Question

va rog foarte mult ajutor !!
$(\cos(x) + \: isin(x)) = a \\ ( \cos(2x) + i \sin(2x)) = b \\ ( \cos(3x) + i \sin(3x)) = c \\ a \times b \times c = i$
calculati x care apartine cadranului 1

Answer 1

[tex]\text{Formula lui Moivre zice asa:}\\ \text{Daca avem un numar complex z}=r(\cos \alpha+i\sin \alpha),\text{atunci:}\\ \boxed{z^n=r^n(\cos n\alpha+i\sin n\alpha)}\\ \text{Avem ca:}\\ b=\cos2x+i\sin2x=(\cos x+i\sin x)^2=a^2\\ c=\cos 3x+i\sin 3x=a^3\\ a\cdot b\cdot c=i\\ a\cdot a^2\cdot a^3=i\\ a^6=i\\ \cos 6x+i\sin 6x=i\\ \text{De aici deducem ca :}\\ cos 6x =0\ si\ \sin 6x=1\text{ adica }6x=\dfrac{\pi}{2}\Rightarrow \boxed{x=\dfrac{\pi}{12}}[/tex]