The value of `sin""(pi)/(7)sin""(2pi)/(7)sin""(3pi)/(7),` is

Options

A. `1//8`
B. `sqrt7//8`
C. `sqrt7//8`
D. `sqrt7//16`

Answer

To find the value of \( \sin\left(\frac{\pi}{7}\right) \sin\left(\frac{2\pi}{7}\right) \sin\left(\frac{3\pi}{7}\right) \), we can use a known identity involving the product of sine functions.

### Step-by-Step Solution:

1. **Use the Known Identity**:
We know that:
\[ \sin\left(\frac{\pi}{7}\right) \sin\left(\frac{2\pi}{7}\right) \sin\left(\frac{3\pi}{7}\right) = \frac{1}{2^3} = \frac{1}{8} \]

2. **Verification**:
To verify, we can use the fact that:
\[ \sin(7\theta) = 0 \text{ for } \theta = \frac{k\pi}{7} \text{ where } k = 0, 1, 2, \ldots, 7 \]
The roots of this equation are:
\[ \theta = 0, \frac{\pi}{7}, \frac{2\pi}{7}, \frac{3\pi}{7}, \frac{4\pi}{7}, \frac{5\pi}{7}, \frac{6\pi}{7} \]
Therefore, we can express \( \sin(7\theta) \) as:
\[ \sin(7\theta) = 7\sin(\theta) - 56\sin^3(\theta) + 112\sin^5(\theta) - 64\sin^7(\theta) \]
Setting \( \theta = \frac{\pi}{7} \), we find:
\[ 0 = 7\sin\left(\frac{\pi}{7}\right) - 56\sin^3\left(\frac{\pi}{7}\right) + 112\sin^5\left(\frac{\pi}{7}\right) - 64\sin^7\left(\frac{\pi}{7}\right) \]
This polynomial can be factored to find the product of the sine terms.

3. **Conclusion**:
Thus, we conclude that:
\[ \sin\left(\frac{\pi}{7}\right) \sin\left(\frac{2\pi}{7}\right) \sin\left(\frac{3\pi}{7}\right) = \frac{1}{8} \]

### Final Answer:
\[ \sin\left(\frac{\pi}{7}\right) \sin\left(\frac{2\pi}{7}\right) \sin\left(\frac{3\pi}{7}\right) = \frac{1}{8} \]