For a complex number Z. If `arg(Z) in (-pi, pi]`, then `arg{1+cos.(6pi)/(7)+isin.(6pi)/(7)}` is (here `i^(2)=-1`)

Options

A. `(3pi)/(7)`
B. `(2pi)/(7)`
C. `-(2pi)/(7)`
D. `-(3pi)/(7)`

Answer

To solve the problem, we need to find the argument of the complex number \( \omega = 1 + \cos\left(\frac{6\pi}{7}\right) + i \sin\left(\frac{6\pi}{7}\right) \).

### Step-by-Step Solution:

1. **Identify the Complex Number**:
We have:
\[ \omega = 1 + \cos\left(\frac{6\pi}{7}\right) + i \sin\left(\frac{6\pi}{7}\right) \]

2. **Use Trigonometric Identities**:
We can use the identity:
\[ 1 + \cos \theta = 2 \cos^2\left(\frac{\theta}{2}\right) \]
Here, let \(\theta = \frac{6\pi}{7}\). Therefore:
\[ 1 + \cos\left(\frac{6\pi}{7}\right) = 2 \cos^2\left(\frac{6\pi/7}{2}\right) = 2 \cos^2\left(\frac{3\pi}{7}\right) \]

3. **Express \(\sin\) in Terms of \(\cos\)**:
We also know:
\[ \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \]
Thus:
\[ \sin\left(\frac{12\pi}{7}\right) = 2 \sin\left(\frac{6\pi}{7}\right) \cos\left(\frac{6\pi}{7}\right) \]
This means:
\[ \sin\left(\frac{6\pi}{7}\right) = \sin\left(\pi - \frac{6\pi}{7}\right) = \sin\left(\frac{\pi}{7}\right) \]

4. **Combine Terms**:
Now we can rewrite \(\omega\):
\[ \omega = 2 \cos^2\left(\frac{3\pi}{7}\right) + i \sin\left(\frac{6\pi}{7}\right) \]
This can be factored as:
\[ \omega = 2 \cos\left(\frac{3\pi}{7}\right) \left(\cos\left(\frac{3\pi}{7}\right) + i \frac{\sin\left(\frac{6\pi}{7}\right)}{2 \cos\left(\frac{3\pi}{7}\right)}\right) \]

5. **Convert to Polar Form**:
The expression inside the parentheses can be recognized as being in the polar form \( r(\cos \theta + i \sin \theta) \), where:
\[ r = \sqrt{\left(\cos\left(\frac{3\pi}{7}\right)\right)^2 + \left(\frac{\sin\left(\frac{6\pi}{7}\right)}{2 \cos\left(\frac{3\pi}{7}\right)}\right)^2} \]
The argument of \(\omega\) is then:
\[ \arg(\omega) = \arg\left(2 \cos\left(\frac{3\pi}{7}\right)\right) + \arg\left(\cos\left(\frac{3\pi}{7}\right) + i \sin\left(\frac{6\pi}{7}\right)\right) \]

6. **Final Argument Calculation**:
The argument of \(\omega\) simplifies to:
\[ \arg(\omega) = \frac{3\pi}{7} \]

### Conclusion:
Thus, the argument of the complex number \( \omega \) is:
\[ \arg(\omega) = \frac{3\pi}{7} \]