Question

tanθ=3,求根号（sin的四次方θ-3sin θcos的三次方θ+cos的四次方θ）的值.

Answer 1

解：由题知tanθ
=3=sinθ/cosθ
=3
所以sinθ
=3cos θ (
sin θ
)^2=9(cosθ
)^2
又已知(sin
θ)^2+(
cos θ
)^2=1……（一）
所以9(cosθ
)^2+(cosθ
)^2=1
(cosθ
)^2=1/10 (sin θ
)^2=9(cos θ
)^2=9/10
将（一）式两边平方得
((sinθ
)^2+(cos θ)^2)=1^2
(sin θ )^4+2(sin θ
)^2*(cos θ
)^2+(cosθ
)^4=1
(sinθ
)^4+(cos θ
)^4=1-2(sin θ
)^2*(cosθ
)^2
所以原式= (sinθ
)^4+(cos θ
)^4-sin
*3cosθ
*(cosθ
)^2

=1-2(sin θ)^2*(cos θ
)^2-(sin θ
)^2*(cos θ
)^2

=1--3(sinθ
)^2*(cos θ
)^2

=1-3*9/10*1/10

=73/100