Principal argument of complex number `z=(sqrt3+i)/(sqrt3-i)` equal

Options

A. `-pi/3`
B. `pi/3`
C. `pi/6`
D. None of these

Explanation

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

### Step 1: Multiply by the Conjugate
To simplify the complex number, we multiply the numerator and the denominator by the conjugate of the denominator:

\[ z = \frac{\sqrt{3} + i}{\sqrt{3} - i} \cdot \frac{\sqrt{3} + i}{\sqrt{3} + i} \]

### Step 2: Simplify the Denominator
The denominator becomes:

\[ (\sqrt{3} - i)(\sqrt{3} + i) = \sqrt{3}^2 - i^2 = 3 - (-1) = 3 + 1 = 4 \]

### Step 3: Simplify the Numerator
The numerator becomes:

\[ (\sqrt{3} + i)(\sqrt{3} + i) = \sqrt{3}^2 + 2\sqrt{3}i + i^2 = 3 + 2\sqrt{3}i - 1 = 2 + 2\sqrt{3}i \]

### Step 4: Combine the Results
Now we can combine the results:

\[ z = \frac{2 + 2\sqrt{3}i}{4} = \frac{1}{2} + \frac{\sqrt{3}}{2}i \]

### Step 5: Identify \( a \) and \( b \)
Here, we have:

\[ a = \frac{1}{2}, \quad b = \frac{\sqrt{3}}{2} \]

### Step 6: Calculate the Argument
The argument of a complex number \( z = a + bi \) can be calculated using the formula:

\[ \text{arg}(z) = \tan^{-1}\left(\frac{b}{a}\right) \]

Substituting the values of \( a \) and \( b \):

\[ \text{arg}(z) = \tan^{-1}\left(\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}\right) = \tan^{-1}(\sqrt{3}) \]

### Step 7: Determine the Angle
We know that:

\[ \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \]

Thus, the principal argument is:

\[ \text{arg}(z) = \frac{\pi}{3} \]

### Final Answer
The principal argument of the complex number \( z = \frac{\sqrt{3} + i}{\sqrt{3} - i} \) is:

\[ \frac{\pi}{3} \]

---