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

Options

A. `pi/3`
B. `(2pi)/3`
C. `(7pi)/6`
D. `(4pi)/3`

Explanation

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

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

### Step 2: Rationalize the denominator
To simplify, we multiply 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: Expand the numerator and denominator
Using the formula \((a + b)(a - b) = a^2 - b^2\) for the denominator:
\[ (1 + i\sqrt{3})(1 - i\sqrt{3}) = 1^2 - (i\sqrt{3})^2 = 1 - (-3) = 1 + 3 = 4 \]

For the numerator, we use the formula \((a - b)^2 = a^2 - 2ab + b^2\):
\[ (1 - i\sqrt{3})^2 = 1^2 - 2(1)(i\sqrt{3}) + (i\sqrt{3})^2 = 1 - 2i\sqrt{3} + (-3) = -2 - 2i\sqrt{3} \]

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

### Step 5: Identify the real and imaginary parts
From the expression \(\frac{-1}{2} - i\frac{\sqrt{3}}{2}\), we identify:
- Real part \(x = -\frac{1}{2}\)
- Imaginary part \(y = -\frac{\sqrt{3}}{2}\)

### Step 6: Find the modulus \(r\)
The modulus \(r\) is given by:
\[ r = \sqrt{x^2 + y^2} = \sqrt{\left(-\frac{1}{2}\right)^2 + \left(-\frac{\sqrt{3}}{2}\right)^2} = \sqrt{\frac{1}{4} + \frac{3}{4}} = \sqrt{1} = 1 \]

### Step 7: Find the argument \(\theta\)
The argument \(\theta\) can be found using:
\[ \cos \theta = \frac{x}{r} = -\frac{1}{2}, \quad \sin \theta = \frac{y}{r} = -\frac{\sqrt{3}}{2} \]

### Step 8: Determine the quadrant
Since both \(x\) and \(y\) are negative, the point lies in the third quadrant.

### Step 9: Find the angle
In the third quadrant, the angle corresponding to \(\cos \theta = -\frac{1}{2}\) and \(\sin \theta = -\frac{\sqrt{3}}{2}\) is:
\[ \theta = \pi + \frac{\pi}{3} = \frac{4\pi}{3} \]

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

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