Đáp án:
\[a + b = 7\]
Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
E = \dfrac{{\sin 2x + 2\sin 3x + \sin 4x}}{{\sin 3x + 2\sin 4x + \sin 5x}}\\
= \dfrac{{\left( {\sin 4x + \sin 2x} \right) + 2\sin 3x}}{{\left( {\sin 5x + \sin 3x} \right) + 2\sin 4x}}\\
= \dfrac{{2.\sin \dfrac{{4x + 2x}}{2}.\cos \dfrac{{4x - 2x}}{2} + 2\sin 3x}}{{2.\sin \dfrac{{5x + 3x}}{2}.\cos \dfrac{{5x - 3x}}{2} + 2\sin 4x}}\\
= \dfrac{{2\sin 3x.\cos x + 2\sin 3x}}{{2\sin 4x.\cos x + 2\sin 4x}}\\
= \dfrac{{\sin 3x.\left( {2\cos x + 2} \right)}}{{\sin 4x\left( {2\cos x + 2} \right)}}\\
= \dfrac{{\sin 3x}}{{\sin 4x}}\\
\Rightarrow a = 3;\,\,\,\,b = 4\\
\Rightarrow a + b = 7
\end{array}\)
Vậy \(a + b = 7\)
