If i z ^4+1=0, then z can take the value - Vidyadip

MCQ

If $i z ^4+1=0$, then z can take the value

A
$\frac{1+i}{\sqrt{2}}$
✓
$\cos \frac{\pi}{8}+ i \sin \frac{\pi}{8}$
C
$\frac{1}{4 i}$
D
i

✓

Answer

Correct option: B.

$\cos \frac{\pi}{8}+ i \sin \frac{\pi}{8}$

(B)
$i z^4=-1 \Rightarrow z^4=\frac{-1}{i} \Rightarrow z^4=i \Rightarrow z=(i)^{\frac{1}{4}}$
$\Rightarrow z=(0+i)^{\frac{1}{4}} \Rightarrow z=\left(\cos \frac{\pi}{2}+i \sin \frac{\pi}{2}\right)^{\frac{1}{4}}$
$\Rightarrow z =\cos \frac{\pi}{8}+ i \sin \frac{\pi}{8}$ (using DeMoivre's theorem)

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