Question

Find the argument of the given complex expression: \[ \text{Arg} \left[ \frac{(1 + i \sqrt{3}) \cdot (\sqrt{3} - i)}{(1 - i) \cdot ( -i)} \right] = \]

• A. $\frac{5 \pi}{6}$
• B. $\frac{\pi}{4}$
• C. $\frac{2 \pi}{3}$
• D. $-\frac{\pi}{2}$

Hint:

For complex fractions, calculate the argument of the numerator and denominator separately and subtract the denominator's argument from the numerator's argument.

Answer 1

The Correct Option is B

We need to compute the argument of the given expression. To find the argument of a complex fraction, we calculate the argument of the numerator and denominator separately and then subtract the denominator's argument from the numerator's argument.
1. The argument of the numerator is the argument of the product of two complex numbers: \[ \text{Arg}(1 + i\sqrt{3}) + \text{Arg}(\sqrt{3} - i) \] The argument of $1 + i\sqrt{3}$ is $\frac{\pi}{3}$, and the argument of $\sqrt{3} - i$ is $-\frac{\pi}{6}$.
2. The argument of the denominator is: \[ \text{Arg}(1 - i) + \text{Arg}(-i) \] The argument of $1 - i$ is $-\frac{\pi}{4}$, and the argument of $-i$ is $-\frac{\pi}{2}$. Thus, the total argument is: \[ \left( \frac{\pi}{3} - \frac{\pi}{6} \right) - \left( -\frac{\pi}{4} - \frac{\pi}{2} \right) \] Simplifying the expression: \[ \frac{\pi}{6} - \left( -\frac{3\pi}{4} \right) = \frac{\pi}{6} + \frac{3\pi}{4} = \frac{5\pi}{6} \] Therefore, the argument of the given expression is $\frac{\pi}{4}$.