If ` z = (-4 +2 sqrt3i)/(5 +sqrt3i)`, then the value of `arg(z)` is

Options

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

Answer

To find the value of \( \arg(z) \) for the complex number \( z = \frac{-4 + 2\sqrt{3}i}{5 + \sqrt{3}i} \), we will simplify the expression and then calculate the argument.

### Step 1: Multiply by the Conjugate
To simplify \( z \), we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \( 5 + \sqrt{3}i \) is \( 5 - \sqrt{3}i \).

\[ z = \frac{(-4 + 2\sqrt{3}i)(5 - \sqrt{3}i)}{(5 + \sqrt{3}i)(5 - \sqrt{3}i)} \]

### Step 2: Calculate the Denominator
The denominator can be calculated using the difference of squares:

\[ (5 + \sqrt{3}i)(5 - \sqrt{3}i) = 5^2 - (\sqrt{3}i)^2 = 25 - 3(-1) = 25 + 3 = 28 \]

### Step 3: Calculate the Numerator
Now we calculate the numerator:

\[ (-4 + 2\sqrt{3}i)(5 - \sqrt{3}i) = -20 + 4\sqrt{3}i + 10\sqrt{3}i - 6i^2 \]

Since \( i^2 = -1 \), we have:

\[ -20 + 14\sqrt{3}i + 6 = -14 + 14\sqrt{3}i \]

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

\[ z = \frac{-14 + 14\sqrt{3}i}{28} = \frac{-14}{28} + \frac{14\sqrt{3}}{28}i = -\frac{1}{2} + \frac{\sqrt{3}}{2}i \]

### Step 5: Find the Argument
The complex number \( z \) can be expressed in the form \( x + yi \) where \( x = -\frac{1}{2} \) and \( y = \frac{\sqrt{3}}{2} \). The argument \( \arg(z) \) is given by:

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

### Step 6: Determine the Angle
The angle \( \tan^{-1}(-\sqrt{3}) \) corresponds to an angle in the fourth quadrant. The reference angle for \( \sqrt{3} \) is \( \frac{\pi}{3} \), so:

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

Since \( z \) is in the second quadrant (as \( x < 0 \) and \( y > 0 \)), we add \( \pi \) to the negative angle:

\[ \arg(z) = \pi - \frac{\pi}{3} = \frac{3\pi}{3} - \frac{\pi}{3} = \frac{2\pi}{3} \]

### Final Result
Thus, the value of \( \arg(z) \) is:

\[ \arg(z) = \frac{2\pi}{3} \]