If z is a complex number such that `|z| =4` and arg (z) `= ( 5pi)/( 6)`, then what is z equal to ?

Options

A. A)`2 sqrt( 3) + 2i`
B. B)`2 sqrt( 3) -2i`
C. C)`2 sqrt( 3)-3i`
D. D)`- sqrt( 3) +i`

Answer

To find the complex number \( z \) given that \( |z| = 4 \) and \( \arg(z) = \frac{5\pi}{6} \), we can use the polar form of complex numbers.

### Step-by-step Solution:

1. **Understanding the Polar Form**:
The polar form of a complex number \( z \) can be expressed as:
\[ z = r(\cos \theta + i \sin \theta) \]
where \( r \) is the modulus (magnitude) and \( \theta \) is the argument (angle).

2. **Substituting the Given Values**:
Here, we know:
- \( r = |z| = 4 \)
- \( \theta = \arg(z) = \frac{5\pi}{6} \)

Substituting these values into the polar form:
\[ z = 4 \left( \cos\left(\frac{5\pi}{6}\right) + i \sin\left(\frac{5\pi}{6}\right) \right) \]

3. **Calculating the Cosine and Sine**:
We need to find \( \cos\left(\frac{5\pi}{6}\right) \) and \( \sin\left(\frac{5\pi}{6}\right) \):
- \( \cos\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2} \)
- \( \sin\left(\frac{5\pi}{6}\right) = \frac{1}{2} \)

4. **Substituting Back**:
Now substituting these values back into the equation for \( z \):
\[ z = 4 \left( -\frac{\sqrt{3}}{2} + i \cdot \frac{1}{2} \right) \]

5. **Simplifying**:
Distributing the 4:
\[ z = 4 \cdot -\frac{\sqrt{3}}{2} + 4 \cdot i \cdot \frac{1}{2} \]
\[ z = -2\sqrt{3} + 2i \]

6. **Final Answer**:
Thus, the complex number \( z \) is:
\[ z = -2\sqrt{3} + 2i \]

### Summary:
The complex number \( z \) is \( -2\sqrt{3} + 2i \).