The solution set of the equation `4 "sin" theta "cos" theta- 2 "cos" theta -2sqrt(3) "sin" theta + sqrt(3) =0 " in the interval" (0, 2 pi)`, is

Options

A. `{ (3pi)/(4), (7pi)/(4)}`
B. `{ (pi)/(3), (5pi)/(3)}`
C. `{ (3pi)/(4),pi, pi/3, (5pi)/(4)}`
D. `{ (pi)/(6), (5pi)/6, (11pi)/(6)}`

Answer

To solve the equation \( 4 \sin \theta \cos \theta - 2 \cos \theta - 2\sqrt{3} \sin \theta + \sqrt{3} = 0 \) in the interval \( (0, 2\pi) \), we can follow these steps:

### Step 1: Rearrange the Equation
We start with the equation:
\[ 4 \sin \theta \cos \theta - 2 \cos \theta - 2\sqrt{3} \sin \theta + \sqrt{3} = 0 \]

### Step 2: Factor the Equation
We can factor out common terms. Notice that we can group the terms:
\[ (4 \sin \theta \cos \theta - 2\sqrt{3} \sin \theta) + (-2 \cos \theta + \sqrt{3}) = 0 \]
Now, we can factor out \( 2 \cos \theta \) from the first part and \( -\sqrt{3} \) from the second part:
\[ 2 \cos \theta (2 \sin \theta - 1) - \sqrt{3} (2 \sin \theta - 1) = 0 \]
This gives us:
\[ (2 \sin \theta - 1)(2 \cos \theta - \sqrt{3}) = 0 \]

### Step 3: Set Each Factor to Zero
Now we set each factor equal to zero:
1. \( 2 \sin \theta - 1 = 0 \)
2. \( 2 \cos \theta - \sqrt{3} = 0 \)

### Step 4: Solve for \( \theta \)
**For the first equation:**
\[ 2 \sin \theta - 1 = 0 \implies \sin \theta = \frac{1}{2} \]
The solutions for \( \sin \theta = \frac{1}{2} \) in the interval \( (0, 2\pi) \) are:
\[ \theta = \frac{\pi}{6}, \frac{5\pi}{6} \]

**For the second equation:**
\[ 2 \cos \theta - \sqrt{3} = 0 \implies \cos \theta = \frac{\sqrt{3}}{2} \]
The solutions for \( \cos \theta = \frac{\sqrt{3}}{2} \) in the interval \( (0, 2\pi) \) are:
\[ \theta = \frac{\pi}{6}, \frac{11\pi}{6} \]

### Step 5: Combine the Solutions
Now we combine all the solutions we found:
- From \( \sin \theta = \frac{1}{2} \): \( \theta = \frac{\pi}{6}, \frac{5\pi}{6} \)
- From \( \cos \theta = \frac{\sqrt{3}}{2} \): \( \theta = \frac{\pi}{6}, \frac{11\pi}{6} \)

The unique solutions in the interval \( (0, 2\pi) \) are:
\[ \theta = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{11\pi}{6} \]

### Final Answer
Thus, the solution set of the equation is:
\[ \left\{ \frac{\pi}{6}, \frac{5\pi}{6}, \frac{11\pi}{6} \right\} \]