Toán Lớp 11: nghiệm của pt sin^2 x+căn3 sinx cosx=1
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Toán Lớp 11: nghiệm của pt sin^2 x+căn3 sinx cosx=1
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\quad \sin^2x + \sqrt3\sin x\cos x = 1\\
\Leftrightarrow \dfrac{1 – \cos2x}{2} + \dfrac{\sqrt3}{2}\sin2x = 1\\
\Leftrightarrow \dfrac{\sqrt3}{2}\sin2x – \dfrac12\cos2x = \dfrac12\\
\Leftrightarrow \sin\left(2x – \dfrac{\pi}{6}\right) = \sin\dfrac{\pi}{6}\\
\Leftrightarrow \left[\begin{array}{l}2x – \dfrac{\pi}{6} = \dfrac{\pi}{6} +k2\pi\\2x – \dfrac{\pi}{6} = \dfrac{5\pi}{6} + k2\pi\end{array}\right.\\
\Leftrightarrow \left[\begin{array}{l}x = \dfrac{\pi}{6} +k\pi\\x = \dfrac{\pi}{2} + k\pi\end{array}\right.\quad (k\in\Bbb Z)\\
\text{Vậy}\ S = \left\{\dfrac{\pi}{6} +k\pi;\ \dfrac{\pi}{2} + k\pi\ \Bigg|\ k\in\Bbb Z\right\}
\end{array}\)