Question

高中数学题：设sinθ+cosθ=√2/3,π/2<θ<π,求sinθ^3+cosθ^3及tanθ-cotθ

为什么tanθ-cotθ的符号是负的

Answer 1

π/2<θ<π,说明sinθ>0,cosθ<0,
sinθ+cosθ=√2/3,
(sinθ+cosθ)^2=2/9,
又sinθ^2+cosθ^2=1,
(sinθ-cosθ)^2=16/9,
因为sinθ>0,cosθ<0,
所以sinθ-cosθ>0,
sinθ-cosθ=4/3,
又sinθ+cosθ=√2/3,
解得sinθ=(√2+4)/6,cosθ=(√2-4)/6,
sinθ^3+cosθ^3=(25√2)/54,
tanθ=sinθ/cosθ=-(4√2+9)/7,cotθ=1/tanθ=(4√2-9)/7,
tanθ-cotθ=-(8√2)/7..

由于π/2<θ<π,又sinθ+cosθ=√2/3,
π/2<θ<3π/4,
因此tanθ-cotθ<0..