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. \(\frac{\pi}{6}\)

Answer 1

The Correct Option is B

We are given the complex number \( z = \frac{8 + 4i}{1 + 3i} \). To find the principal argument, we first express \( z \) in polar form. We multiply the numerator and denominator by the conjugate of the denominator \( 1 - 3i \) to simplify: \[ z = \frac{(8 + 4i)(1 - 3i)}{(1 + 3i)(1 - 3i)} = \frac{(8 + 4i)(1 - 3i)}{1 + 9} = \frac{(8 + 4i)(1 - 3i)}{10} \] Now, expand the numerator: \[ (8 + 4i)(1 - 3i) = 8 - 24i + 4i - 12i^2 = 8 - 20i + 12 = 20 - 20i \] Thus, \[ z = \frac{20 - 20i}{10} = 2 - 2i \] Now, express the complex number in polar form \( z = r(\cos \theta + i \sin \theta) \), where \[ r = \sqrt{2^2 + (-2)^2} = \sqrt{8} = 2\sqrt{2}, \quad \theta = \tan^{-1}\left(\frac{-2}{2}\right) = \tan^{-1}(-1) \] Since \( \tan^{-1}(-1) = -\frac{\pi}{4} \)

The correct option is (B) : \(\frac{-\pi}{4}\)

Answer 2

First, simplify the complex number z:

\(z = \frac{8+4i}{1+3i}\)

Multiply the numerator and denominator by the conjugate of the denominator (1-3i):

\(z = \frac{(8+4i)(1-3i)}{(1+3i)(1-3i)}\)

\(z = \frac{8 - 24i + 4i - 12i^2}{1 - 9i^2}\)

Since \(i^2 = -1\), we have:

\(z = \frac{8 - 20i + 12}{1 + 9}\)

\(z = \frac{20 - 20i}{10}\)

\(z = 2 - 2i\)

Now, find the principal argument of z = 2 - 2i. The principal argument, arg(z), is the angle θ such that z = r(cosθ + isinθ), where r is the modulus of z and -π < θ ≤ π.

The complex number is in the fourth quadrant (since the real part is positive and the imaginary part is negative). Thus, the principal argument will be negative.

We can find the argument using the arctangent function:

\(\theta = \arctan\left(\frac{\text{Im}(z)}{\text{Re}(z)}\right) = \arctan\left(\frac{-2}{2}\right) = \arctan(-1)\)

Since the complex number is in the fourth quadrant, the principal argument is \(-\frac{\pi}{4}\).

Therefore, the principal argument of the complex number z is equal to \(-\frac{\pi}{4}\).