Question

The principal argument of the complex number \( z = \frac{8+4i}{1+3i} \) is equal to

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

Hint:

Always convert a complex number to its standard $a + ib$ form before trying to find its argument. Pay close attention to the signs of the real and imaginary parts to correctly identify the quadrant, as this determines whether the argument is positive or negative.

Answer 1

The Correct Option is B

Step 1: Rationalize the Denominator
\[ z = \frac{8+4i}{1+3i} \times \frac{1-3i}{1-3i} \]

Step 2: Simplify
\[ z = \frac{(8+4i)(1-3i)}{(1+3i)(1-3i)} \] Denominator: \[ = 1 - (3i)^2 = 1 + 9 = 10 \] Numerator: \[ = 8 - 24i + 4i - 12i^2 \] \[ = 8 - 20i + 12 = 20 - 20i \] \[ z = \frac{20 - 20i}{10} = 2 - 2i \]

Step 3: Find Argument
\[ \tan\theta = \frac{-2}{2} = -1 \] \[ \theta = \tan^{-1}(1) = \frac{\pi}{4} \] Since $z$ lies in 4th quadrant: \[ \theta = -\frac{\pi}{4} \]

Step 4: Final Answer
\[ \boxed{-\frac{\pi}{4}} \]