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 follow these steps:

### Step 1: Multiply by the Conjugate of the Denominator
To simplify the expression, we multiply the numerator and the denominator by the conjugate of the denominator.

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

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

\[ (5 + \sqrt{3}i)(5 - \sqrt{3}i) = 5^2 - (\sqrt{3})^2 = 25 - 3 = 22 \]

### Step 3: Simplify the Numerator
Now, we simplify the numerator:

\[ (-4 + 2\sqrt{3}i)(5 - \sqrt{3}i) = -20 + 4\sqrt{3}i + 10\sqrt{3}i - 2\sqrt{3}(-1) \]

This simplifies to:

\[ -20 + 14\sqrt{3}i + 2 = -18 + 14\sqrt{3}i \]

### Step 4: Combine the Results
Now, we can write \( z \) as:

\[ z = \frac{-18 + 14\sqrt{3}i}{22} = \frac{-9 + 7\sqrt{3}i}{11} \]

### Step 5: Identify Real and Imaginary Parts
From the expression \( z = \frac{-9}{11} + \frac{7\sqrt{3}}{11}i \), we identify:

- Real part \( a = \frac{-9}{11} \)
- Imaginary part \( b = \frac{7\sqrt{3}}{11} \)

### Step 6: Calculate the Argument
The argument \( \arg(z) \) can be calculated using the formula:

\[ \arg(z) = \tan^{-1}\left(\frac{b}{a}\right) = \tan^{-1}\left(\frac{\frac{7\sqrt{3}}{11}}{\frac{-9}{11}}\right) = \tan^{-1}\left(\frac{7\sqrt{3}}{-9}\right) \]

### Step 7: Determine the Quadrant
Since the real part is negative and the imaginary part is positive, \( z \) lies in the second quadrant. The reference angle can be found using:

\[ \tan^{-1}\left(-\frac{7\sqrt{3}}{9}\right) \]

### Step 8: Final Argument Calculation
In the second quadrant, the argument is given by:

\[ \arg(z) = \pi + \tan^{-1}\left(-\frac{7\sqrt{3}}{9}\right) = \pi - \tan^{-1}\left(\frac{7\sqrt{3}}{9}\right) \]

Using the known values, we can find that:

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

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

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

---