Toán Lớp 11: 1. cos 2x 5 sin.x+2=0
2. cos2x –3.cosx =1
3. 2 sinx-cos²x-2=0
4 cos2x-4 sin²x-1=0
20 điểm ạ,Ai giải nhanh em cho 5 sao và cảm ơn ạ,đang gấp lắm ạ
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Toán Lớp 11: 1. cos 2x 5 sin.x+2=0
2. cos2x –3.cosx =1
3. 2 sinx-cos²x-2=0
4 cos2x-4 sin²x-1=0
20 điểm ạ,Ai giải nhanh em cho 5 sao và cảm ơn ạ,đang gấp lắm ạ
Comments ( 1 )
1,\\
\left[ \begin{array}{l}
x = – \dfrac{\pi }{6} + k2\pi \\
x = \dfrac{{7\pi }}{6} + k2\pi
\end{array} \right.\,\,\,\left( {k \in Z} \right)\\
2,\\
\left[ \begin{array}{l}
x = \dfrac{{2\pi }}{3} + k2\pi \\
x = – \dfrac{{2\pi }}{3} + k2\pi
\end{array} \right.\,\,\,\,\left( {k \in Z} \right)\\
3,\\
x = \dfrac{\pi }{2} + k2\pi \,\,\,\left( {k \in Z} \right)\\
4,\\
x = k\pi \,\,\,\,\,\left( {k \in Z} \right)
\end{array}\)
1,\\
\cos 2x + 5\sin + 2 = 0\\
\Leftrightarrow \left( {1 – 2{{\sin }^2}x} \right) + 5\sin x + 2 = 0\\
\Leftrightarrow – 2{\sin ^2}x + 5\sin x + 3 = 0\\
\Leftrightarrow 2{\sin ^2}x – 5\sin x – 3 = 0\\
\Leftrightarrow \left( {2{{\sin }^2}x – 6\sin x} \right) + \left( {\sin x – 3} \right) = 0\\
\Leftrightarrow 2\sin x.\left( {\sin x – 3} \right) + \left( {\sin x – 3} \right) = 0\\
\Leftrightarrow \left( {\sin x – 3} \right)\left( {2\sin x + 1} \right) = 0\\
\Leftrightarrow \left[ \begin{array}{l}
\sin x – 3 = 0\\
2\sin x + 1 = 0
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
\sin x = 3\\
\sin x = – \dfrac{1}{2}
\end{array} \right.\\
– 1 \le \sin x \le 1 \Rightarrow \sin x = – \dfrac{1}{2}\\
\sin x = – \dfrac{1}{2} \Leftrightarrow \left[ \begin{array}{l}
x = – \dfrac{\pi }{6} + k2\pi \\
x = \pi – \left( { – \dfrac{\pi }{6}} \right) + k2\pi
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
x = – \dfrac{\pi }{6} + k2\pi \\
x = \dfrac{{7\pi }}{6} + k2\pi
\end{array} \right.\,\,\,\left( {k \in Z} \right)\\
2,\\
\cos 2x – 3\cos x = 1\\
\Leftrightarrow \left( {2{{\cos }^2}x – 1} \right) – 3\cos x – 1 = 0\\
\Leftrightarrow 2{\cos ^2}x – 3\cos x – 2 = 0\\
\Leftrightarrow \left( {2{{\cos }^2}x – 4\cos x} \right) + \left( {\cos x – 2} \right) = 0\\
\Leftrightarrow 2\cos x.\left( {\cos x – 2} \right) + \left( {\cos x – 2} \right) = 0\\
\Leftrightarrow \left( {\cos x – 2} \right)\left( {2\cos x + 1} \right) = 0\\
\Leftrightarrow \left[ \begin{array}{l}
\cos x – 2 = 0\\
2\cos x + 1 = 0
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
\cos x = 2\\
\cos x = – \dfrac{1}{2}
\end{array} \right.\\
– 1 \le \cos x \le 1 \Rightarrow \cos x = – \dfrac{1}{2}\\
\cos x = – \dfrac{1}{2} \Leftrightarrow \left[ \begin{array}{l}
x = \dfrac{{2\pi }}{3} + k2\pi \\
x = – \dfrac{{2\pi }}{3} + k2\pi
\end{array} \right.\,\,\,\,\left( {k \in Z} \right)\\
3,\\
2\sin x – {\cos ^2}x – 2 = 0\\
\Leftrightarrow 2\sin x – \left( {1 – {{\sin }^2}x} \right) – 2 = 0\\
\Leftrightarrow 2\sin x – 1 + {\sin ^2}x – 2 = 0\\
\Leftrightarrow {\sin ^2}x + 2\sin x – 3 = 0\\
\Leftrightarrow \left( {{{\sin }^2}x – \sin x} \right) + \left( {3\sin x – 3} \right) = 0\\
\Leftrightarrow \sin x\left( {\sin x – 1} \right) + 3.\left( {\sin x – 1} \right) = 0\\
\Leftrightarrow \left( {\sin x – 1} \right)\left( {\sin x + 3} \right) = 0\\
\Leftrightarrow \left[ \begin{array}{l}
\sin x – 1 = 0\\
\sin x + 3 = 0
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
\sin x = 1\\
\sin x = – 3
\end{array} \right.\\
– 1 \le \sin x \le 1 \Rightarrow \sin x = 1 \Leftrightarrow x = \dfrac{\pi }{2} + k2\pi \,\,\,\left( {k \in Z} \right)\\
4,\\
\cos 2x – 4{\sin ^2}x – 1 = 0\\
\Leftrightarrow \left( {1 – 2{{\sin }^2}x} \right) – 4{\sin ^2}x – 1 = 0\\
\Leftrightarrow 1 – 2{\sin ^2}x – 4{\sin ^2}x – 1 = 0\\
\Leftrightarrow – 6{\sin ^2}x = 0\\
\Leftrightarrow {\sin ^2}x = 0\\
\Leftrightarrow \sin x = 0\\
\Leftrightarrow x = k\pi \,\,\,\,\,\left( {k \in Z} \right)
\end{array}\)