Giải phương trình: \(\sin 4x - \cos 4x = 1 + 4\sqrt 2 \sin \left( {x - \frac{\pi }{4}} \right)\)
\(\sin 4x - c{\rm{os}}4x = 1 + 4\sqrt 2 \sin \left( {x - \frac{\pi }{4}} \right)\)
\( \Leftrightarrow 0 = c{\rm{os}}4x + \left( {1 - \sin 4x} \right) + 4\left( {{\mathop{\rm s}\nolimits} {\rm{inx}} - \cos x} \right)\)
\( \Leftrightarrow 0 = \left( {c{\rm{o}}{{\rm{s}}^2}2x - {{\sin }^2}2x} \right) + {\left( {c{\rm{os}}2x - \sin 2x} \right)^2} - 4\left( {\cos x - {\mathop{\rm s}\nolimits} {\rm{inx}}} \right)\)
\( \Leftrightarrow 0 = 2\cos 2x\left( {c{\rm{os}}2x - \sin 2x} \right) - 4\left( {\cos x - {\mathop{\rm s}\nolimits} {\rm{inx}}} \right)\)
\( \Leftrightarrow 0 = 2\left( {\cos x - {\mathop{\rm s}\nolimits} {\rm{inx}}} \right)\left( {\cos x + {\mathop{\rm s}\nolimits} {\rm{inx}}} \right)\left( {\cos 2x - {\mathop{\rm s}\nolimits} {\rm{in2x}}} \right) - 4\left( {\cos x - {\mathop{\rm s}\nolimits} {\rm{inx}}} \right)\)
   
\( \Leftrightarrow 2\sqrt 2 c{\rm{os}}\left( {x + \frac{\pi }{4}} \right)\left[ {c{\rm{os}}\left( {x + \frac{\pi }{2}} \right) + c{\rm{os}}3x - 2} \right] = 0\)
$a)$ \(c{\rm{os}}\left( {x + \frac{\pi }{4}} \right) = 0 \Leftrightarrow x = \frac{\pi }{4} + k\pi \)  (TM)
$b)$ \(\left\{ \begin{array}{l}
c{\rm{os}}\left( {x + \frac{\pi }{2}} \right) = 1\\
c{\rm{os}}3x = 1
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
x =  - \frac{\pi }{2} + 2k\pi \\
c{\rm{os}}\left( { - \frac{{3\pi }}{2} + 6k\pi } \right) = 1
\end{array} \right.\) (VN)

Đáp số: \(x = \frac{\pi }{4} + k\pi \)
cảm ơn bạn nhìu nhé –  nhonhungchat 28-06-12 12:14 AM

Bạn cần đăng nhập để có thể gửi đáp án

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