已知f(x)=sin^4x+cos^4x+2sin^3xcosx-sinxcosx-3/4,求f(x)的最小正周期

来源:学生作业帮助网 编辑:六六作业网 时间:2024/12/26 03:38:08
已知f(x)=sin^4x+cos^4x+2sin^3xcosx-sinxcosx-3/4,求f(x)的最小正周期已知f(x)=sin^4x+cos^4x+2sin^3xcosx-sinxcosx-3

已知f(x)=sin^4x+cos^4x+2sin^3xcosx-sinxcosx-3/4,求f(x)的最小正周期
已知f(x)=sin^4x+cos^4x+2sin^3xcosx-sinxcosx-3/4,求f(x)的最小正周期

已知f(x)=sin^4x+cos^4x+2sin^3xcosx-sinxcosx-3/4,求f(x)的最小正周期

具体的解答过程在下面的图上

f(x)=sin^4x+cos^4x+2sin^3xcosx-sinxcosx-3/4
=(cos^2x+sin^2x)^2-2sin^2xcos^2x+2sin^3xcosx-sinxcosx-3/4
=sinxcosx(2sin^2-2sinxcosx-1)+1/4
=-1/2sin2x[2sinxcosx+2cos^2x-1)+1/4
=-1/2sin2x(s...

全部展开

f(x)=sin^4x+cos^4x+2sin^3xcosx-sinxcosx-3/4
=(cos^2x+sin^2x)^2-2sin^2xcos^2x+2sin^3xcosx-sinxcosx-3/4
=sinxcosx(2sin^2-2sinxcosx-1)+1/4
=-1/2sin2x[2sinxcosx+2cos^2x-1)+1/4
=-1/2sin2x(sin2x+cos2x)+1/4
=-1/2[1/2(cos4x-sin4x+1)]+1/4
=-1/4(cos4x-sin4x)+1/2
=-√2/4cos(4x+∏/4)+1/4
4x=2∏,x=∏/2
f(x)的最小正周期∏/2

收起