Question

If \(z = \frac{-2}{1 + \sqrt{3}i}\) then the value of \(\arg(z)\) is

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

Hint:

Argument depends on quadrant: for \((-, +)\), \(\arg = \pi - \tan^{-1}(y/|x|)\).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Rationalize and find argument.
Step 2: Detailed Explanation:
\(z = \frac{-2}{1 + \sqrt{3}i} \cdot \frac{1 - \sqrt{3}i}{1 - \sqrt{3}i} = \frac{-2(1 - \sqrt{3}i)}{1 + 3} = \frac{-1 + \sqrt{3}i}{2}\)
\(z = -\frac{1}{2} + i\frac{\sqrt{3}}{2}\)
\(\tan\theta = (\sqrt{3}/2)/(-1/2) = -\sqrt{3}\), but in II quadrant \(\rightarrow \theta = 2\pi/3\)
Step 3: Final Answer:
\(\arg(z) = 2\pi/3\).