若3sinθ-cosθ=0,则（cos^2）θ+1/2sin2θ是?

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若3sinθ-cosθ=0,则（cos^2）θ+1/2sin2θ是?
若3sinθ-cosθ=0,则（cos^2）θ+1/2sin2θ是?

若3sinθ-cosθ=0,则（cos^2）θ+1/2sin2θ是?
3sinθ-cosθ=0
tanθ=1/3
(cosθ)^2+(1/2)sin(2θ)=(1/2)[cos(2θ)+1]+(1/2)sin(2θ)
=(1/2)+(1/2){[1-(tanθ)^2]/[1+(tanθ)^2]+2tanθ/[ 1+(tanθ)^2]}
=(1/2)+(1/2){[1-(tanθ)^2+2tanθ]/[ 1+(tanθ)^2]}
=(1/2){[1+(tanθ)^2+1-(tanθ)^2+2tanθ]/[ 1+(tanθ)^2]}
=[1+tanθ]/[ 1+(tanθ)^2]
=[1+1/3]/[ 1+1/9]
=6/5

3sinθ-cosθ=0,
两边平方 1-6sinθcosθ=0
3sin2θ=1 sin2θ=1/3
（cos^2）θ+1/2sin2θ (代一个cosθ=3sinθ)
=3sincosθ+1/2sin2θ
=3/2sin2θ+1/2sin2θ
=2sin2θ
=2*(1/3)
=2/3

cosθ=3sinθ
（cos^2）θ+（sin^2）θ=1
sinθ=根号10/10
cosθ=3根号10/10
sin2θ=2sinθ*cosθ=3/5
原式=9/10+3/10=6/5

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