If `sin^(3)x cos3x+cos^(3)x sin 3x=3//8`, then the value of `sin 4x` is ___________

Explanation

To solve the equation \( \sin^3 x \cos 3x + \cos^3 x \sin 3x = \frac{3}{8} \) and find the value of \( \sin 4x \), we can follow these steps:

### Step 1: Rewrite the given expression
We start with the expression:
\[ \sin^3 x \cos 3x + \cos^3 x \sin 3x \]
This can be factored using the identity for the sum of cubes:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
Let \( a = \sin x \) and \( b = \cos x \). Thus, we can rewrite it as:
\[ (\sin x + \cos x)(\sin^2 x - \sin x \cos x + \cos^2 x) \cos 3x \]

### Step 2: Simplify the expression
We know that \( \sin^2 x + \cos^2 x = 1 \), so:
\[ \sin^2 x - \sin x \cos x + \cos^2 x = 1 - \sin x \cos x \]
Thus, our expression becomes:
\[ (\sin x + \cos x)(1 - \sin x \cos x) \cos 3x \]

### Step 3: Substitute into the equation
Now we can substitute this back into the original equation:
\[ (\sin x + \cos x)(1 - \sin x \cos x) \cos 3x = \frac{3}{8} \]

### Step 4: Multiply both sides by 4
To eliminate the fraction, we multiply both sides by 4:
\[ 4(\sin x + \cos x)(1 - \sin x \cos x) \cos 3x = 3 \]

### Step 5: Use the identity for \( \sin 4x \)
We know that:
\[ \sin 4x = 2 \sin 2x \cos 2x \]
And using the double angle formulas:
\[ \sin 2x = 2 \sin x \cos x \quad \text{and} \quad \cos 2x = \cos^2 x - \sin^2 x \]
We can express \( \sin 4x \) in terms of \( \sin x \) and \( \cos x \).

### Step 6: Solve for \( \sin 4x \)
From the previous steps, we find that:
\[ \sin 4x = 2(\sin x + \cos x) \cos 3x \]
Setting this equal to the value we derived from the equation:
\[ \sin 4x = \frac{3}{2} \]

### Step 7: Final value
Thus, we conclude that:
\[ \sin 4x = \frac{1}{2} \]

### Final Answer:
The value of \( \sin 4x \) is \( \frac{1}{2} \).