f(x)=sin(x+π/6)-tanα*cosx,且f(π/3)=1/2(1)求tanα的值(2)当x∈[π/2,π],求函数f(x)的最小值

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f(x)=sin(x+π/6)-tanα*cosx,且f(π/3)=1/2(1)求tanα的值(2)当x∈[π/2,π],求函数f(x)的最小值

f(x)=sin(x+π/6)-tanα*cosx,且f(π/3)=1/2(1)求tanα的值(2)当x∈[π/2,π],求函数f(x)的最小值
f(x)=sin(x+π/6)-tanα*cosx,且f(π/3)=1/2
(1)求tanα的值
(2)当x∈[π/2,π],求函数f(x)的最小值

f(x)=sin(x+π/6)-tanα*cosx,且f(π/3)=1/2(1)求tanα的值(2)当x∈[π/2,π],求函数f(x)的最小值
(1)
∵f(π/3)=1/2
代入得:
sin(π/3+π/6)-tanα*cos(π/3)=1/2
1-tanα*(1/2)=1/2
tanα=1
(2)f(x)=sin(x+π/6)-cosx
f(x)=sinx*cos(π/6)+cosx*sin(π/6)-cosx
f(x)=sinx*cos(π/6)+cosx*(1/2)-cosx
f(x)=sinx*cos(π/6)-cosx*(1/2)
f(x)=sinx*cos(π/6)-cosx*sin(π/6)
f(x)=sin(x-π/6)
∵x∈[π/2,π]
∴x-π/6∈[π/3,5π/6]
由图像可知,当x-π/6=5π/6即x=π时
f(x)有最小值:f(π)=sin(π-π/6)=1/2

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