The principal solution `cotx=sqrt(3)` are

Options

A. `(pi)/(6), (5pi)/(6)`
B. `(7pi)/(6), (11pi)/(6)`
C. `(pi)/(6), (7pi)/(6)`
D. `(5pi)/(6), (11pi)/(6)`

Answer

To solve the equation \( \cot x = \sqrt{3} \), we will follow these steps:

### Step 1: Understand the cotangent function
The cotangent function is defined as:
\[ \cot x = \frac{\cos x}{\sin x} \]
We need to find the angles \( x \) where \( \cot x = \sqrt{3} \).

### Step 2: Identify the reference angle
We know that:
\[ \cot \frac{\pi}{6} = \sqrt{3} \]
Thus, the reference angle for \( \cot x = \sqrt{3} \) is \( \frac{\pi}{6} \).

### Step 3: Determine the quadrants
The cotangent function is positive in the first and third quadrants. Therefore, we will find solutions in both quadrants.

### Step 4: Find the solutions in the first quadrant
In the first quadrant, the solution is:
\[ x = \frac{\pi}{6} \]

### Step 5: Find the solutions in the third quadrant
In the third quadrant, we can find the angle by adding \( \pi \) to the reference angle:
\[ x = \pi + \frac{\pi}{6} = \frac{6\pi}{6} + \frac{\pi}{6} = \frac{7\pi}{6} \]

### Step 6: State the principal solutions
The principal solutions for \( \cot x = \sqrt{3} \) in the interval \( [0, 2\pi) \) are:
\[ x = \frac{\pi}{6}, \quad x = \frac{7\pi}{6} \]

### Final Answer
The principal solutions of \( \cot x = \sqrt{3} \) are:
\[ x = \frac{\pi}{6}, \quad x = \frac{7\pi}{6} \]

---