For any integer n, arg of
z ` = (( sqrt(3) + i)^(4 n + 1))/(( 1 - i sqrt(3))^(4 n))` is

Options

A. `pi//6`
B. `pi//3`
C. `pi//2`
D. `2 pi//3`

Answer

To find the argument of the complex number \( z = \frac{(\sqrt{3} + i)^{4n + 1}}{(1 - i\sqrt{3})^{4n}} \), we will follow these steps:

### Step 1: Find the argument of the numerator
The numerator is \( (\sqrt{3} + i)^{4n + 1} \).

1. **Convert \( \sqrt{3} + i \) to polar form**:
- The modulus \( r \) is given by \( r = \sqrt{(\sqrt{3})^2 + 1^2} = \sqrt{3 + 1} = \sqrt{4} = 2 \).
- The argument \( \theta \) is given by \( \theta = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6} \).

2. **Express in polar form**:
\[ \sqrt{3} + i = 2 \left(\cos\frac{\pi}{6} + i\sin\frac{\pi}{6}\right) = 2 e^{i\frac{\pi}{6}} \]

3. **Raise to the power \( 4n + 1 \)**:
\[ (\sqrt{3} + i)^{4n + 1} = (2 e^{i\frac{\pi}{6}})^{4n + 1} = 2^{4n + 1} e^{i\frac{(4n + 1)\pi}{6}} \]

4. **Argument of the numerator**:
\[ \text{arg}((\sqrt{3} + i)^{4n + 1}) = \frac{(4n + 1)\pi}{6} \]

### Step 2: Find the argument of the denominator
The denominator is \( (1 - i\sqrt{3})^{4n} \).

1. **Convert \( 1 - i\sqrt{3} \) to polar form**:
- The modulus \( r \) is given by \( r = \sqrt{1^2 + (-\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 \).
- The argument \( \theta \) is given by \( \theta = \tan^{-1}\left(\frac{-\sqrt{3}}{1}\right) = -\frac{\pi}{3} \).

2. **Express in polar form**:
\[ 1 - i\sqrt{3} = 2 \left(\cos\left(-\frac{\pi}{3}\right) + i\sin\left(-\frac{\pi}{3}\right)\right) = 2 e^{-i\frac{\pi}{3}} \]

3. **Raise to the power \( 4n \)**:
\[ (1 - i\sqrt{3})^{4n} = (2 e^{-i\frac{\pi}{3}})^{4n} = 2^{4n} e^{-i\frac{4n\pi}{3}} \]

4. **Argument of the denominator**:
\[ \text{arg}((1 - i\sqrt{3})^{4n}) = -\frac{4n\pi}{3} \]

### Step 3: Combine the arguments
Now we can find the argument of \( z \):
\[ \text{arg}(z) = \text{arg}((\sqrt{3} + i)^{4n + 1}) - \text{arg}((1 - i\sqrt{3})^{4n}) \]
\[ \text{arg}(z) = \frac{(4n + 1)\pi}{6} - \left(-\frac{4n\pi}{3}\right) \]
\[ = \frac{(4n + 1)\pi}{6} + \frac{4n\pi}{3} \]

### Step 4: Simplify the expression
To combine the fractions, we need a common denominator:
\[ \frac{4n\pi}{3} = \frac{8n\pi}{6} \]
So,
\[ \text{arg}(z) = \frac{(4n + 1)\pi}{6} + \frac{8n\pi}{6} = \frac{(4n + 1 + 8n)\pi}{6} = \frac{(12n + 1)\pi}{6} \]

Thus, the final result is:
\[ \text{arg}(z) = \frac{(12n + 1)\pi}{6} \]