求y=sin^2(x)+3sin(x)cos(x)-2cos^2(x)的最大值

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求y=sin^2(x)+3sin(x)cos(x)-2cos^2(x)的最大值
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求y=sin^2(x)+3sin(x)cos(x)-2cos^2(x)的最大值
求y=sin^2(x)+3sin(x)cos(x)-2cos^2(x)的最大值

求y=sin^2(x)+3sin(x)cos(x)-2cos^2(x)的最大值
y==sin^2(x)+3sin(x)cos(x)-2cos^2(x)
=cos^2(x)+sin^2(x)+3sin(x)cos(x)-3cos^2(x)
=1+3sin(x)cos(x)-3cos^2(x)
=1+3/2sin(2x)-3/2[cos(2x)+1]
=3/2sin(2x)-3/2cos(2x)-1/2
=3/根号2[sin(2x)cosπ/4-cos(2x)sinπ/4]-1/2
=3根号2/2 sin(2x-π/4) -1/2
sin(2x-π/4)的最大值为1
∴y=3根号2/2 sin(2x-π/4) -1/2的最大值:
3根号2/2 - 1/2