The argument of `(1 + i sqrt""3) // (1 - i sqrt""3)` is

Options

A. ` 60^(@)`
B. `120^(@)`
C. `210^(@)`
D. `240^(@)`

Answer

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 expression
We start with the expression:
\[ z = \frac{1 + i \sqrt{3}}{1 - i \sqrt{3}} \]

### Step 2: Rationalize the denominator
To simplify this expression, 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
Using the formula for the square of a binomial, we expand the numerator:
\[ (1 + i \sqrt{3})^2 = 1^2 + 2(1)(i \sqrt{3}) + (i \sqrt{3})^2 = 1 + 2i \sqrt{3} - 3 = -2 + 2i \sqrt{3} \]

### Step 4: Expand the denominator
Using the difference of squares formula, we expand the denominator:
\[ (1 - i \sqrt{3})(1 + i \sqrt{3}) = 1^2 - (i \sqrt{3})^2 = 1 - (-3) = 1 + 3 = 4 \]

### Step 5: Combine the results
Now we can combine the results from the numerator and denominator:
\[ z = \frac{-2 + 2i \sqrt{3}}{4} = \frac{-1 + i \sqrt{3}}{2} \]

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

### Step 7: Determine the quadrant
Since \(a < 0\) and \(b > 0\), the complex number lies in the second quadrant.

### Step 8: Calculate the argument
The argument \(\theta\) can be calculated using the formula for the argument in the second quadrant:
\[ \theta = \pi - \tan^{-1}\left(\frac{b}{a}\right) = \pi - \tan^{-1}\left(\frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}}\right) = \pi - \tan^{-1}(-\sqrt{3}) \]

### Step 9: Simplify the argument
Since \(\tan^{-1}(-\sqrt{3})\) corresponds to \(-\frac{\pi}{3}\), we have:
\[ \theta = \pi - \left(-\frac{\pi}{3}\right) = \pi + \frac{\pi}{3} = \frac{3\pi}{3} + \frac{\pi}{3} = \frac{4\pi}{3} \]

### Step 10: Final result
Thus, the argument of the complex number \(\frac{1 + i \sqrt{3}}{1 - i \sqrt{3}}\) is:
\[ \theta = \frac{2\pi}{3} \text{ (or } 120^\circ\text{)} \]