题目内容
已知O为原点,向量



(1)求证:(



(2)求tan∠AOB的最大值及相应x值.
【答案】分析:(1)先求出
,再由(
)•
=0×2+2sinx×0=0可证.
(2)由tan∠AOB=tan(∠AOC-∠BOC),根据两角差的正切公式可得答案.
解答:解:(1)∵0<x<
,∴3sinx>sinx,∴
又
∴(
)•
=0×2+2sinx×0=0
∴(
)⊥
.
(2)tan∠AOC=
,tan
∵
,∴
⊥
,0<
.
∴tan∠AOB=tan(∠AOC-∠BOC)
=
=
=
(当tanx=
即x=
时取“=”)
所以tan∠AOB的最大值为
,相应的x=
点评:本题主要考查向量垂直和点乘之间的关系以及三角的正切求值问题.



(2)由tan∠AOB=tan(∠AOC-∠BOC),根据两角差的正切公式可得答案.
解答:解:(1)∵0<x<


又

∴(


∴(


(2)tan∠AOC=


∵




∴tan∠AOB=tan(∠AOC-∠BOC)
=


=

(当tanx=


所以tan∠AOB的最大值为


点评:本题主要考查向量垂直和点乘之间的关系以及三角的正切求值问题.

练习册系列答案
相关题目