三角函数求解：若tanx=1/2,求sin^2x+2sinxsin(π/2-x)+3sin^2(3π/2-x)

三角函数求解：若tanx=1/2,求sin^2x+2sinxsin(π/2-x)+3sin^2(3π/2-x)
三角函数求解：若tanx=1/2,求sin^2x+2sinxsin(π/2-x)+3sin^2(3π/2-x)

三角函数求解：若tanx=1/2,求sin^2x+2sinxsin(π/2-x)+3sin^2(3π/2-x)
tanx = 1/2 = sinx / cosx = sinx / sqrt(1-sin^2x)
sin^2 x / (1-sin^2x) = 1/4
sin x = sqrt(1/5)
cos x = sqrt(4/5)
sin^2x+2sinxsin(π/2-x)+3sin^2(3π/2-x) = 1/5 + 2*2/5 + 3*4/5 = 3.4

因为tanx=1/2 即sinx/cosx=1/2
所以cosx=2sinx
由 sin^2x+cos^2x =1
sin^2x+(2sinx)^2x=1
解得cosx=2√5/5 sinx=√5/5
sin^2x+2sinxsin(π/2-x)+3sin^2(3π/2-x)
=sin^2x+2sinxcosx+3cos^2x
=(√5/5)^2+2(√5/5)(2√5/5)+3(2√5/5 )^2
=17/5

tanx=1/2，sinx/cosx=1/2→sinx=cosx/2①.
sin^2x+cos^2x=1②。联立两式得:cos^2x=4/5
sin^2x+2sinxsin(π/2-x)+3sin^2(3π/2-x)=(1-cos2x)/2+2sinxcosx+3【1-cos2（3π/2-x）】/2
=1/2-cos2x/2+cos^2x+3/2+3cos2x/2
=2+cos^2x+cos2x=2+cos^2x+2cos^2x-1=3cos^2x+1=3*4/5+1=3.4