6:若函数f(x)=asin(x+3分之π)+sin(x-6分之π)是偶函数,则a的值为多少
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6:若函数f(x)=asin(x+3分之π)+sin(x-6分之π)是偶函数,则a的值为多少
答:
f(x)=asin(x+π/3)+sin(x-π/6)
f(-x)=asin(-x+π/3)+sin(-x-π/6)
=-asin(x-π/3)-sin(x+π/6)=f(x)
=asin(x+π/3)+sin(x-π/6)
所以:
-asinxcosπ/3+acosxsinπ/3-sinxcosπ/6-cosxsinπ/6
=asinxcosπ/3+acosxsinπ/3+sinxcosπ/6-cosxsinπ/6
所以:2asinxcosπ/3=-2sinxcosπ/6
对任意x都成立,则有:
acosπ/3=-cosπ/6
a=-√3
f(x)=asin(x+π/3)+sin(x-π/6)
f(-x)=asin(-x+π/3)+sin(-x-π/6)
=-asin(x-π/3)-sin(x+π/6)=f(x)
=asin(x+π/3)+sin(x-π/6)
所以:
-asinxcosπ/3+acosxsinπ/3-sinxcosπ/6-cosxsinπ/6
=asinxcosπ/3+acosxsinπ/3+sinxcosπ/6-cosxsinπ/6
所以:2asinxcosπ/3=-2sinxcosπ/6
对任意x都成立,则有:
acosπ/3=-cosπ/6
a=-√3
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