Question

已知函数f(x)=2√3sinx-2cosx(2)若f(x)=0,求2cos^2-sinx-1/√2sin(x+π/4)的值

Answer 1

f(x)=2√3sinx-2cosx
若f(x)=0, 即 2√3sinx-2cosx=0
∴√3sinx-cosx=0
∴tanx=√3/3
∴x=2kπ+π/6,或x=2kπ+7π/6，k∈Z
当x=2kπ+π/6,k∈Z,时
cosx=√3/2,sinx=1/2
(2cos²x-sinx-1)/√2sin(x+π/4)
=(2cos²x-sinx-1)/(sinx+cosx)
=(3/2-1/2-1)/(sinx+cosx)
=0
当x=2kπ+7π/6,k∈Z,时
cosx=-√3/2,sinx=-1/2
(2cos²x-sinx-1)/√2sin(x+π/4)
=(2cos²x-sinx-1)/(sinx+cosx)
=(3/2+1/2-1)/(-1/2-√3/2)
=-4/(√3+1)
=-2(√3-1)
=2-2√3