Question

a=alfa
Calculati tg a/2, daca 2cosa-sina+1=0

Answer 1

$\displaystyle\\ \text{Calculati tg}\frac{\alpha}{2},\text{ daca }~2\cos\alpha-\sin \alpha+1=0\\\\ \text{Transformam sin si cos in functie de tangenta unghiului pe jumatate.}\\\\ \text{tg}\frac{\alpha}{2}=t\\\\ \sin\alpha=\frac{2t}{1+t^2}\\\\ \cos\alpha=\frac{1-t^2}{1+t^2}\\\\ 2\cos\alpha-\sin\alpha+1=0\\\\ 2\cdot\frac{1-t^2}{1+t^2}-\frac{2t}{1+t^2}+1=0~~\Big|\cdot (1+t^2)\\\\ 2(1-t^2)-2t+1+t^2=0\\\\ 2-2t^2-2t+1+t^2=0\\\\ -2t^2+ t^2-2t+2+1=0\\\\ -t^2-2t+3=0~~\Big|\cdot (-1)\\\\ t^2+2t-3=0$

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$\displaystyle\\ t^2+2t-3=0\\\\ t_{12}=\frac{-2\pm\sqrt{2^2+4\cdot 3}}{2}=\frac{-2\pm\sqrt{16}}{2}=\frac{-2\pm4}{2}=-1\pm 2\\\\ t_1=-1+2=\boxed{\bf 1}\\\\ t_2=-1-2=\boxed{\bf -3}\\\\ \Longrightarrow~~\boxed{\text{tg}\frac{\alpha}{2} \in \{-3;~1\}}$

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