Find the argument of `- sqrt(3) +i`

Options

A. `2`
B. `(-2pi)/(3)`
C. `(5pi)/(6)`
D. None of these

Answer

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

### Step 1: Identify the real and imaginary parts
The complex number can be expressed as:
\[ z = -\sqrt{3} + i \]
Here, the real part \(a = -\sqrt{3}\) and the imaginary part \(b = 1\).

### Step 2: Calculate the modulus \(r\)
The modulus \(r\) of a complex number \(z = a + bi\) is given by:
\[ r = \sqrt{a^2 + b^2} \]
Substituting the values:
\[ r = \sqrt{(-\sqrt{3})^2 + (1)^2} = \sqrt{3 + 1} = \sqrt{4} = 2 \]

### Step 3: Find the cosine and sine values
Using the definitions of cosine and sine in terms of the argument \(\theta\):
\[ \cos \theta = \frac{a}{r} \quad \text{and} \quad \sin \theta = \frac{b}{r} \]
Substituting the values:
\[ \cos \theta = \frac{-\sqrt{3}}{2} \quad \text{and} \quad \sin \theta = \frac{1}{2} \]

### Step 4: Determine the angle \(\theta\)
From the values of \(\cos \theta\) and \(\sin \theta\):
- \(\cos \theta = -\frac{\sqrt{3}}{2}\) corresponds to angles in the second quadrant.
- \(\sin \theta = \frac{1}{2}\) corresponds to \(30^\circ\) or \(\frac{\pi}{6}\).

Since the cosine is negative and sine is positive, the angle \(\theta\) is in the second quadrant:
\[ \theta = 180^\circ - 30^\circ = 150^\circ \]
In radians, this is:
\[ \theta = \frac{150 \times \pi}{180} = \frac{5\pi}{6} \]

### Final Answer
Thus, the argument of the complex number \(-\sqrt{3} + i\) is:
\[ \boxed{\frac{5\pi}{6}} \]