Question

cum calculez z la puterea 4 = i ? numere complexe

Answer 1

Sriem sub forma trigonometrica: $z=r(\cos\alpha+i\sin\alpha)$

Acum, evident ca $r=1$ .

Atunci avem:  $z^4=\cos 4\alpha+i\sin 4\alpha$

Iar ecuatia noastra este: $\cos 4\alpha+i\sin 4\alpha=i$

De unde rezulta:

$\cos 4\alpha=0\\ \sin 4\alpha=1$

De unde ne dam seama imediat ca:

$4\alpha=\frac{\pi}{2}\\ \\ \\ \alpha=\dfrac{\pi}{8}.$

Asa ca solutia va fi:

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