[tex]\displaystyle\\
\text{Calculati tg}\frac{\alpha}{2},\text{ daca }~2\cos\alpha-\sin \alpha+1=0\\\\
\text{Transformam sin si cos in functie de tangenta unghiului pe jumatate.}\\\\
\text{tg}\frac{\alpha}{2}=t\\\\
\sin\alpha=\frac{2t}{1+t^2}\\\\
\cos\alpha=\frac{1-t^2}{1+t^2}\\\\
2\cos\alpha-\sin\alpha+1=0\\\\
2\cdot\frac{1-t^2}{1+t^2}-\frac{2t}{1+t^2}+1=0~~\Big|\cdot (1+t^2)\\\\
2(1-t^2)-2t+1+t^2=0\\\\
2-2t^2-2t+1+t^2=0\\\\
-2t^2+ t^2-2t+2+1=0\\\\
-t^2-2t+3=0~~\Big|\cdot (-1)\\\\
t^2+2t-3=0[/tex]
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[tex]\displaystyle\\
t^2+2t-3=0\\\\
t_{12}=\frac{-2\pm\sqrt{2^2+4\cdot 3}}{2}=\frac{-2\pm\sqrt{16}}{2}=\frac{-2\pm4}{2}=-1\pm 2\\\\
t_1=-1+2=\boxed{\bf 1}\\\\
t_2=-1-2=\boxed{\bf -3}\\\\
\Longrightarrow~~\boxed{\text{tg}\frac{\alpha}{2} \in \{-3;~1\}}[/tex]
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