{3} \]
Thus, we can write:
\[ \frac{\pi}{3\sqrt{3}} \sin 2x = \frac{\pi}{3} + n\pi \quad (n \in \mathbb{Z}) \]

### Step 3: Isolate \(\sin 2x\)
Now, we will isolate \(\sin 2x\):
\[ \sin 2x = \frac{\pi}{3} \cdot \frac{3\sqrt{3}}{\pi} + n\pi \cdot \frac{3\sqrt{3}}{\pi} \]
This simplifies to:
\[ \sin 2x = \sqrt{3} + 3\sqrt{3}n \]

### Step 4: Consider the case \(n = 0\)
To find the least positive solution, we start with \(n = 0\):
\[ \sin 2x = \sqrt{3} \]

### Step 5: Solve for \(2x\)
The sine function equals \(\frac{\sqrt{3}}{2}\) at:
\[ 2x = \frac{\pi}{3} + 2k\pi \quad \text{or} \quad 2x = \frac{2\pi}{3} + 2k\pi \quad (k \in \mathbb{Z}) \]

### Step 6: Solve for \(x\)
Dividing by 2, we find:
\[ x = \frac{\pi}{6} + k\pi \quad \text{or} \quad x = \frac{\pi}{3} + k\pi \]

### Step 7: Find the least positive solution
The least positive solution occurs when \(k = 0\):
\[ x = \frac{\pi}{6} \]

### Conclusion
Thus, the least positive solution of the equation is:
\[ \boxed{\frac{\pi}{6}} \]