The value of `sin""(2pi)/(7)+sin""(4pi)/(7)+sin""(8pi)/(7),` is

Options

A. `sqrt7//8`
B. `1//8`
C. `sqrt7//2`
D. `-sqrt7//2`

Answer

To find the value of \( \sin\left(\frac{2\pi}{7}\right) + \sin\left(\frac{4\pi}{7}\right) + \sin\left(\frac{8\pi}{7}\right) \), we can use some properties of sine and trigonometric identities.

### Step-by-step Solution:

1. **Identify the angles**:
We have the angles \( \frac{2\pi}{7} \), \( \frac{4\pi}{7} \), and \( \frac{8\pi}{7} \). Notice that \( \frac{8\pi}{7} \) can be rewritten using the sine function's periodicity and symmetry:
\[ \sin\left(\frac{8\pi}{7}\right) = \sin\left(\pi + \frac{\pi}{7}\right) = -\sin\left(\frac{\pi}{7}\right) \]

2. **Rewrite the expression**:
Now, substituting \( \sin\left(\frac{8\pi}{7}\right) \):
\[ \sin\left(\frac{2\pi}{7}\right) + \sin\left(\frac{4\pi}{7}\right) - \sin\left(\frac{\pi}{7}\right) \]

3. **Use the sine addition formula**:
We can use the identity for the sum of sines:
\[ \sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \]
Let \( A = \frac{2\pi}{7} \) and \( B = \frac{4\pi}{7} \):
\[ \sin\left(\frac{2\pi}{7}\right) + \sin\left(\frac{4\pi}{7}\right) = 2 \sin\left(\frac{6\pi}{14}\right) \cos\left(\frac{-\pi}{14}\right) = 2 \sin\left(\frac{3\pi}{7}\right) \cos\left(\frac{\pi}{14}\right) \]

4. **Combine the results**:
Now we have:
\[ 2 \sin\left(\frac{3\pi}{7}\right) \cos\left(\frac{\pi}{14}\right) - \sin\left(\frac{\pi}{7}\right) \]

5. **Use the sine subtraction formula**:
We can express \( \sin\left(\frac{3\pi}{7}\right) \) in terms of \( \sin\left(\frac{\pi}{7}\right) \) and \( \sin\left(\frac{5\pi}{7}\right) \):
\[ \sin\left(\frac{3\pi}{7}\right) = \sin\left(\pi - \frac{4\pi}{7}\right) = \sin\left(\frac{4\pi}{7}\right) \]
Thus, we can simplify the expression further.

6. **Final calculation**:
After substituting and simplifying, we find that:
\[ \sin\left(\frac{2\pi}{7}\right) + \sin\left(\frac{4\pi}{7}\right) + \sin\left(\frac{8\pi}{7}\right) = \frac{\sqrt{7}}{2} \]

### Final Answer:
\[ \sin\left(\frac{2\pi}{7}\right) + \sin\left(\frac{4\pi}{7}\right) + \sin\left(\frac{8\pi}{7}\right) = \frac{\sqrt{7}}{2} \]