Question

若sinθ+cosθ=根号2求下列各式值 (1)sinθ的三次方+cosθ的三次方 (2)tanθ+cotθ

Answer 1

sinθ+cosθ=√2
两边平方得1+2sinθcosθ=2
sinθcosθ=1/2
(1)
sin³θ+cos³θ=(sinθ+cosθ)(sin²θ-sinθcosθ+cos²θ)=√2*(1-1/2)=√2/2
(2)
tanθ+cotθ=sinθ/cosθ+cosθ/sinθ=1/sinθcosθ=2

Answer 2

sinθ+cosθ=根号2

（sinQ^2+cosQ^2）^2=1
由这2个可解出 sinqcosq=1/2
sinq^3+cos^3=(sinq+cosq)(sinq^2-sinqcosq+cosq^2)=根号2（1-1/2)=(根号2)/2