the value of `sin(pi/7)+sin((2pi)/7)+sin((3pi)/7)` is

Options

A. `cot""(pi)/(14)`
B. `1/2cot""(pi)/(14)`
C. `tan""(pi)/(14)`
D. `1/2tan""(pi)/(14)`

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 trigonometric identities to simplify the expression step by step.

### Step-by-Step Solution:

1. **Write the Expression**:
\[ S = \sin\left(\frac{\pi}{7}\right) + \sin\left(\frac{2\pi}{7}\right) + \sin\left(\frac{3\pi}{7}\right) \]

2. **Use the Sum-to-Product Identity**:
We can combine the first two sine terms using the identity:
\[ \sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \]
Let \( A = \frac{\pi}{7} \) and \( B = \frac{2\pi}{7} \):
\[ \sin\left(\frac{\pi}{7}\right) + \sin\left(\frac{2\pi}{7}\right) = 2 \sin\left(\frac{\frac{\pi}{7} + \frac{2\pi}{7}}{2}\right) \cos\left(\frac{\frac{\pi}{7} - \frac{2\pi}{7}}{2}\right) \]
Simplifying gives:
\[ = 2 \sin\left(\frac{3\pi}{14}\right) \cos\left(-\frac{\pi}{14}\right) = 2 \sin\left(\frac{3\pi}{14}\right) \cos\left(\frac{\pi}{14}\right) \]

3. **Combine with the Third Term**:
Now we add \( \sin\left(\frac{3\pi}{7}\right) \):
\[ S = 2 \sin\left(\frac{3\pi}{14}\right) \cos\left(\frac{\pi}{14}\right) + \sin\left(\frac{3\pi}{7}\right) \]

4. **Rewrite \( \sin\left(\frac{3\pi}{7}\right) \)**:
We can express \( \sin\left(\frac{3\pi}{7}\right) \) using the identity:
\[ \sin\left(\frac{3\pi}{7}\right) = \sin\left(\pi - \frac{3\pi}{7}\right) = \sin\left(\frac{4\pi}{7}\right) \]

5. **Use the Sum-to-Product Identity Again**:
Now we can combine \( \sin\left(\frac{3\pi}{14}\right) \) and \( \sin\left(\frac{4\pi}{7}\right) \):
\[ S = 2 \sin\left(\frac{3\pi}{14}\right) \cos\left(\frac{\pi}{14}\right) + \sin\left(\frac{4\pi}{7}\right) \]

6. **Final Combination**:
We can use the identities again to combine the terms:
\[ S = 2 \sin\left(\frac{3\pi}{14}\right) \cos\left(\frac{\pi}{14}\right) + 2 \sin\left(\frac{4\pi}{14}\right) \cos\left(\frac{3\pi}{14}\right) \]

7. **Use the Final Identity**:
Finally, we can combine these using the sine addition formula:
\[ S = \frac{1}{2} \cos\left(\frac{\pi}{14}\right) \]

### Conclusion:
The value of \( \sin\left(\frac{\pi}{7}\right) + \sin\left(\frac{2\pi}{7}\right) + \sin\left(\frac{3\pi}{7}\right) \) is:
\[ \frac{1}{2} \cos\left(\frac{\pi}{14}\right) \]