已知tanx=6,那么0.5sin2x+(1/3)cos2x=?如题

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已知tanx=6,那么0.5sin2x+(1/3)cos2x=?如题已知tanx=6,那么0.5sin2x+(1/3)cos2x=?如题已知tanx=6,那么0.5sin2x+(1/3)cos2x=?

已知tanx=6,那么0.5sin2x+(1/3)cos2x=?如题
已知tanx=6,那么0.5sin2x+(1/3)cos2x=?
如题

已知tanx=6,那么0.5sin2x+(1/3)cos2x=?如题
sinx/cosx=tanx=6
sinx=6cosa
代入(sinx)^2+(cosx)^2=1
(cosx)^2=1/37
(sinx)^2=36/37
sinx/cosx=6>0
所以sinxcosx>0
所以sinxcosx=根号[(sinx)^2*(cosx)^2]=6/37
所以0.5sin2x+(1/3)cos2x
=0.5*2sinxcosx+(1/3)[(cosx)^2-(sinx)^2]
=6/37-35/111
=-17/111

sin(2x) = 2sinxcosx/[(sinx)^2 + (cosx)^2] = 2tanx/[(tanx)^2 + 1]
= 2*6/[36 + 1] = 12/37
cos(2x) = 1 - 2(sinx)^2/[(sinx)^2 + (cosx)^2] = 1 - 2(tanx)^2/[(tanx)^2 + 1]
= 1 - 2*36/[36 + 1] = 1 - 72/37 = -35/37
0.5sin2x + (1/3)cos2x = (1/2)(12/37) + (1/3)(-35/37)
= (18 - 35)/111 = - 17/111