For a complex number z, if `arg(z) in (-pi, pi], ` then `arg(1+cos.(6pi)/(5)+I sin.(6pi)/(5))` is (here `i^(2)-1`)

Options

A. `(3pi)/(5)`
B. `(2pi)/(5)`
C. `-(2pi)/(5)`
D. `-(3pi)/(5)`

Answer

To find the argument of the complex number \( z = 1 + \cos\left(\frac{6\pi}{5}\right) + i \sin\left(\frac{6\pi}{5}\right) \), we can follow these steps:

### Step 1: Simplify the Complex Number
We start with the expression:
\[ z = 1 + \cos\left(\frac{6\pi}{5}\right) + i \sin\left(\frac{6\pi}{5}\right) \]

### Step 2: Use Trigonometric Identities
Using the cosine and sine values:
- \(\cos\left(\frac{6\pi}{5}\right) = -\cos\left(\frac{\pi}{5}\right)\)
- \(\sin\left(\frac{6\pi}{5}\right) = -\sin\left(\frac{\pi}{5}\right)\)

Thus, we can rewrite \( z \):
\[ z = 1 - \cos\left(\frac{\pi}{5}\right) - i \sin\left(\frac{\pi}{5}\right) \]

### Step 3: Combine Real and Imaginary Parts
Now, we can express \( z \) as:
\[ z = (1 - \cos\left(\frac{\pi}{5}\right)) - i \sin\left(\frac{\pi}{5}\right) \]

### Step 4: Identify Real and Imaginary Parts
Let:
- Real part \( a = 1 - \cos\left(\frac{\pi}{5}\right) \)
- Imaginary part \( b = -\sin\left(\frac{\pi}{5}\right) \)

### Step 5: Calculate the Argument
The argument of a complex number \( z = a + bi \) is given by:
\[ \arg(z) = \tan^{-1}\left(\frac{b}{a}\right) \]
In our case:
\[ \arg(z) = \tan^{-1}\left(\frac{-\sin\left(\frac{\pi}{5}\right)}{1 - \cos\left(\frac{\pi}{5}\right)}\right) \]

### Step 6: Simplify the Argument
Using the identity \( 1 - \cos\theta = 2\sin^2\left(\frac{\theta}{2}\right) \):
\[ 1 - \cos\left(\frac{\pi}{5}\right) = 2\sin^2\left(\frac{\pi}{10}\right) \]
Thus, we can rewrite:
\[ \arg(z) = \tan^{-1}\left(\frac{-\sin\left(\frac{\pi}{5}\right)}{2\sin^2\left(\frac{\pi}{10}\right)}\right) \]

### Step 7: Determine the Quadrant
Since \( a > 0 \) and \( b < 0 \), the complex number lies in the fourth quadrant. Therefore, the argument will be negative.

### Step 8: Final Argument Calculation
To find the argument in the range \( (-\pi, \pi] \):
\[ \arg(z) = -\frac{2\pi}{5} \]

### Conclusion
Thus, the argument of the complex number \( z \) is:
\[ \arg(z) = -\frac{2\pi}{5} \]