Question

Amplitude of \( \frac{1 + \sqrt{3}i}{\sqrt{3} + 1} \) is:

• A. \( \frac{\pi}{6} \)
• B. \( \frac{\pi}{4} \)
• C. \( \frac{\pi}{3} \)
• D. \( \frac{\pi}{2} \)

Hint:

To find the amplitude of a complex number, express the number in the form \( a + bi \) and use the arctangent function.

Answer 1

The Correct Option is A

Step 1: The amplitude (or argument) of a complex number \( z = a + bi \) is given by: \[ \text{Amplitude of } z = \arg(z) = \tan^{-1} \left( \frac{b}{a} \right). \] Step 2: For \( \frac{1 + \sqrt{3}i}{\sqrt{3} + 1} \), we calculate the argument using the formula. The result is \( \frac{\pi}{6} \).

Final Answer: \[ \boxed{\frac{\pi}{6}} \]