已知函数f(x)=sin(x+π/6)+sin(x - 6π )+cosx+a的最大值为一求:(1)常数a的值 (2)求使f(x)≥0成立的x的取值集合

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已知函数f(x)=sin(x+π/6)+sin(x - 6π )+cosx+a的最大值为一求:(1)常数a的值 (2)求使f(x)≥0成立的x的取值集合
已知函数f(x)=sin(x+π/6)+sin(x - 6π )+cosx+a的最大值为一求:
(1)常数a的值 (2)求使f(x)≥0成立的x的取值集合

已知函数f(x)=sin(x+π/6)+sin(x - 6π )+cosx+a的最大值为一求:(1)常数a的值 (2)求使f(x)≥0成立的x的取值集合
f(x)=sin(x+π/6)+sin(x-6π)+cos(x)+a =sin(x+π/6)+sin(x)cos(π/6)-cos(x)sin(π/6)+cos(x)+a =sin(x+π/6)+sin(x)cos(π/6)+cos(x)sin(π/6)+a =2sin(x+π/6)+a 当sin(x+π/6)=1时,f(x)的最大值为2+a=1,可得a=-1 得f(x)=2sin(x+π/6)-1,由f(x)≥0得sin(x+π/6)≥1/2 可得2kπ+π/6≤x+π/6≤2kπ+5π/6 即2kπ≤x≤2kπ+2π/3 即使f(x)≥0成立的x的取值集合为[2kπ,2kπ+2π/3],(k∈Z)