Question

CAREVA CINE ARE HABAR DE ASTA DAU COROANA

Determinați forma trigonometrica a numerelor complexe z cu proprietatea z = - 1/z ?

Answer 1

$\displaystyle z \neq 0. \\ \\ z=-\frac{1}{z} \Leftrightarrow z^2=-1 \Rightarrow z \in \{-i;i \}. \\ \\ Forma~trigonometrica: \\ \\ -i=\cos \frac{3 \pi}{2}+i \sin \frac{3 \pi}{2}. \\ \\ i=\cos \frac{ \pi}{2}+i \sin \frac{ \pi }{2}.$

Answer 2

z=-1/z

z²=-1=cosπ+isinπ
z1,2 =√(cosπ+isinπ)
cum modulul este 1, √1=1 iar afixele, cf. formueide extragere a radicalului,  vor fi (π+0)/2 =π/2 si,
respectiv, (π+2π)/2=3π/2 asadar
z1= cosπ/2 +isinπ/2=i
z2=cos3π/2+isin3π/2=-i

Altfel
z²=-1=i²
z²-i²=0
(z-i)(z+i)=0
z1=i=cosπ/2 +isinπ/2
z2=-i=cos3π/2+isin3π/2