Toán Lớp 11: 6cos² 2x+ 5sin2x-7=0
Sin²x+2sinx+1=0
Cos2x+sinx+1=0
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Toán Lớp 11: 6cos² 2x+ 5sin2x-7=0
Sin²x+2sinx+1=0
Cos2x+sinx+1=0
Comments ( 1 )
1,\\
\left[ \begin{array}{l}
x = \dfrac{\pi }{{12}} + k\pi \\
x = \dfrac{{5\pi }}{{12}} + k\pi \\
x = \dfrac{1}{2}\arcsin \dfrac{1}{3} + k\pi \\
x = \dfrac{\pi }{2} – \dfrac{1}{2}\arcsin \dfrac{1}{3} + k\pi
\end{array} \right.\,\,\left( {k \in Z} \right)\\
2,\\
x = – \dfrac{\pi }{2} + k2\pi \,\,\,\left( {k \in Z} \right)\\
3,\\
\left[ \begin{array}{l}
x = \arcsin \dfrac{{1 – \sqrt {17} }}{4} + k2\pi \\
x = \pi – \arcsin \dfrac{{1 – \sqrt {17} }}{4} + k2\pi
\end{array} \right.\,\,\,\,\left( {k \in Z} \right)
\end{array}\)
1,\\
6{\cos ^2}2x + 5\sin 2x – 7 = 0\\
\Leftrightarrow 6.\left( {1 – {{\sin }^2}2x} \right) + 5\sin 2x – 7 = 0\\
\Leftrightarrow 6 – 6{\sin ^2}2x + 5\sin 2x – 7 = 0\\
\Leftrightarrow – 6{\sin ^2}2x + 5\sin 2x – 1 = 0\\
\Leftrightarrow 6{\sin ^2}2x – 5\sin 2x + 1 = 0\\
\Leftrightarrow \left( {2\sin 2x – 1} \right)\left( {3\sin 2x – 1} \right) = 0\\
\Leftrightarrow \left[ \begin{array}{l}
2\sin 2x – 1 = 0\\
3\sin 2x – 1 = 0
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
\sin 2x = \dfrac{1}{2}\\
\sin 2x = \dfrac{1}{3}
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
2x = \dfrac{\pi }{6} + k2\pi \\
2x = \dfrac{{5\pi }}{6} + k2\pi \\
2x = \arcsin \dfrac{1}{3} + k2\pi \\
2x = \pi – \arcsin \dfrac{1}{3} + k2\pi
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
x = \dfrac{\pi }{{12}} + k\pi \\
x = \dfrac{{5\pi }}{{12}} + k\pi \\
x = \dfrac{1}{2}\arcsin \dfrac{1}{3} + k\pi \\
x = \dfrac{\pi }{2} – \dfrac{1}{2}\arcsin \dfrac{1}{3} + k\pi
\end{array} \right.\,\,\left( {k \in Z} \right)\\
2,\\
{\sin ^2}x + 2\sin x + 1 = 0\\
\Leftrightarrow {\sin ^2}x + 2.\sin x.1 + {1^2} = 0\\
\Leftrightarrow {\left( {\sin x + 1} \right)^2} = 0\\
\Leftrightarrow \sin x + 1 = 0\\
\Leftrightarrow \sin x = – 1\\
\Leftrightarrow x = – \dfrac{\pi }{2} + k2\pi \,\,\,\left( {k \in Z} \right)\\
3,\\
\cos 2x + \sin x + 1 = 0\\
\Leftrightarrow 1 – 2{\sin ^2}x + \sin x + 1 = 0\\
\Leftrightarrow – 2{\sin ^2}x + \sin x + 2 = 0\\
\Leftrightarrow 2{\sin ^2}x – \sin x – 2 = 0\\
\Leftrightarrow \left[ \begin{array}{l}
\sin x = \dfrac{{1 + \sqrt {17} }}{4}\\
\sin x = \dfrac{{1 – \sqrt {17} }}{4}
\end{array} \right.\\
– 1 \le \sin x \le 1 \Rightarrow \sin x = \dfrac{{1 – \sqrt {17} }}{4}\\
\sin x = \dfrac{{1 – \sqrt {17} }}{4}\\
\Leftrightarrow \left[ \begin{array}{l}
x = \arcsin \dfrac{{1 – \sqrt {17} }}{4} + k2\pi \\
x = \pi – \arcsin \dfrac{{1 – \sqrt {17} }}{4} + k2\pi
\end{array} \right.\,\,\,\,\left( {k \in Z} \right)
\end{array}\)