If x and y are positive acute angles such that `sin(2x + 3y) = (sqrt(3))/(2)` and `cos(4x - 3y) = (sqrt(3))/(2)`, then what is the value of `tan(6x - 3y)` ?

Options

A. 0
B. 1
C. `1//sqrt""3`
D. `sqrt""3`

Answer

To solve the problem step by step, we need to analyze the given equations and find the value of \( \tan(6x - 3y) \).

### Step 1: Analyze the first equation
We have:
\[ \sin(2x + 3y) = \frac{\sqrt{3}}{2} \]
The value of \( \sin \theta = \frac{\sqrt{3}}{2} \) corresponds to angles:
\[ 2x + 3y = 60^\circ \quad \text{(1)} \]
or
\[ 2x + 3y = 120^\circ \quad \text{(2)} \]

### Step 2: Analyze the second equation
Next, we have:
\[ \cos(4x - 3y) = \frac{\sqrt{3}}{2} \]
The value of \( \cos \theta = \frac{\sqrt{3}}{2} \) corresponds to angles:
\[ 4x - 3y = 30^\circ \quad \text{(3)} \]
or
\[ 4x - 3y = 330^\circ \quad \text{(4)} \]

### Step 3: Solve the equations
We will consider the first pair of equations (1) and (3):
1. \( 2x + 3y = 60^\circ \)
2. \( 4x - 3y = 30^\circ \)

Now, we can add these two equations:
\[ (2x + 3y) + (4x - 3y) = 60^\circ + 30^\circ \]
This simplifies to:
\[ 6x = 90^\circ \]
Thus, we find:
\[ x = 15^\circ \]

### Step 4: Substitute \( x \) back to find \( y \)
Now we can substitute \( x = 15^\circ \) back into equation (1):
\[ 2(15^\circ) + 3y = 60^\circ \]
This simplifies to:
\[ 30^\circ + 3y = 60^\circ \]
Subtracting \( 30^\circ \) from both sides gives:
\[ 3y = 30^\circ \]
Thus, we find:
\[ y = 10^\circ \]

### Step 5: Calculate \( \tan(6x - 3y) \)
Now we need to find \( \tan(6x - 3y) \):
\[ 6x - 3y = 6(15^\circ) - 3(10^\circ) = 90^\circ - 30^\circ = 60^\circ \]
Thus,
\[ \tan(6x - 3y) = \tan(60^\circ) = \sqrt{3} \]

### Final Answer
The value of \( \tan(6x - 3y) \) is:
\[ \sqrt{3} \]