Question

Let \( z_1 = 1 + i\sqrt{3} \) and \( z_2 = 1 + i \), then \( \arg\left( \frac{z_1}{z_2} \right) \) is

• A. \( \frac{5\pi}{12} \)
• B. \( \frac{7\pi}{12} \)
• C. \( \frac{11\pi}{12} \)
• D. \( \frac{3\pi}{12} \)
• E. Not defined

Hint:

Argument of quotient = difference of arguments.

Answer 1

The Correct Option is D

Concept: \[ \arg\left(\frac{z_1}{z_2}\right) = \arg(z_1) - \arg(z_2) \]

Step 1: Find argument of \(z_1\). \[ z_1 = 1 + i\sqrt{3} \Rightarrow \tan \theta = \frac{\sqrt{3}}{1} = \sqrt{3} \Rightarrow \theta = \frac{\pi}{3} \]

Step 2: Find argument of \(z_2\). \[ z_2 = 1 + i \Rightarrow \tan \theta = 1 \Rightarrow \theta = \frac{\pi}{4} \]

Step 3: Subtract arguments. \[ \arg\left(\frac{z_1}{z_2}\right) = \frac{\pi}{3} - \frac{\pi}{4} = \frac{4\pi - 3\pi}{12} = \frac{\pi}{12} \]