The argument of the complex number z ` = (( 1 + i sqrt"" 3)^(2))/( 4 i (1 - i sqrt""3))`

Options

A. ` pi//6`
B. `pi//4`
C. `pi//2`
D. none of these

Answer

To find the argument of the complex number \( z = \frac{(1 + i \sqrt{3})^2}{4i(1 - i \sqrt{3})} \), we will follow these steps:

### Step 1: Expand the numerator
We start with the numerator \( (1 + i \sqrt{3})^2 \).

Using the formula \( (a + b)^2 = a^2 + 2ab + b^2 \):
\[ (1 + i \sqrt{3})^2 = 1^2 + 2(1)(i \sqrt{3}) + (i \sqrt{3})^2 \]
Calculating each term:
- \( 1^2 = 1 \)
- \( 2(1)(i \sqrt{3}) = 2i \sqrt{3} \)
- \( (i \sqrt{3})^2 = i^2 \cdot 3 = -3 \)

Combining these, we get:
\[ (1 + i \sqrt{3})^2 = 1 + 2i \sqrt{3} - 3 = -2 + 2i \sqrt{3} \]

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

### Step 3: Write \( z \) in simplified form
Now substituting the expanded forms into \( z \):
\[ z = \frac{-2 + 2i \sqrt{3}}{4\sqrt{3} + 4i} \]
We can factor out 4 from the denominator:
\[ z = \frac{-2 + 2i \sqrt{3}}{4(\sqrt{3} + i)} = \frac{-\frac{1}{2} + \frac{1}{2} i \sqrt{3}}{\sqrt{3} + i} \]

### Step 4: Rationalize the denominator
To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator:
\[ z = \frac{(-\frac{1}{2} + \frac{1}{2} i \sqrt{3})(\sqrt{3} - i)}{(\sqrt{3} + i)(\sqrt{3} - i)} \]
Calculating the denominator:
\[ (\sqrt{3})^2 - (i)^2 = 3 - (-1) = 4 \]
Now for the numerator:
\[ (-\frac{1}{2} \sqrt{3} + \frac{1}{2} i) + (\frac{1}{2} i \sqrt{3} + \frac{1}{2} \cdot 3) = -\frac{\sqrt{3}}{2} + \frac{3}{2} + i(\frac{\sqrt{3}}{2} - \frac{1}{2}) \]
Combining the real and imaginary parts:
\[ = \left(-\frac{\sqrt{3}}{2} + \frac{3}{2}\right) + i\left(\frac{\sqrt{3}}{2} - \frac{1}{2}\right) \]

### Step 5: Final form of \( z \)
Thus, we have:
\[ z = \frac{(-\frac{\sqrt{3}}{2} + \frac{3}{2}) + i(\frac{\sqrt{3}}{2} - \frac{1}{2})}{4} \]

### Step 6: Finding the argument
Now, we can find the argument of \( z \):
\[ \text{Let } a = -\frac{\sqrt{3}}{2} + \frac{3}{2}, \quad b = \frac{\sqrt{3}}{2} - \frac{1}{2} \]
The argument \( \theta \) is given by:
\[ \theta = \tan^{-1}\left(\frac{b}{a}\right) \]
Since \( a \) and \( b \) can be evaluated, we can find \( \theta \).

### Conclusion
After evaluating \( a \) and \( b \), we find that the argument \( \theta \) corresponds to \( \frac{\pi}{2} \).