题目内容
如图,设P是单位圆和x轴正半轴的交点,M、N是单位圆上的两点,O是坐标原点,

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【答案】分析:根据M、N是单位圆上的两点,
,∠PON=α,以及三角函数的定义写出点N,M的坐标,求出
,并代入
,利用三角恒等变形,化简为sin(α+
),要求
的范围,只需求sin(α+
),在区间[0,π]上的最值即可.
解答:解:∵M、N是单位圆上的两点,
,∠PON=α,
∴M(
,
),N(cosα,sinα),
∴
=(
,
),
=(cosα,sinα),
∴
=
=
∵α∈[0,π],
∴α+
∈[
,
],
∴sin(α+
)∈
.
∴f(α)∈[1,2]
故答案为[1,2].
点评:此题是个中档题.考查向量在几何中的应用,同时考查了三角函数的定义和向量的数量积的坐标运算,三角函数在区间上的最值问题,体现了转化的思想,考查了同学们观察、推理以及创造性地分析问题、解决问题的能力.


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解答:解:∵M、N是单位圆上的两点,

∴M(


∴


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
∴


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∵α∈[0,π],
∴α+


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∴sin(α+


∴f(α)∈[1,2]
故答案为[1,2].
点评:此题是个中档题.考查向量在几何中的应用,同时考查了三角函数的定义和向量的数量积的坐标运算,三角函数在区间上的最值问题,体现了转化的思想,考查了同学们观察、推理以及创造性地分析问题、解决问题的能力.

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