Question

If $ i = \sqrt{-1} $ then  $\text{Arg}\left[ \frac{(1+i)^{2025}}{1+i^{2022}} \right]=$

• A. \(\frac{-π}{4}\)
• B. \(\frac{π}{4}\)
• C. \(\frac{3π}{4}\)
• D. \(\frac{-3π}{4}\)

Answer 1

The Correct Option is A

To solve the problem, we need to evaluate the argument (principal value) of the complex expression:

$\text{Arg}\left(\frac{(1+i)^{2025}}{(1-i)^{2022}}\right)$

1. Expressing Complex Numbers in Polar Form:
We know:
$1+i = \sqrt{2} \text{cis}\left(\frac{\pi}{4}\right)$
$1-i = \sqrt{2} \text{cis}\left(-\frac{\pi}{4}\right)$

2. Apply Exponent Rules:
Using the property: $(r \text{cis} \theta)^n = r^n \text{cis}(n\theta)$
So, $(1+i)^{2025} = (\sqrt{2})^{2025} \text{cis}\left(\frac{2025\pi}{4}\right)$
$(1-i)^{2022} = (\sqrt{2})^{2022} \text{cis}\left(-\frac{2022\pi}{4}\right)$

3. Divide the Expressions:
$\frac{(1+i)^{2025}}{(1-i)^{2022}} = \frac{(\sqrt{2})^{2025}}{(\sqrt{2})^{2022}} \cdot \text{cis}\left(\frac{2025\pi}{4} + \frac{2022\pi}{4}\right)$
= $(\sqrt{2})^3 \cdot \text{cis}\left(\frac{4047\pi}{4}\right)$

4. Compute the Argument:
We only need the angle (argument) part: $\text{Arg} = \frac{4047\pi}{4}$
Now reduce the angle to its principal value in $(-\pi, \pi]$.

Divide $4047$ by $8$: $4047 \div 8 = 505$ remainder $7$
So, $ \frac{4047\pi}{4} = \frac{8 \cdot 505 + 7}{4} \pi = 505 \cdot 2\pi + \frac{7\pi}{4}$

Since $2\pi$ is a full rotation, the angle reduces to $\frac{7\pi}{4}$
But $\frac{7\pi}{4}$ lies in the 4th quadrant, and we must bring it into $(-\pi, \pi]$ range.

$ \frac{7\pi}{4} - 2\pi = \frac{7\pi - 8\pi}{4} = -\frac{\pi}{4}$

Final Answer:
The value of the argument is $ { -\frac{\pi}{4} } $