已知f(x)=5cos²x+sin²x-(4根号3)sinxcosx(1)化简f(x)的解析式,并求f(x)的最小正周期(2)当x∈[-π/6,π/4]时,求f(x)的值域
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![已知f(x)=5cos²x+sin²x-(4根号3)sinxcosx(1)化简f(x)的解析式,并求f(x)的最小正周期(2)当x∈[-π/6,π/4]时,求f(x)的值域](/uploads/image/z/3877574-14-4.jpg?t=%E5%B7%B2%E7%9F%A5f%28x%29%3D5cos%26%23178%3Bx%2Bsin%26%23178%3Bx-%EF%BC%884%E6%A0%B9%E5%8F%B73%EF%BC%89sinxcosx%281%29%E5%8C%96%E7%AE%80f%EF%BC%88x%EF%BC%89%E7%9A%84%E8%A7%A3%E6%9E%90%E5%BC%8F%2C%E5%B9%B6%E6%B1%82f%EF%BC%88x%EF%BC%89%E7%9A%84%E6%9C%80%E5%B0%8F%E6%AD%A3%E5%91%A8%E6%9C%9F%EF%BC%882%EF%BC%89%E5%BD%93x%E2%88%88%5B-%CF%80%2F6%2C%CF%80%2F4%5D%E6%97%B6%2C%E6%B1%82f%EF%BC%88x%EF%BC%89%E7%9A%84%E5%80%BC%E5%9F%9F)
已知f(x)=5cos²x+sin²x-(4根号3)sinxcosx(1)化简f(x)的解析式,并求f(x)的最小正周期(2)当x∈[-π/6,π/4]时,求f(x)的值域
已知f(x)=5cos²x+sin²x-(4根号3)sinxcosx
(1)化简f(x)的解析式,并求f(x)的最小正周期
(2)当x∈[-π/6,π/4]时,求f(x)的值域
已知f(x)=5cos²x+sin²x-(4根号3)sinxcosx(1)化简f(x)的解析式,并求f(x)的最小正周期(2)当x∈[-π/6,π/4]时,求f(x)的值域
f(x) = 5cos²x + sin²x-4√3sinxcosx
= 4cos²x + cos²x + sin²x - 2√3sin2x
= 2(cos2x+1) + 1 - 2√3sin2x
= -2√3sin2x+2cos2x+3
= -4(sin2xcosπ/6-cos2xsinπ/6) + 3
= -4sin(2x-π/6) + 3
最小正周期:2π/2 = π
x∈[-π/6,π/4]时
2x-π/6∈[-π/2,π/3]
sin(2x-π/6)∈[-1,√3/2]
-4sin(2x-π/6)∈[-2√3,4]
f(x)= -4sin(2x-π/6) + 3的值域[3-2√3,7]
1
f(x)=5*(1+cos2x)/2+(1-cos2x)/2-2√3sin2x
=2cos2x-2√3sin2x+3
=4[cos(2x)*(1/2)-sin(2x)*(√3/2)]+3
=4[cos(2x)cos(π/3)-sin(2x)sin(π/3)]+3
=4cos(2x+π/3)+3
(1) T=2π/2=π
(2) x∈[-π/6,π/4]
2x+π/3∈[0,5π/6]
cos(2x+π/3)∈[-√3/2,1]
所以 y∈[-2√3+3,7]