已知向量m=(cosα-根号2/3,-1)向量n=(sinα,1),向量m与向量n为共线向量,且α∈[π/2,0]

题目

已知向量m=(cosα-根号2/3,-1)向量n=(sinα,1),向量m与向量n为共线向量,且α∈[π/2,0]
(1)求sinα+cosα的值
(2)求sin2α/sinα-cosα的值

答案

∵向量m∥向量n,∴(cosα-√2/3)*1-(-1)sinα=0.
(1)sinα+cosα=√2/3.
(2)(sin2α/sinα)-cosα=2sinαcosα/sinα-cosα.
= 2cosα-cosα
=cosα α∈[0,π/2]
由(1)^2,得：1+2sinαcosα=2/9.
2sinαcosα=-7/9.
2√(1-cos^2α)cosα=-7/9.
4(1-cos^2α)cos^2α=49/81.
4cos^4α-4cos^2α+49/81=0
4(cos^2α-1/2)^2-1+49/81=0.
4(cos^2α-1/2)^2=32/81.
(cos^2α-1/2)^2=8/81.
cos^2α=1/2±2√2/9.
cosα=±√[(9±4√2)/18] ∵α∈[0,π/2] ∴cosα＞0.
∴ (sin2α/sinα-cosαα=cosα=√[(9±4√2)/18].