The sum of all the values of x between 0 and `4pi` which satisfy the equation `sinxsqrt(8cos^(2)x)=1` is `kpi`, then the value of `(k)/(5)` is equal to

Answer

To solve the equation \( \sin x \sqrt{8 \cos^2 x} = 1 \) for \( x \) in the interval \( [0, 4\pi] \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \sin x \sqrt{8 \cos^2 x} = 1 \]
This can be rewritten as:
\[ \sqrt{8} \sin x \cos x = 1 \]
Since \( \sqrt{8} = 2\sqrt{2} \), we have:
\[ 2\sqrt{2} \sin x \cos x = 1 \]

### Step 2: Use the double angle identity
Using the identity \( \sin 2x = 2 \sin x \cos x \), we can express the equation as:
\[ \sqrt{2} \sin 2x = 1 \]
Dividing both sides by \( \sqrt{2} \):
\[ \sin 2x = \frac{1}{\sqrt{2}} \]

### Step 3: Solve for \( 2x \)
The general solution for \( \sin \theta = \frac{1}{\sqrt{2}} \) is:
\[ \theta = \frac{\pi}{4} + n\pi \quad \text{for } n \in \mathbb{Z} \]
Thus, we have:
\[ 2x = \frac{\pi}{4} + n\pi \]
Dividing by 2 gives:
\[ x = \frac{\pi}{8} + \frac{n\pi}{2} \]

### Step 4: Find values of \( x \) in the interval \( [0, 4\pi] \)
To find the values of \( x \) in the interval \( [0, 4\pi] \), we will substitute integer values for \( n \):

- For \( n = 0 \):
\[ x = \frac{\pi}{8} \]
- For \( n = 1 \):
\[ x = \frac{\pi}{8} + \frac{\pi}{2} = \frac{\pi}{8} + \frac{4\pi}{8} = \frac{5\pi}{8} \]
- For \( n = 2 \):
\[ x = \frac{\pi}{8} + \pi = \frac{\pi}{8} + \frac{8\pi}{8} = \frac{9\pi}{8} \]
- For \( n = 3 \):
\[ x = \frac{\pi}{8} + \frac{3\pi}{2} = \frac{\pi}{8} + \frac{12\pi}{8} = \frac{13\pi}{8} \]
- For \( n = 4 \):
\[ x = \frac{\pi}{8} + 2\pi = \frac{\pi}{8} + \frac{16\pi}{8} = \frac{17\pi}{8} \]
- For \( n = 5 \):
\[ x = \frac{\pi}{8} + \frac{5\pi}{2} = \frac{\pi}{8} + \frac{20\pi}{8} = \frac{21\pi}{8} \]
- For \( n = 6 \):
\[ x = \frac{\pi}{8} + 3\pi = \frac{\pi}{8} + \frac{24\pi}{8} = \frac{25\pi}{8} \]
- For \( n = 7 \):
\[ x = \frac{\pi}{8} + \frac{7\pi}{2} = \frac{\pi}{8} + \frac{28\pi}{8} = \frac{29\pi}{8} \]

### Step 5: Filter values within the interval
Now we filter the values to keep only those in the interval \( [0, 4\pi] \):
- \( \frac{\pi}{8} \)
- \( \frac{5\pi}{8} \)
- \( \frac{9\pi}{8} \)
- \( \frac{13\pi}{8} \)
- \( \frac{17\pi}{8} \)
- \( \frac{21\pi}{8} \)
- \( \frac{25\pi}{8} \)
- \( \frac{29\pi}{8} \)

### Step 6: Sum the values
Now we sum these values:
\[ S = \frac{\pi}{8} + \frac{5\pi}{8} + \frac{9\pi}{8} + \frac{13\pi}{8} + \frac{17\pi}{8} + \frac{21\pi}{8} + \frac{25\pi}{8} + \frac{29\pi}{8} \]
This can be simplified as:
\[ S = \frac{(1 + 5 + 9 + 13 + 17 + 21 + 25 + 29)\pi}{8} \]
Calculating the sum of the coefficients:
\[ 1 + 5 + 9 + 13 + 17 + 21 + 25 + 29 = 120 \]
Thus:
\[ S = \frac{120\pi}{8} = 15\pi \]

### Step 7: Find \( k \) and compute \( \frac{k}{5} \)
From the problem, we have \( S = k\pi \) where \( k = 15 \). Therefore:
\[ \frac{k}{5} = \frac{15}{5} = 3 \]

### Final Answer
The value of \( \frac{k}{5} \) is \( \boxed{3} \).