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BITSAT 2012
Exam
BITSAT
Year
2012
Subject
Mathematics
Chapter
Complex Number

BITSAT 2012 · Mathematics · Complex Number

The amplitude of (π)/(5)+i (1- (π)/(5))

The amplitude of \sin \frac{\pi}{5}+i \left(1-\cos$\frac{\pi}{5}\right)

Options

  1. Aπ/5\pi / 5
  2. B2π/52 \pi / 5
  3. Cπ/10\pi / 10
  4. Dπ/15\pi / 15

Correct answer

C. π/10\pi / 10

Step-by-step solution

sin⁡π5+i(1−cos⁡π5)=2sin⁡π10cos⁡π10+i2sin⁡2π10=2sin⁡π10(cos⁡π10+isin⁡π10)Foramplitude,\sin \frac{\pi}{5}+\mathrm{i}\left(1-\cos \frac{\pi}{5}\right)=2 \sin \frac{\pi}{10} \cos \frac{\pi}{10}+\mathrm{i} 2 \sin ^{2} \frac{\pi}{10}=2 \sin \frac{\pi}{10} \left(\cos \frac{\pi}{10}+\mathrm{i} \sin \frac{\pi}{10}\right) For amplitude,\tan \theta=\frac{\pi}{\sin \frac{\pi}{10}}{\cos \frac{\pi}{10}}=\tan \frac{\pi}{10}\Rightarrow \theta=\frac{\pi}{10}$$
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