One of the general solutions of `sqrt(3) cos theta -3 sin theta =4 sin 2 theta cos 3 theta` is

Options

A. `(3n pm 1) pi//12, AA n in Z`
B. `(4n pm 1) pi//9, AA n in Z`
C. `(3n pm 1) pi//9, AA n in Z`
D. `(3n pm 1) pi//3, AA n in Z`

Explanation

To solve the equation \( \sqrt{3} \cos \theta - 3 \sin \theta = 4 \sin 2\theta \cos 3\theta \), we will follow these steps:

### Step 1: Rewrite the Right Side
We start with the equation:
\[ \sqrt{3} \cos \theta - 3 \sin \theta = 4 \sin 2\theta \cos 3\theta \]
Using the double angle identity for sine, \( \sin 2\theta = 2 \sin \theta \cos \theta \), we can rewrite the right side:
\[ 4 \sin 2\theta \cos 3\theta = 4(2 \sin \theta \cos \theta) \cos 3\theta = 8 \sin \theta \cos \theta \cos 3\theta \]

### Step 2: Rearrange the Equation
Now, we can rearrange the equation:
\[ \sqrt{3} \cos \theta - 3 \sin \theta - 8 \sin \theta \cos \theta \cos 3\theta = 0 \]

### Step 3: Factor Out Common Terms
We can factor out \( \sin \theta \):
\[ \sqrt{3} \cos \theta - \sin \theta (3 + 8 \cos \theta \cos 3\theta) = 0 \]

### Step 4: Set Each Factor to Zero
This gives us two cases to consider:
1. \( \sqrt{3} \cos \theta = 0 \)
2. \( 3 + 8 \cos \theta \cos 3\theta = 0 \)

### Step 5: Solve the First Case
For the first case:
\[ \sqrt{3} \cos \theta = 0 \implies \cos \theta = 0 \]
The general solution for \( \cos \theta = 0 \) is:
\[ \theta = \frac{\pi}{2} + n\pi, \quad n \in \mathbb{Z} \]

### Step 6: Solve the Second Case
For the second case:
\[ 3 + 8 \cos \theta \cos 3\theta = 0 \implies 8 \cos \theta \cos 3\theta = -3 \]
This can be rearranged to:
\[ \cos \theta \cos 3\theta = -\frac{3}{8} \]

### Step 7: Use Trigonometric Identities
Using the identity \( \cos A \cos B = \frac{1}{2} [\cos(A+B) + \cos(A-B)] \):
\[ \cos \theta \cos 3\theta = \frac{1}{2} [\cos(4\theta) + \cos(2\theta)] = -\frac{3}{8} \]
Multiplying through by 2:
\[ \cos(4\theta) + \cos(2\theta) = -\frac{3}{4} \]

### Step 8: Solve the Equation
This equation can be solved using numerical or graphical methods or further trigonometric identities. However, we can also find specific angles that satisfy this equation.

### Final General Solutions
Combining the solutions from both cases, we have:
1. From \( \cos \theta = 0 \):
\[ \theta = \frac{\pi}{2} + n\pi, \quad n \in \mathbb{Z} \]
2. From \( \cos \theta \cos 3\theta = -\frac{3}{8} \):
This requires further analysis or numerical methods to find specific angles.

### Conclusion
Thus, one of the general solutions of the given equation is:
\[ \theta = \frac{\pi}{2} + n\pi, \quad n \in \mathbb{Z} \]