Tot numere complexe

[Adriana Nistor]

Scrieti sub forma trigonometrica numarul:

\( z=\frac{(1-i\sqrt{3})(\cos x+i \sin x)}{\cos x+ \sin x+ i (\cos x- \sin x)} \), unde \( x\in R \).

[Virgil Nicula]

Scrieti sub forma trigonometrica numarul \( z=\frac{(1-i\sqrt{3})(\cos x+i \sin x)}{\cos x+ \sin x+ i (\cos x- \sin x)} \), unde \( x\in R^* \) .

Dem. \( \cos x+ \sin x+ i (\cos x- \sin x)=\sqrt 2\cdot\left(\cos\frac {\pi}{4}+i\sin\frac {\pi}{4}\right)\cdot\left(\cos x-i\sin x\right)\ \Longrightarrow\ \arg\left(1-i\sqrt 3\right)=2\pi-\frac {\pi}{3} \)

(argument redus !) si \( |z|=\sqrt 2 \) si \( \mathrm{Arg}(z)=\left[\left(2\pi -\frac {\pi}{3}\right)+x\right]-\left(\frac {\pi}{4}-x\right)+2\pi Z=2x-\frac {7\pi}{12}+2\pi Z \) (argument extins !).