若F(X)=ASIN(X+PI/4)+3SIN(X—PI/4)是偶函数,则A=
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若F(X)=ASIN(X+PI/4)+3SIN(X—PI/4)是偶函数,则A=
∵f(x)=Asin(x+π/4)+3sin(x-π/4)是偶函数
∴f(-x)=f(x)
即Asin(-x+π/4)+3sin(-x-π/4)=Asin(x+π/4)+3sin(x-π/4)
A[sin(-x+π/4)-sin(x+π/4)]=3[sin(x-π/4)-sin(-x-π/4)]
2Acosπ/4sin(-x)=6cos(-π/4)sinx
-2Acosπ/4sinx=6cos(π/4)sinx
∴-2A=6
∴A=-3
∴f(-x)=f(x)
即Asin(-x+π/4)+3sin(-x-π/4)=Asin(x+π/4)+3sin(x-π/4)
A[sin(-x+π/4)-sin(x+π/4)]=3[sin(x-π/4)-sin(-x-π/4)]
2Acosπ/4sin(-x)=6cos(-π/4)sinx
-2Acosπ/4sinx=6cos(π/4)sinx
∴-2A=6
∴A=-3
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