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NDA 2022
Exam
NDA
Year
2022
Subject
Mathematics
Chapter
Complex Number

NDA 2022 · Mathematics · Complex Number

What is the principal argument of (1)/(1+i) where i= -1 ?

What is the principal argument of 11+i\frac{1}{1+i} where i=−1i=\sqrt{-1} ?

Options

  1. A−3π4-\frac{3 \pi}{4}
  2. B−π4-\frac{\pi}{4}
  3. Cπ4\frac{\pi}{4}
  4. D3π4\frac{3 \pi}{4}

Correct answer

B. −π4-\frac{\pi}{4}

Step-by-step solution

Let Z=11+i×1−i1−i=1−i1+1=12−12itan⁡θ=∣−1212∣=1=tan⁡π4∴θ=π4\mathrm{Z}=\frac{1}{1+\mathrm{i}} \times \frac{1-\mathrm{i}}{1-\mathrm{i}}=\frac{1-\mathrm{i}}{1+1}=\frac{1}{2}-\frac{1}{2} \mathrm{i}\tan \theta=\left|\frac{-\frac{1}{2}}{\frac{1}{2}}\right|=1=\tan \frac{\pi}{4}\therefore \theta=\frac{\pi}{4} But z lies in 4th 4^{\text{th }} quadrant ∴\therefore Principal argument =−π4=\frac{-\pi}{4}
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