求函数f(x)=(3+5sinx)/√(5+4cosx+3sinx)的值域

题目

求函数f(x)=(3+5sinx)/√(5+4cosx+3sinx)的值域

答案

关键是化成半角sinx=2sin(x/2)*cos(x/2) cosx=cos^2(x/2)-sin^2(x/2)
原式分子=3sin^2(x/2)+3cos^2(x/2)+10sin(x/2)cos(x/2)
=[sin(x/2)+3cos(x/2)][3sin(x/2)+cos(x/2)]
分母=5sin^2(x/2)+5cos^2(x/2)+4cos^2(x/2)-4sin^2(x/2)+6sin(x/2)*cos(x/2) 的开平方
=| 3cos(x/2)+sin(x/2) |
则原式=+- 10^0.5[cos(x/2 - a)]
过程自己完善一下