θ∈﹙0,π﹚,且sinθ+cosθ=1/3,求sinθ,cosθ的值

Question

θ∈﹙0,π﹚,且sinθ+cosθ=1/3,求sinθ,cosθ的值

Answer 1

sinθ+cosθ=1/3,cosθ=1/3-sinθ
(sinθ)^2+(cosθ)^2=1
(sinθ)^2+(1/3-sinθ)^2=1
(sinθ)^2+1/9-2/3*sinθ+(sinθ)^2=1
2(sinθ)^2-2/3*sinθ-8/9=0
(sinθ)^2-1/3*(sinθ)-4/9=0
(sinθ-1/6)^2-1/36-4/9=0
(sinθ-1/6)^2=17/36 θ∈﹙0,π﹚,sinθ>0
sinθ=1/6+（根号17）/6=(1+根号17）/6,或sinθ=1/6-（根号17）/6=(1-根号17）/6