równanie trygonometryczne

[Maciek32]

Rozwiąż równanie
\(6\cos x+2\sqrt{3}\sin x + \frac{3}{\tg x}+\sqrt{3}=0 \)

[eresh]

Rozwiąż równanie
\(6\cos x+2\sqrt{3}\sin x + \frac{3}{\tg x}+\sqrt{3}=0 \)

\(6\cos x+2\sqrt{3}\sin x + \frac{3}{\tg x}+\sqrt{3}=0\\
6\cos x+2\sqrt{3}\sin x + \frac{3\cos x}{\sin x}+\sqrt{3}=0\\
6\sin x\cos x+2\sqrt{3}\sin^2x+3\cos x+\sqrt{3}\sin x=0\\
2\sin x(3\cos x+\sqrt{3})+3\cos x+\sqrt{3}\sin x=0\\
(3\cos x+\sqrt{3}\sin x)(2\sin x+1)=0\\
3\cos x+\sqrt{3}\sin x=0\;\;\;\vee\;\;\;2\sin x+1=0\\
\sqrt{3}\sin x=-3\cos x\;\;\;\vee\;\;\;\sin x=-\frac{1}{2}\\
\tg x=-\frac{3}{\sqrt{3}}=-\sqrt{3}\;\;\;\vee\;\;\sin x=-\frac{1}{2}\\
x=-\frac{\pi}{3}+k\pi\;\;\vee\;\;x=-\frac{\pi}{6}+2k\pi\;\;\;\vee\;\;\;x=-\frac{5\pi}{6}+2k\pi\;\;\;k\in\mathbb{Z}
\)