Determine the smallest positive value of `(32x)/(pi)` which satisfy the equation `sqrt(1+sin 2x)-sqrt(2)cos 3x=0`

Answer

To solve the equation \( \sqrt{1 + \sin 2x} - \sqrt{2} \cos 3x = 0 \) and determine the smallest positive value of \( \frac{32x}{\pi} \), we can follow these steps:

### Step 1: Rewrite the equation
Starting with the given equation:
\[ \sqrt{1 + \sin 2x} = \sqrt{2} \cos 3x \]

### Step 2: Square both sides
To eliminate the square root, we square both sides:
\[ 1 + \sin 2x = 2 \cos^2 3x \]

### Step 3: Use the identity for cosine
We know that \( \cos^2 3x = 1 - \sin^2 3x \). Substituting this into the equation gives:
\[ 1 + \sin 2x = 2(1 - \sin^2 3x) \]
\[ 1 + \sin 2x = 2 - 2 \sin^2 3x \]

### Step 4: Rearrange the equation
Rearranging the equation, we have:
\[ \sin 2x + 2 \sin^2 3x = 1 \]

### Step 5: Use the double angle identity
Recall that \( \sin 2x = 2 \sin x \cos x \). Thus, we can rewrite the equation as:
\[ 2 \sin x \cos x + 2 \sin^2 3x = 1 \]

### Step 6: Factor out the 2
Dividing the entire equation by 2 gives:
\[ \sin x \cos x + \sin^2 3x = \frac{1}{2} \]

### Step 7: Solve for \( \sin x \) and \( \cos x \)
Using the identity \( \sin^2 3x = 1 - \cos^2 3x \), we can express \( \sin^2 3x \) in terms of \( \cos 3x \) and substitute back into the equation.

### Step 8: Set up the trigonometric equation
We can set up the trigonometric equation based on the identities and solve for \( x \).

### Step 9: Find the general solution
From the derived equations, we can find the general solutions for \( x \):
\[ x - \frac{\pi}{4} = 2n\pi \pm 3x \]
This leads to two equations:
1. \( 2x = 2n\pi + \frac{\pi}{4} \)
2. \( 2x = 2n\pi - \frac{\pi}{4} \)

### Step 10: Solve for \( x \)
From the first equation:
\[ x = n\pi + \frac{\pi}{8} \]
From the second equation:
\[ x = n\pi - \frac{\pi}{8} \]

### Step 11: Find the smallest positive value
To find the smallest positive value, we can start with \( n = 0 \):
- From \( x = 0 + \frac{\pi}{8} = \frac{\pi}{8} \)
- From \( x = 0 - \frac{\pi}{8} \) (not positive)

Thus, the smallest positive value of \( x \) is:
\[ x = \frac{\pi}{8} \]

### Step 12: Calculate \( \frac{32x}{\pi} \)
Now, we substitute \( x \) into \( \frac{32x}{\pi} \):
\[ \frac{32 \cdot \frac{\pi}{8}}{\pi} = \frac{32}{8} = 4 \]

### Final Answer
The smallest positive value of \( \frac{32x}{\pi} \) is:
\[ \boxed{4} \]