Đáp án:
\[\begin{array}{l}
56,\,\,\,D\\
57,\,\,\,\,C
\end{array}\]
Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
56,\\
{\sin ^2}x - \sqrt 3 \sin x.\cos x - 1 = 0\\
\Leftrightarrow \left( {{{\sin }^2}x - 1} \right) - \sqrt 3 \sin x.\cos x = 0\\
\Leftrightarrow - {\cos ^2}x - \sqrt 3 \sin x.\cos x = 0\\
\Leftrightarrow - \cos x.\left( {\cos x + \sqrt 3 \sin x} \right) = 0\\
\Leftrightarrow \left[ \begin{array}{l}
\cos x = 0\\
\cos x + \sqrt 3 \sin x = 0
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
\cos x = 0\\
\dfrac{1}{2}\cos x + \dfrac{{\sqrt 3 }}{2}\sin x = 0
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
\cos x = 0\\
\cos x.\cos \dfrac{\pi }{3} + \sin x.\sin \dfrac{\pi }{3} = 0
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
\cos x = 0\\
\cos \left( {x - \dfrac{\pi }{3}} \right) = 0
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
x = \dfrac{\pi }{2} + k\pi \\
x - \dfrac{\pi }{3} = - \dfrac{\pi }{2} + k\pi
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
x = \dfrac{\pi }{2} + k\pi \\
x = - \dfrac{\pi }{6} + k\pi
\end{array} \right.\,\,\,\,\,\left( {k \in Z} \right)\\
57,\\
2{\sin ^2}x - \sqrt 3 \sin x.\cos x + {\cos ^2}x = 1\\
\Leftrightarrow 2{\sin ^2}x - \sqrt 3 \sin x.\cos x + {\cos ^2}x - 1 = 0\\
\Leftrightarrow 2{\sin ^2}x - \sqrt 3 \sin x.\cos x - {\sin ^2}x = 0\\
\Leftrightarrow {\sin ^2}x - \sqrt 3 \sin x.\cos x = 0\\
\Leftrightarrow 2\sin x.\left( {\dfrac{1}{2}\sin x - \dfrac{{\sqrt 3 }}{2}\cos x} \right) = 0\\
\Leftrightarrow \left[ \begin{array}{l}
\sin x = 0\\
\dfrac{1}{2}\sin x - \dfrac{{\sqrt 3 }}{2}\cos x = 0
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
\sin x = 0\\
\sin x.\cos \dfrac{\pi }{3} - \cos x.\sin \dfrac{\pi }{3} = 0
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
\sin x = 0\\
\sin \left( {x - \dfrac{\pi }{3}} \right) = 0
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
x = k\pi \\
x - \dfrac{\pi }{3} = k\pi
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
x = k\pi \\
x = \dfrac{\pi }{3} + k\pi
\end{array} \right.\,\,\,\,\,\left( {k \in Z} \right)
\end{array}\)
