Toán Lớp 11: giúp em 5 câu này với e vote cho Ạ
1) 3sin²x – cosx + 1 = 0
2) cos²2x – 4sin2x + 4 = 0
3) sin²2x + ( 1 – √3 )cos2x + √3 – 1 = 0
4) tan²2x + ( 1 + √3 ) tan2x + √3 = 0
5) cos2x – 3cosx + 2 =0
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Toán Lớp 11: giúp em 5 câu này với e vote cho Ạ
1) 3sin²x – cosx + 1 = 0
2) cos²2x – 4sin2x + 4 = 0
3) sin²2x + ( 1 – √3 )cos2x + √3 – 1 = 0
4) tan²2x + ( 1 + √3 ) tan2x + √3 = 0
5) cos2x – 3cosx + 2 =0
Comments ( 1 )
1,\\
3{\sin ^2}x – \cos x + 1 = 0\\
\Leftrightarrow 3.\left( {1 – {{\cos }^2}x} \right) – \cos x + 1 = 0\\
\Leftrightarrow 3 – 3{\cos ^2}x – \cos x + 1 = 0\\
\Leftrightarrow – 3{\cos ^2}x – \cos x + 4 = 0\\
\Leftrightarrow 3{\cos ^2}x + \cos x – 4 = 0\\
\Leftrightarrow \left( {3\cos x + 4} \right)\left( {\cos x – 1} \right) = 0\\
\Leftrightarrow \left[ \begin{array}{l}
3\cos x + 4 = 0\\
\cos x – 1 = 0
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
\cos x = – \dfrac{4}{3}\\
\cos x = 1
\end{array} \right.\\
– 1 \le \cos x \le 1 \Rightarrow \cos x = 1 \Leftrightarrow x = k2\pi \,\,\,\,\,\left( {k \in Z} \right)\\
2,\\
{\cos ^2}2x – 4\sin 2x + 4 = 0\\
\Leftrightarrow \left( {1 – {{\sin }^2}2x} \right) – 4\sin 2x + 4 = 0\\
\Leftrightarrow 1 – {\sin ^2}2x – 4\sin 2x + 4 = 0\\
\Leftrightarrow – {\sin ^2}2x – 4\sin 2x + 5 = 0\\
\Leftrightarrow {\sin ^2}2x + 4\sin 2x – 5 = 0\\
\Leftrightarrow \left( {\sin 2x – 1} \right)\left( {\sin 2x + 5} \right) = 0\\
\Leftrightarrow \left[ \begin{array}{l}
\sin 2x – 1 = 0\\
\sin 2x + 5 = 0
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
\sin 2x = 1\\
\sin 2x = – 5
\end{array} \right.\\
– 1 \le \sin 2x \le 1 \Rightarrow \sin 2x = 1\\
\Leftrightarrow 2x = \dfrac{\pi }{2} + k2\pi \\
\Leftrightarrow x = \dfrac{\pi }{4} + k\pi \,\,\,\,\,\,\,\,\,\left( {k \in Z} \right)\\
3,\\
{\sin ^2}2x + \left( {1 – \sqrt 3 } \right)\cos 2x + \sqrt 3 – 1 = 0\\
\Leftrightarrow \left( {1 – {{\cos }^2}2x} \right) + \left( {1 – \sqrt 3 } \right)\cos 2x + \sqrt 3 – 1 = 0\\
\Leftrightarrow 1 – {\cos ^2}2x + \cos 2x – \sqrt 3 \cos 2x + \sqrt 3 -1 = 0\\
\Leftrightarrow – {\cos ^2}2x + \cos 2x – \sqrt 3 \cos 2x + \sqrt 3 = 0\\
\Leftrightarrow \left( {{{\cos }^2}2x – \cos 2x} \right) + \left( {\sqrt 3 \cos 2x – \sqrt 3 } \right) = 0\\
\Leftrightarrow \cos 2x.\left( {\cos 2x – 1} \right) + \sqrt 3 \left( {\cos 2x – 1} \right) = 0\\
\Leftrightarrow \left( {\cos 2x – 1} \right)\left( {\cos 2x + \sqrt 3 } \right) = 0\\
\Leftrightarrow \left[ \begin{array}{l}
\cos 2x – 1 = 0\\
\cos 2x + \sqrt 3 = 0
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
\cos 2x = 1\\
\cos 2x = – \sqrt 3
\end{array} \right.\\
– 1 \le \cos 2x \le 1 \Rightarrow \cos 2x = 1\\
\Leftrightarrow 2x = k2\pi \\
\Leftrightarrow x = k\pi \,\,\,\,\,\left( {k \in Z} \right)\\
4,\\
{\tan ^2}2x + \left( {1 + \sqrt 3 } \right)\tan 2x + \sqrt 3 = 0\\
\Leftrightarrow {\tan ^2}2x + \tan 2x + \sqrt 3 .\tan 2x + \sqrt 3 = 0\\
\Leftrightarrow \left( {{{\tan }^2}2x + \tan 2x} \right) + \left( {\sqrt 3 .\tan 2x + \sqrt 3 } \right) = 0\\
\Leftrightarrow \tan 2x.\left( {\tan 2x + 1} \right) + \sqrt 3 .\left( {\tan 2x + 1} \right) = 0\\
\Leftrightarrow \left( {\tan 2x + 1} \right)\left( {\tan 2x + \sqrt 3 } \right) = 0\\
\Leftrightarrow \left[ \begin{array}{l}
\tan 2x + 1 = 0\\
\tan 2x + \sqrt 3 = 0
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
\tan 2x = – 1\\
\tan 2x = – \sqrt 3
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
2x = – \dfrac{\pi }{4} + k\pi \\
2x = – \dfrac{\pi }{3} + k\pi
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
x = – \dfrac{\pi }{8} + \dfrac{{k\pi }}{2}\\
x = – \dfrac{\pi }{6} + \dfrac{{k\pi }}{2}
\end{array} \right.\,\,\,\,\,\,\left( {k \in Z} \right)\\
5,\\
\cos 2x – 3\cos x + 2 = 0\\
\Leftrightarrow \left( {2{{\cos }^2}x – 1} \right) – 3\cos x + 2 = 0\\
\Leftrightarrow 2{\cos ^2}x – 1 – 3\cos x + 2 = 0\\
\Leftrightarrow 2{\cos ^2}x – 3\cos x + 1 = 0\\
\Leftrightarrow \left( {\cos x – 1} \right)\left( {2\cos x – 1} \right) = 0\\
\Leftrightarrow \left[ \begin{array}{l}
\cos x – 1 = 0\\
2\cos x – 1 = 0
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
\cos x = 1\\
\cos x = \dfrac{1}{2}
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
x = k2\pi \\
x = \pm \dfrac{\pi }{3} + k2\pi
\end{array} \right.\,\,\,\,\left( {k \in Z} \right)
\end{array}\)