Solve the following equation: `\ 3cos^2theta-2\ sqrt(3)sinthetacostheta-3sin^2theta=0`

Explanation

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

### Step 1: Rearrange 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 out common terms
We can rearrange the equation by grouping terms:
\[ 3\cos^2\theta - 3\sin^2\theta - 2\sqrt{3}\sin\theta\cos\theta = 0 \]
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 \( \cos 2\theta = \cos^2\theta - \sin^2\theta \) and \( \sin 2\theta = 2\sin\theta\cos\theta \), we can rewrite the equation:
\[ 3\cos 2\theta - \sqrt{3}\sin 2\theta = 0 \]

### Step 4: Isolate the trigonometric functions
Rearranging gives:
\[ 3\cos 2\theta = \sqrt{3}\sin 2\theta \]

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

### 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 general solution for \(\tan x = \sqrt{3}\) is:
\[ 2\theta = n\pi + \frac{\pi}{3}, \quad n \in \mathbb{Z} \]

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

### Final Answer
Thus, the general solution for the given equation is:
\[ \theta = \frac{n\pi}{2} + \frac{\pi}{6}, \quad n \in \mathbb{Z} \]