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

To find the value of \(\arg(z)\) for \(z = \frac{-2}{1 + \sqrt{3}i}\), we need to first express \(z\) in a standard form of a complex number and then determine the argument.

1. Rationalize the Denominator: The given complex number is \(z = \frac{-2}{1 + \sqrt{3}i}\).
   • To rationalize the denominator, multiply both the numerator and the denominator by the conjugate of the denominator \(1 - \sqrt{3}i\).
   \(z = \frac{-2(1 - \sqrt{3}i)}{(1 + \sqrt{3}i)(1 - \sqrt{3}i)}\)
2. Simplify the Denominator:
   • The denominator simplifies using the identity \(a^2 - b^2 = (a+b)(a-b)\). Here, \(a = 1\) and \(b = \sqrt{3}i\).
   \((1 + \sqrt{3}i)(1 - \sqrt{3}i) = 1^2 - (\sqrt{3}i)^2 = 1 - (-3) = 4\)
3. Simplify the Numerator:
   • Compute the multiplication in the numerator:
   Numerator = \(-2(1 - \sqrt{3}i) = -2 + 2\sqrt{3}i\)
4. Resulting Complex Number:
   • Combine the results:
   \(z = \frac{-2 + 2\sqrt{3}i}{4} = -\frac{1}{2} + \frac{\sqrt{3}}{2}i\)
5. Identify Argument:
   • The standard form of a complex number is \(x + yi\), and the argument \(\theta\) is given by \(\tan\theta = \frac{y}{x}\).
   • Here, \(-\frac{1}{2}\) is the real part, and \(\frac{\sqrt{3}}{2}\) is the imaginary part.
   \(\tan\theta = \frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = -\sqrt{3}\)
6. Calculate the Argument:
   • The angle whose tangent is -\sqrt{3} is \(-\pi/3\) in the fourth quadrant, but since the point (-\frac{1}{2}, \frac{\sqrt{3}}{2})\) lies in the second quadrant, we add \(\pi\):
   \(\theta = \pi - \pi/3 = 2\pi/3\)
7. Conclusion:
   • Thus, the value of \(\arg(z)\) is \(2\pi/3\).

Therefore, the correct answer is \(2\pi/3\).