MT【279】分母为根式的两个函数

函数$f(x)=\dfrac{3+5\sin x}{\sqrt{5+4\cos x+3\sin x}}$的值域是____

分析:注意到$f(x)=\sqrt{10}\dfrac{5\sin x+3}{\sqrt{(5\sin x+3)^2+(5\cos x+4)^2}}$
令$m=5\sin x+3,n=5\cos x+4$则$m^2+n^2=25$

故由几何意义$\sqrt{10}\dfrac{m}{\sqrt{m^2+n^2}}=(-\dfrac{4\sqrt{10}}{5},\sqrt{10}]$

练习:
函数$y=\dfrac{|(\cos \alpha+\sqrt{2}\sin\alpha)t-\sqrt{2}|}{\sqrt{t^2-2\sqrt{2}t\cos\alpha +2}},(t\in R,\alpha\in(0,\dfrac{\pi}{2}))$的最大值是_____

分析:$y=\dfrac{|(\cos \alpha-\sqrt{2})+\sqrt{2}t\sin\alpha|}{(\cos \alpha-\sqrt{2})^2+(t\sin\alpha)^2}$
令$m=\cos \alpha-\sqrt{2},n=t\sin\alpha$ 则$y=\dfrac{|m+\sqrt{2}n|}{\sqrt{m^2+n^2}}$,

由几何意义,$(1,\sqrt{2})$到直线$mx+ny=0$距离最大为$\sqrt{3}$

注:也可以用向量求最大值.$a=(m,n),b=(1,\sqrt{2})$