Toán Lớp 11: Giải phương trình (sin$\frac{x}{2}$ – cos $\frac{x}{2}$) ² = sin ²x – 3sinx +2
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Toán Lớp 11: Giải phương trình (sin$\frac{x}{2}$ – cos $\frac{x}{2}$) ² = sin ²x – 3sinx +2
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{\left( {\sin \dfrac{x}{2} – \cos \dfrac{x}{2}} \right)^2} = {\sin ^2}x – 3\sin x + 2\\
\Leftrightarrow {\sin ^2}\dfrac{x}{2} – 2\sin \dfrac{x}{2}.\cos \dfrac{x}{2} + {\cos ^2}\dfrac{x}{2}\\
= {\sin ^2}x – 3\sin x + 2\\
\Leftrightarrow \left( {{{\sin }^2}\dfrac{x}{2} + {{\cos }^2}\dfrac{x}{2}} \right) – 2\sin \dfrac{x}{2}.\cos \dfrac{x}{2}\\
= {\sin ^2}x – 3\sin x + 2\\
\Leftrightarrow 1 – \sin \left( {2.\dfrac{x}{2}} \right) = {\sin ^2}x – 3\sin x + 2\\
\Leftrightarrow 1 – \sin x = {\sin ^2}x – 3\sin x + 2\\
\Leftrightarrow {\sin ^2}x – 2\sin x + 1 = 0\\
\Leftrightarrow {\left( {\sin x – 1} \right)^2} = 0\\
\Leftrightarrow \sin x = 1\\
\Leftrightarrow x = \dfrac{\pi }{2} + k2\pi \left( {k \in Z} \right)\\
Vậy\,x = \dfrac{\pi }{2} + k2\pi \left( {k \in Z} \right)
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