Đáp án:
\[m + n = 10\]
Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
\sin 2x + \sin 4x + \sin 6x\\
= \left( {\sin 2x + \sin 6x} \right) + \sin 4x\\
= 2.\sin \dfrac{{2x + 6x}}{2}.\cos \dfrac{{2x - 6x}}{2} + \sin 4x\\
= 2.\sin 4x.\cos \left( { - 2x} \right) + \sin 4x\\
= 2\sin 4x.\cos 2x + \sin 4x\\
= 2\sin 4x.\left( {\cos 2x + \dfrac{1}{2}} \right)\\
= 2\sin 4x.\left( {\cos 2x + \cos \dfrac{\pi }{3}} \right)\\
= 2.sin4x.2.\cos \dfrac{{2x + \dfrac{\pi }{3}}}{2}.\cos \dfrac{{2x - \dfrac{\pi }{3}}}{2}\\
= 4\sin 4x.\cos \left( {x + \dfrac{\pi }{6}} \right).\cos \left( {x - \dfrac{\pi }{6}} \right)\\
\Rightarrow m = 4;\,\,\,n = 6\\
\Rightarrow m + n = 10
\end{array}\)
Vậy \(m + n = 10\)
