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Complex Numbers — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsComplex Numbers2 Marks
MCQ
If $z=1-\cos \alpha+i \sin \alpha$, then amp $z=$
  • A
    $\frac{\alpha}{2}$
  • B
    $-\frac{\alpha}{2}$
  • C
    $\frac{\pi}{2}+\frac{\alpha}{2}$
  • ✓
    $\frac{\pi}{2}-\frac{\alpha}{2}$
✓

Answer

Correct option: D.
$\frac{\pi}{2}-\frac{\alpha}{2}$
(D)
$\operatorname{amp}(z)=\tan ^{-1}\left(\frac{\sin \alpha}{1-\cos \alpha}\right)$
$=\tan ^{-1}\left(\cot \frac{\alpha}{2}\right)$
$=\tan ^{-1}\left\{\tan \left(\frac{\pi}{2}-\frac{\alpha}{2}\right)\right\}=\frac{\pi}{2}-\frac{\alpha}{2}$

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