函数$f(x)=dfrac{3+5sin x}{sqrt{5+4cos x+3sin x}}$的值域是____
分析:注意到$f(x)=sqrt{10}dfrac{5sin x+3}{sqrt{(5sin x+3)^2+(5cos x+4)^2}}$
令$m=5sin x+3,n=5cos x+4$则$m^2+n^2=25$
故由几何意义$sqrt{10}dfrac{m}{sqrt{m^2+n^2}}=(-dfrac{4sqrt{10}}{5},sqrt{10}]$
练习:
函数$y=dfrac{|(cos alpha+sqrt{2}sinalpha)t-sqrt{2}|}{sqrt{t^2-2sqrt{2}tcosalpha +2}},(tin R,alphain(0,dfrac{pi}{2}))$的最大值是_____
分析:$y=dfrac{|(cos alpha-sqrt{2})+sqrt{2}tsinalpha|}{(cos alpha-sqrt{2})^2+(tsinalpha)^2}$
令$m=cos alpha-sqrt{2},n=tsinalpha$ 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.$a=(m,n),b=(1,sqrt{2})$