The value of `sin(pi/14)sin((3pi)/14)sin((5pi)/14)` is

Options

A. 1
B. `1//4`
C. `1//8`
D. `sqrt2//7`

Answer

To find the value of \( \sin\left(\frac{\pi}{14}\right) \sin\left(\frac{3\pi}{14}\right) \sin\left(\frac{5\pi}{14}\right) \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \sin\left(\frac{\pi}{14}\right) \sin\left(\frac{3\pi}{14}\right) \sin\left(\frac{5\pi}{14}\right) \]
To simplify, we multiply and divide by \( 2 \cos\left(\frac{\pi}{14}\right) \):
\[ = \frac{2 \cos\left(\frac{\pi}{14}\right) \sin\left(\frac{\pi}{14}\right) \sin\left(\frac{3\pi}{14}\right) \sin\left(\frac{5\pi}{14}\right)}{2 \cos\left(\frac{\pi}{14}\right)} \]

### Step 2: Substitute \( \theta \)
Let \( \theta = \frac{\pi}{14} \). Then, we have:
\[ = \frac{2 \cos(\theta) \sin(\theta) \sin(3\theta) \sin(5\theta)}{2 \cos(\theta)} \]
This simplifies to:
\[ = \sin(\theta) \sin(3\theta) \sin(5\theta) \]

### Step 3: Use the identity for \( 2 \sin A \sin B \)
Next, we use the identity \( 2 \sin A \sin B = \cos(A - B) - \cos(A + B) \):
\[ = \frac{1}{2} \left( \cos(3\theta - \theta) - \cos(3\theta + \theta) \right) \sin(5\theta) \]
This becomes:
\[ = \frac{1}{2} \left( \cos(2\theta) - \cos(4\theta) \right) \sin(5\theta) \]

### Step 4: Substitute back for \( \theta \)
Now, substituting back \( \theta = \frac{\pi}{14} \):
\[ = \frac{1}{2} \left( \cos\left(\frac{2\pi}{14}\right) - \cos\left(\frac{4\pi}{14}\right) \right) \sin\left(\frac{5\pi}{14}\right) \]
This simplifies to:
\[ = \frac{1}{2} \left( \cos\left(\frac{\pi}{7}\right) - \cos\left(\frac{2\pi}{7}\right) \right) \sin\left(\frac{5\pi}{14}\right) \]

### Step 5: Use the identity for \( \sin 2A \)
Now, we can rewrite \( \sin(5\theta) \) as:
\[ = \frac{1}{4} \sin(6\theta) \text{ (using } 2 \sin A \cos A = \sin(2A)\text{)} \]

### Step 6: Final simplification
Using the identity for sine:
\[ = \frac{1}{8} \sin(7\theta - \theta) = \frac{1}{8} \left( \sin(7\theta) \cos(\theta) - \cos(7\theta) \sin(\theta) \right) \]
Since \( 7\theta = \frac{\pi}{2} \), we know:
\[ \sin\left(\frac{\pi}{2}\right) = 1 \quad \text{and} \quad \cos\left(\frac{\pi}{2}\right) = 0 \]
Thus, the expression simplifies to:
\[ \frac{\cos(\theta)}{8 \cos(\theta)} = \frac{1}{8} \]

### Conclusion
The value of \( \sin\left(\frac{\pi}{14}\right) \sin\left(\frac{3\pi}{14}\right) \sin\left(\frac{5\pi}{14}\right) \) is:
\[ \boxed{\frac{1}{8}} \]