1613

在 ${\displaystyle \mathrm{\Delta }ABC}$ 中, ${\displaystyle \cos A\cdot (3\sin B+4\sin C)}$ 的最小值为

最小值下, ${\displaystyle \cos A<0}$. 则根据辅助角公式和均值不等式 $$\begin{array}{rl}\text{原式}& =\cos A(3\sin B+4(\sin B\cos A+\sin A\cos B))\\ & =\cos A(\sin B(3+4\cos A)+\cos B(4\sin A))\\ & \ge \cos A\sqrt{(3+4\cos A{)}^2+(4\sin A{)}^2}\\ & =-\frac{1}{12}\sqrt{(-12\cos A)(-12\cos A)(25+24\cos A)}\\ & \ge -\frac{1}{12}\sqrt{{\left(\frac{-12\cos A-12\cos A+25+24\cos A}{3}\right)}^3}\\ & =-\frac{125\sqrt{3}}{108}.\end{array}$$