Question

已知sinα+sinβ=1/4,cosα+cos1/3,求tan(α+β)与sin(α+β)的值

Answer 1

已知sinα+sinβ=1/4,cosα+cosβ=1/3,求tan(α+β)与sin(α+β)的值
令θ=(α+β)/2,γ=(α-β)/2
sinα+sinβ=1/4,则sin(θ+γ)+sin(θ-γ)=1/4
sinθcosγ+cosθsinγ+sinθcosγ-cosθsinγ=1/4
sinθcosγ=1/8 (1式)
cosα+cosβ=1/3,则cos(θ+γ)+cos(θ-γ)=1/3
cosθcosγ+sinθsinγ+cosθcosγ-sinθsinγ=1/3
cosθcosγ=1/6 (2式)
(1式)除以(2式)=tanθ=6/8=3/4
tan(α+β)=tan(2θ)=2tanθ/(1-tanθ^2)=24/7
sin(α+β)=sin(2θ)=2tanθ/(1+tanθ^2)=2*3*4/25=24/25