The argument of `(1-isqrt(3))/(1+isqrt(3))`, is

Options

A. `pi/3`
B. `(2pi)/3`
C. `(7pi)/6`
D. `-(2pi)/3`

Explanation

To find the argument of the complex number \(\frac{1 - i\sqrt{3}}{1 + i\sqrt{3}}\), we can follow these steps:

### Step 1: Write the complex number in standard form
We start with the complex number:
\[ z = \frac{1 - i\sqrt{3}}{1 + i\sqrt{3}} \]

### Step 2: Multiply the numerator and denominator by the conjugate of the denominator
To simplify, we multiply both the numerator and the denominator by the conjugate of the denominator:
\[ z = \frac{(1 - i\sqrt{3})(1 - i\sqrt{3})}{(1 + i\sqrt{3})(1 - i\sqrt{3})} \]

### Step 3: Simplify the denominator
The denominator simplifies as follows:
\[ (1 + i\sqrt{3})(1 - i\sqrt{3}) = 1^2 - (i\sqrt{3})^2 = 1 - (-3) = 1 + 3 = 4 \]

### Step 4: Expand the numerator
Now, we expand the numerator:
\[ (1 - i\sqrt{3})(1 - i\sqrt{3}) = 1 - 2i\sqrt{3} + (i\sqrt{3})^2 = 1 - 2i\sqrt{3} - 3 = -2 - 2i\sqrt{3} \]

### Step 5: Combine results
Now we can write \(z\) as:
\[ z = \frac{-2 - 2i\sqrt{3}}{4} = -\frac{1}{2} - \frac{i\sqrt{3}}{2} \]

### Step 6: Identify the real and imaginary parts
From the expression \(-\frac{1}{2} - \frac{i\sqrt{3}}{2}\), we identify:
- Real part \(x = -\frac{1}{2}\)
- Imaginary part \(y = -\frac{\sqrt{3}}{2}\)

### Step 7: Calculate the argument
The argument \(\theta\) of a complex number \(z = x + iy\) is given by:
\[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \]
Substituting the values:
\[ \theta = \tan^{-1}\left(\frac{-\frac{\sqrt{3}}{2}}{-\frac{1}{2}}\right) = \tan^{-1}(\sqrt{3}) \]

### Step 8: Determine the angle
Since both \(x\) and \(y\) are negative, the complex number lies in the third quadrant. The reference angle for \(\tan^{-1}(\sqrt{3})\) is \(\frac{\pi}{3}\). Therefore, the argument in the third quadrant is:
\[ \theta = \pi + \frac{\pi}{3} = \frac{3\pi}{3} + \frac{\pi}{3} = \frac{4\pi}{3} \]

### Final Answer
Thus, the argument of the complex number \(\frac{1 - i\sqrt{3}}{1 + i\sqrt{3}}\) is:
\[ \frac{4\pi}{3} \]