For what values of x does the equation `4sinx+3sin2x-2sin4x=2sqrt(3)` hold?

Options

A. `(pi)/(6)`
B. `(pi)/(4)`
C. `(pi)/(3)`
D. `(pi)/(2)`

Explanation

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

### Step 1: Substitute \( x = \frac{\pi}{6} \)

We start by substituting \( x = \frac{\pi}{6} \) into the equation.

\[ 4\sin\left(\frac{\pi}{6}\right) + 3\sin\left(2 \cdot \frac{\pi}{6}\right) - 2\sin\left(4 \cdot \frac{\pi}{6}\right) \]

### Step 2: Calculate \( \sin\left(\frac{\pi}{6}\right) \)

We know that:

\[ \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \]

Substituting this value into the equation gives:

\[ 4 \cdot \frac{1}{2} = 2 \]

### Step 3: Calculate \( \sin\left(\frac{\pi}{3}\right) \)

Next, we calculate \( \sin\left(2 \cdot \frac{\pi}{6}\right) = \sin\left(\frac{\pi}{3}\right) \):

\[ \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \]

Substituting this value gives:

\[ 3 \cdot \frac{\sqrt{3}}{2} = \frac{3\sqrt{3}}{2} \]

### Step 4: Calculate \( \sin\left(\frac{2\pi}{3}\right) \)

Next, we calculate \( \sin\left(4 \cdot \frac{\pi}{6}\right) = \sin\left(\frac{2\pi}{3}\right) \):

\[ \sin\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2} \]

Substituting this value gives:

\[ -2 \cdot \frac{\sqrt{3}}{2} = -\sqrt{3} \]

### Step 5: Combine all the parts

Now, we combine all the parts we calculated:

\[ 2 + \frac{3\sqrt{3}}{2} - \sqrt{3} \]

### Step 6: Simplify the expression

To simplify:

\[ 2 + \frac{3\sqrt{3}}{2} - \frac{2\sqrt{3}}{2} = 2 + \frac{3\sqrt{3} - 2\sqrt{3}}{2} = 2 + \frac{\sqrt{3}}{2} \]

### Step 7: Find a common denominator

We can express \( 2 \) as \( \frac{4}{2} \):

\[ \frac{4}{2} + \frac{\sqrt{3}}{2} = \frac{4 + \sqrt{3}}{2} \]

### Step 8: Check if it equals \( 2\sqrt{3} \)

Now we check if this is equal to \( 2\sqrt{3} \):

\[ \frac{4 + \sqrt{3}}{2} = 2\sqrt{3} \]

Multiplying both sides by 2 gives:

\[ 4 + \sqrt{3} = 4\sqrt{3} \]

### Step 9: Solve for \( \sqrt{3} \)

Rearranging gives:

\[ 4 = 4\sqrt{3} - \sqrt{3} = 3\sqrt{3} \]

Dividing by \( \sqrt{3} \):

\[ \frac{4}{\sqrt{3}} = 3 \]

### Step 10: Final conclusion

Thus, the original equation holds true for \( x = \frac{\pi}{6} \).