If `3 cos^2 theta - 2 sqrt3 sin theta.cos theta -3 sin^2 theta = 0`, then `theta` =

Options

A. `(n pi)/2 + pi/6`
B. `(n pi)/2 - pi/6`
C. `(n pi)/2 + pi/3`
D. `(npi)/2 - pi/3`

Answer

To solve the equation \(3 \cos^2 \theta - 2 \sqrt{3} \sin \theta \cos \theta - 3 \sin^2 \theta = 0\), we will follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ 3 \cos^2 \theta - 2 \sqrt{3} \sin \theta \cos \theta - 3 \sin^2 \theta = 0 \]

### Step 2: Factor the equation
We can rearrange the equation:
\[ 3 \cos^2 \theta - 3 \sin^2 \theta - 2 \sqrt{3} \sin \theta \cos \theta = 0 \]
This can be factored as:
\[ 3 (\cos^2 \theta - \sin^2 \theta) - 2 \sqrt{3} \sin \theta \cos \theta = 0 \]

### Step 3: Use trigonometric identities
Using the identities:
- \(\cos^2 \theta - \sin^2 \theta = \cos 2\theta\)
- \(2 \sin \theta \cos \theta = \sin 2\theta\)

We can rewrite the equation:
\[ 3 \cos 2\theta - \sqrt{3} \sin 2\theta = 0 \]

### Step 4: Rearranging the equation
Rearranging gives us:
\[ 3 \cos 2\theta = \sqrt{3} \sin 2\theta \]

### Step 5: Divide both sides by \(\cos 2\theta\)
Assuming \(\cos 2\theta \neq 0\), we divide both sides by \(\cos 2\theta\):
\[ 3 = \sqrt{3} \tan 2\theta \]

### Step 6: Solve for \(\tan 2\theta\)
This simplifies to:
\[ \tan 2\theta = \frac{3}{\sqrt{3}} = \sqrt{3} \]

### Step 7: Find the general solution for \(2\theta\)
The solutions for \(\tan 2\theta = \sqrt{3}\) are:
\[ 2\theta = \frac{\pi}{3} + n\pi, \quad n \in \mathbb{Z} \]

### Step 8: Solve for \(\theta\)
Dividing by 2 gives:
\[ \theta = \frac{\pi}{6} + \frac{n\pi}{2}, \quad n \in \mathbb{Z} \]

### Final Answer
Thus, the values of \(\theta\) are:
\[ \theta = \frac{\pi}{6} + \frac{n\pi}{2}, \quad n \in \mathbb{Z} \]

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