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Sin³x-3sinxcos²x+2cos³x=0
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Toán Lớp 11: Giúp em với ạ ????????
Em đang cần gấp lắm ????????????
Sin³x-3sinxcos²x+2cos³x=0
Comments ( 1 )
x = \dfrac{\pi }{4} + k\pi \\
x = \arctan \left( { – 2} \right) + k\pi
\end{array} \right.\,\,\,\,\,\left( {k \in Z} \right)\)
{\sin ^3}x – 3\sin x.{\cos ^2}x + 2{\cos ^3}x = 0\,\,\,\,\,\,\left( 1 \right)\\
\Leftrightarrow \left( {{{\sin }^3}x – \sin x.{{\cos }^2}x} \right) + \left( { – 2\sin x.{{\cos }^2}x + 2{{\cos }^3}x} \right) = 0\\
\Leftrightarrow \sin x.\left( {{{\sin }^2}x – {{\cos }^2}x} \right) – 2{\cos ^2}x.\left( {\sin x – \cos x} \right) = 0\\
\Leftrightarrow \sin x.\left( {\sin x – \cos x} \right)\left( {\sin x + \cos x} \right) – 2{\cos ^2}x.\left( {\sin x – \cos x} \right) = 0\\
\Leftrightarrow \left( {\sin x – \cos x} \right).\left[ {\sin x.\left( {\sin x + \cos x} \right) – 2{{\cos }^2}x} \right] = 0\\
\Leftrightarrow \left( {\sin x – \cos x} \right).\left( {{{\sin }^2}x + \sin x.\cos x – 2{{\cos }^2}x} \right) = 0\\
\Leftrightarrow \left( {\sin x – \cos x} \right).\left[ {\left( {{{\sin }^2}x – \sin x.\cos x} \right) + \left( {2\sin x.\cos x – 2{{\cos }^2}x} \right)} \right] = 0\\
\Leftrightarrow \left( {\sin x – \cos x} \right).\left[ {\sin x.\left( {\sin x – \cos x} \right) + 2\cos x.\left( {\sin x – \cos x} \right)} \right] = 0\\
\Leftrightarrow \left( {\sin x – \cos x} \right).\left( {\sin x – \cos x} \right).\left( {\sin x + 2\cos x} \right) = 0\\
\Leftrightarrow \left[ \begin{array}{l}
\sin x – \cos x = 0\\
\sin x + 2\cos x = 0
\end{array} \right.\,\,\,\,\,\left( * \right)\\
TH1:\,\,\,\,\cos x = 0\\
\left( 1 \right) \Leftrightarrow {\sin ^3}x = 0 \Leftrightarrow \sin x = 0\\
\Rightarrow {\sin ^2}x + {\cos ^2}x = 0\,\,\,\,\,\left( L \right)\\
TH2:\,\,\,\cos x \ne 0\\
\left( * \right) \Leftrightarrow \left[ \begin{array}{l}
\dfrac{{\sin x – \cos x}}{{\cos x}} = 0\\
\dfrac{{\sin x + 2\cos x}}{{\cos x}} = 0
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
\dfrac{{\sin x}}{{\cos x}} – 1 = 0\\
\dfrac{{\sin x}}{{\cos x}} + 2 = 0
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
\tan x = 1\\
\tan x = – 2
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
x = \dfrac{\pi }{4} + k\pi \\
x = \arctan \left( { – 2} \right) + k\pi
\end{array} \right.\,\,\,\,\,\left( {k \in Z} \right)
\end{array}\)