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), (7pi)/(4), (pi)/(3), (5pi)/(3)}`
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: Rearranging 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: Factoring the Equation
We can rearrange the equation to group terms:
\[ 4 \sin \theta \cos \theta - 2\sqrt{3} \sin \theta - 2 \cos \theta + \sqrt{3} = 0 \]

Next, we can factor out common terms. We can take \( 2 \) as a common factor from the first two terms:
\[ 2(2 \sin \theta \cos \theta - \sqrt{3} \sin \theta) - 2 \cos \theta + \sqrt{3} = 0 \]

This can be rewritten as:
\[ 2 \sin \theta (2 \cos \theta - \sqrt{3}) - 2 \cos \theta + \sqrt{3} = 0 \]

### Step 3: Setting Up the Factors
Now, we can set the equation to zero:
\[ 2 \sin \theta (2 \cos \theta - \sqrt{3}) = 2 \cos \theta - \sqrt{3} \]

### Step 4: Solving for \( \sin \theta \) and \( \cos \theta \)
This leads to two cases:

**Case 1:** \( 2 \sin \theta = 0 \)
\[ \sin \theta = 0 \]
The solutions for \( \sin \theta = 0 \) in the interval \( (0, 2\pi) \) are:
\[ \theta = \pi \]

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

### Step 5: Combining Solutions
Now we combine the solutions from both cases:
- From Case 1: \( \theta = \pi \)
- From Case 2: \( \theta = \frac{\pi}{6}, \frac{11\pi}{6} \)

### Final Solution Set
Thus, the complete solution set in the interval \( (0, 2\pi) \) is:
\[ \theta = \frac{\pi}{6}, \pi, \frac{11\pi}{6} \]

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