giải phương trinh lương giac sau:
cos^5(x)+sin^5(x)+cos(2x)+sin(2x)=1+căn2
Ta có:
[TEX]\sin ^5 x \le \sin ^2 x[/TEX]
[TEX]c{\rm{os}}^{\rm{5}} x \le c{\rm{os}}^{\rm{2}} x[/TEX]
[TEX]\sin 2x + c{\rm{os2x = }}\sqrt {\rm{2}} \sin \left( {2x + \frac{\pi }{4}} \right) \le \sqrt 2 [/TEX]
[TEX]\sin ^5 x + c{\rm{os}}^{\rm{5}} x + \sin 2x + c{\rm{os2x}} \le {\rm{1 + }}\sqrt 2 [/TEX]
[TEX]\sin ^5 x + c{\rm{os}}^{\rm{5}} x + \sin 2x + c{\rm{os2x = 1 + }}\sqrt 2 \Leftrightarrow \left\{ \begin{array}{l}sin ^5 x = \sin ^2 x \\ c{\rm{os}}^{\rm{5}} x = c{\rm{os}}^{\rm{2}} x \\ {\mathop{\rm s}\nolimits} {\rm{in2x + cos2x = }}\sqrt {\rm{2}} \\ \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}\left[\begin{array}{l}{\mathop{\rm s}\nolimits} i{\rm{n = 0}} \\ {\rm{sinx = 1}} \\ \end{array} \right. \\ \left[ \begin{array}{l} c{\rm{osx = 0}} \\ {\rm{cosx = 1}} \\
\end{array} \right. \\ 2\sin x.c{\rm{osx + cos}}^{\rm{2}} x - \sin ^2 x = \sqrt {\rm{2}} \\ \end{array} \right.[/TEX]
Vô nghiệm
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