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$2.\;4\cos^2x-11\sin x-11=0\\4-4\sin^2x-11\sin x-11=0\\-4\sin^2x-11\sin x-7=0\\4\sin^2x+11\sin x+7=0\\\sin x=t,\;\sin^2x=t^2,\;t\in[-1;\;1]\\4t^2+11t+7=0\\D=\121-4\cdot4\cdot7=121-112=9\\t_{1,2}=\frac{-11\pm9}4\\t_1=1\\t_2=-4\;-\;He\;nogx.\\\sin x=1\Rightarrow x=\frac\pi2+\pi n,\;n\in\mathbb{Z}\\\\3.\;3\sin^2x+8\sin x\cos x+4\cos^2x=0\;\;\;\div\cos^2x\\3tg^2x+8tg x+4=0\\tgx=t,\;tg^2x=t^2\\3t^2+8t+4=0\\D=64-4\cdot3\cdot4=16\\t_{1,2}=\frac{-8\pm4}6\\t_1=-2,\;t_2=-\frac23$ $\begin{cases}tgx=-2\\tgx=-\frac23\end{cases}\Rightarrow\begin{cases}x_1=-arctg(2)+\pi n\\x=-arctg(\frac23)+\pi n\end{cases},\;n\in\mathbb{Z}\\\\\\4.\;5tgx-12ctgx+11=0\\\frac{5\sin x}{\cos x}-\frac{12\cos x}{\sin x}+11=0\\\frac{5\sin^2x-12\cos^2x+11\sin x\cos x}{\sin x\cos x}=0\\\sin x\cos x\neq0\Rightarrow x\neq\frac\pi2+\pi n,\;x\neq\pi n\\5\sin^2x-12\cos^2x+11\sin x\cos x=0\;\;\div\cos^2x\\5tg^2x+11tgx-12=0\\tgx=t,\;tg^2x=t^2\\5t^2+11t-12=0\\D=121+4\cdot5\cdot12=361$
$t_{1,2}=\frac{-11\pm19}{10}\\t_1=-2,\;t_2=\frac45\\x_1=-arctg(2)+\pi n\\x_2=arctg(\frac45)6+\pi n,\;n\in\mathbb{Z}$