Question

已知sinθ+cosθ=根号2/3(π/2<θ<π),求tanθ的值

Answer 1

解析：
已知sinθ+cosθ=根号2/3，那么：
根号2*sin(θ+π/4)=根号2/3
即sin(θ+π/4)=1/3
又π/2<θ<π，则：3π/4<θ+π/4<5π/4
因为sin(θ+π/4)>0，所以可知3π/4<θ+π/4<π
则cos(θ+π/4)=-根号[1-sin²(θ+π/4)]=-2根号2/3
tan(θ+π/4)=sin(θ+π/4)/cos(θ+π/4)=-根号2/4
所以：tanθ=tan[(θ+π/4)-π/4]
=[tan(θ+π/4) - tan(π/4)]/[1+tan(θ+π/4)tan(π/4)]
=(-根号2/4 -1)/(1-根号2/4)
=(4+根号2)/(4-根号2)
=(4+根号2)²/[(4-根号2)(4+根号2)]
=(18+8根号2)/14
=(9+4根号2)/7

Answer 2

解：tanθ为负值 cosθ为负值
因为sinθ+cosθ=根号2/3 sinθ^2+cosθ^2=1
所以sinθ=1+2*根号14/9
cosθ=-5/2*(1+2*根号14)
所以tanθ=-2*（57+4*根号14）/45
谢谢