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 will follow these steps:

### Step 1: Rewrite the complex number
The complex number can be rewritten as:
\[ z = 1 + \cos\left(\frac{6\pi}{5}\right) + i \sin\left(\frac{6\pi}{5}\right) \]

### Step 2: Use trigonometric identities
Using the trigonometric identity \( 1 + \cos \theta = 2 \cos^2\left(\frac{\theta}{2}\right) \), we can express \( 1 + \cos\left(\frac{6\pi}{5}\right) \) as:
\[ 1 + \cos\left(\frac{6\pi}{5}\right) = 2 \cos^2\left(\frac{6\pi}{10}\right) = 2 \cos^2\left(\frac{3\pi}{5}\right) \]

### Step 3: Express \( \sin\left(\frac{6\pi}{5}\right) \)
Using the identity for sine, we can express \( \sin\left(\frac{6\pi}{5}\right) \) as:
\[ \sin\left(\frac{6\pi}{5}\right) = \sin\left(\frac{6\pi}{5}\right) = 2 \sin\left(\frac{3\pi}{5}\right) \cos\left(\frac{3\pi}{5}\right) \]

### Step 4: Substitute back into the complex number
Now substituting these back into the expression for \( z \):
\[ z = 2 \cos^2\left(\frac{3\pi}{5}\right) + i \cdot 2 \sin\left(\frac{3\pi}{5}\right) \cos\left(\frac{3\pi}{5}\right) \]

### Step 5: Factor out the common terms
Factoring out \( 2 \cos\left(\frac{3\pi}{5}\right) \):
\[ z = 2 \cos\left(\frac{3\pi}{5}\right) \left( \cos\left(\frac{3\pi}{5}\right) + i \sin\left(\frac{3\pi}{5}\right) \right) \]

### Step 6: Identify the polar form
The expression \( \cos\left(\frac{3\pi}{5}\right) + i \sin\left(\frac{3\pi}{5}\right) \) can be recognized as the polar form \( e^{i\theta} \), where \( \theta = \frac{3\pi}{5} \).

### Step 7: Determine the argument
Thus, the argument of the complex number \( z \) is:
\[ \arg(z) = \frac{3\pi}{5} \]

### Final Answer
The argument of \( z \) is \( \frac{3\pi}{5} \).

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