Question

упростите выражение.
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Answer 1

$\frac{Sin\alpha+Sin2\alpha+Sin3\alpha}{Cos\alpha+Cos2\alpha+Cos3\alpha}=\frac{(Sin\alpha+Sin3\alpha)+Sin2\alpha}{(Cos\alpha+Cos3\alpha)+Cos2\alpha}=\frac{2Sin\frac{\alpha+3\alpha}{2}Cos\frac{\alpha-3\alpha}{2}+Sin2\alpha}{2Cos\frac{\alpha+3\alpha}{2}Cos\frac{\alpha-3\alpha}{2}+Cos2\alpha}=\frac{2Sin2\alpha Cos\alpha+Sin2\alpha}{2Cos2\alpha Cos\alpha+Cos2\alpha}=\\=\frac{Sin2\alpha(2Cos\alpha+1)}{Cos2\alpha(2Cos\alpha+1)}=\frac{Sin2\alpha}{Cos2\alpha}=tg2\alpha$