解题方法
1 . 已知函数
,其中
.求证:
(1)
,且
;
(2)
,
,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c4d6363133c710c00b99fafa01dce16.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4166972dec0aa3e8694a44eeb941a08.png)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1948bdb9bfc6493bc0e596d9a0dab5f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/accad8245514b083d7434160085188fd.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b9f295a43c5d78cf9518456fef0abda.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32474ff2d16bb427dc7426e481b20709.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2479b7fa52eafe0e011435864bfe9c37.png)
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解题方法
2 . 定义函数
为“正余弦”函数.结合学过的知识,可以得到该函数的一些性质:容易证明
为该函数的周期,但是否是最小正周期呢?我们继续探究:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/216fe768d8ce994867dde9ad5708d7ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5c4fb17f6c4d2a854d76062ee167c6c.png)
.可得:
也为函数
的周期.但是否为该函数的最小正周期呢?我们可以分区间研究
的单调性:函数
在
是严格减函数,在
上严格增函数,再结合
,可以确定:
的最小正周期为
.进一步我们可以求出该函数的值域了.定义函数
为“余正弦”函数,根据阅读材料的内容,解决下列问题:
(1)求“余正弦”函数的定义域;
(2)判断“余正弦”函数的奇偶性,并说明理由;
(3)探究“余正弦”函数的单调性及最小正周期,说明理由,并求其值域.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7051f44b01baed6574abaca7f3d7b6e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35e2d7c958e99bcd9d7f251c19ee3544.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/216fe768d8ce994867dde9ad5708d7ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5c4fb17f6c4d2a854d76062ee167c6c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa1dd1e3ecbf87b4c4a2b4ab71f5859.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70f5389990c3a0c5373f3bd9fb2454c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7051f44b01baed6574abaca7f3d7b6e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7051f44b01baed6574abaca7f3d7b6e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7051f44b01baed6574abaca7f3d7b6e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44d51992c05a557cf6058664f1f8961e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8386e2f935d78f9137e1d9cb050223e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b429642b4cc19a976d2592c3bf685ae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7051f44b01baed6574abaca7f3d7b6e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70f5389990c3a0c5373f3bd9fb2454c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/430d24c464431cb2900239095f23f9bf.png)
(1)求“余正弦”函数的定义域;
(2)判断“余正弦”函数的奇偶性,并说明理由;
(3)探究“余正弦”函数的单调性及最小正周期,说明理由,并求其值域.
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解题方法
3 . 已知函数
,
.
(1)证明函数
为偶函数,并求出其最大值;
(2)求函数
在
上单调递增区间.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/035ed7b51cfb67529e30f335945ccb72.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78b4d2cdbc9e45cbc12c47747e6f5314.png)
(1)证明函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3aae9c8988f4a48db69cad3308942c9.png)
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2卷引用:江苏省苏州市高新区第一中学2021-2022学年高二下学期期中数学试题
解题方法
4 . 已知函数
为奇函数.
(1)求实数
的值,判断函数
的单调性并用定义证明;
(2)求关于
的不等式
的解集.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a4266ae5b43bea012ec6642dfaab78d.png)
(1)求实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)求关于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d990f5ed1396782ef7bed8e9a243a54f.png)
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2卷引用:广西北海市北京市第八中学北海实验学校2022-2023学年高一下学期期中考试数学试题
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5 . 已知函数
,函数
,设
.
(1)求证:
是函数f(x)的一个周期;
(2)当k=0时,求F(x)在区间
上的最大值;
(3)若函数F(x)在区间
内恰好有奇数个零点,求实数k的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07699d9a8a6a1770b7ae08897abac853.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d4a029e4c4f50133090661fbcf09654.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f3c2be7482719651bcf491949681e05.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1ad72d7565699d1ebb741eb0ce12bac.png)
(2)当k=0时,求F(x)在区间
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e491151109a22b53131ba3203da29837.png)
(3)若函数F(x)在区间
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cbfe8e7fb253685e0e50bae0c5482314.png)
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5卷引用:上海市西南位育中学2020-2021学年高一下学期期中数学试题
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解题方法
6 . 已知正
的边长为
,内切圆圆心为
,点
满足
.
(1)求证:
为定值;
(2)把三个实数
,
,
的最小值记为
,b,c},若
,求
的取值范围;
(3)若
,
,求当
取最大值时,
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41322821ce31416fdac8dd6e0aa41c71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e105760638b22b26ff8bec4354255e4c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f8180faf978008d2bc7704cb69c3c40.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/304010e1253e0fc6f7578c210be321f9.png)
(2)把三个实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c4ac0a523138c4597301dbd6ed3abb5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8bb980e0614df97e69a89948d3b21ccb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c20fc69bb272fc609c2a7c95f888373c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc95236ed98064b97d67045706a21906.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4e7bf9200b351a259ddfc6c0266129d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d380dea30f490babb2aef4edc49afc6.png)
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4卷引用:浙江省温州中学2020-2021学年高一下学期期中数学试题
7 . 已知函数
的部分图象如图所示.
![](https://img.xkw.com/dksih/QBM/2022/1/19/2897858638028800/2905694596751360/STEM/580f71ea-6ef8-42e3-8919-bea0bc788c86.png?resizew=202)
(1)求函数f(x)的解析式:
(2)证明:
,使得
成立.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ea345085e0957f48cb30766604589c.png)
![](https://img.xkw.com/dksih/QBM/2022/1/19/2897858638028800/2905694596751360/STEM/580f71ea-6ef8-42e3-8919-bea0bc788c86.png?resizew=202)
(1)求函数f(x)的解析式:
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b66d64d61aa6daee84d844e1458c009e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce00a50660b3f6071dc14d9b872874e.png)
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3卷引用:广东省茂名市电白区2021-2022学年高一下学期期中数学试题
8 . 在平面直角坐标系中,O为坐标原点,A,B,C三点满足
.
(1)求证:
;
(2)已知
,
,
,
,若
的最小值为
,求
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6983b8b908199ce3ba3e06bd4d8e6c39.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97b0ec538da1d765fa41f8d2cec70e8f.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fdd04ec1b4569f53912cd42767fb32e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2937748aea6427cd517ce39771b19ea0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ce485410257c9c1fae9d87ce3e44cc8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5644835a8f27b3fb66cafe1d92d1ac9c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1987ecbd076d89da5ef1e2561d79d857.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1987ecbd076d89da5ef1e2561d79d857.png)
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9 . 已知向量
,
,函数
.
(1)求函数
的最小正周期及其单调递减区间;
(2)若
,
是函数
的零点,用列举法表示
的值组成的集合;
(3)求证:方程
不存在正实数解.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ecd10cca49bd57253b1d070dec19f99.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ac61a30b3ace8d177cb41b324363c90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dae9f908aabcd9d9c46a0ecdfd1d6c12.png)
(1)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50edf6c7494709b20edf593b04698eec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7128a9468850b07edd4ad47c7d31fded.png)
(3)求证:方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b4d40624fc4d5a669a76185052ee6b8.png)
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10 . 定义:若函数
的定义域为D,且存在非零常数
,对任意
,
恒成立,则称
为线周期函数,
为
的线周期.
(1)下列函数
(其中
表示不超过x的最大整数),是线周期函数的是____________(直接填写序号);
(2)若
为线周期函数,其线周期为
,求证:
为周期函数;
(3)若
为线周期函数,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e02cab1add26335b3cb43d5b54c7c853.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85eabd57a1bc7fde8cd6da81977bca5a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(1)下列函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e5276150cf27c826686b5f9df818dc2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c4f5908d6a1217e493ed7586b6964dd.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be1ce3f01e2b6364f9a9fdaf197d5e29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd182241405bd00367a423e61df292a5.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05312d9a1821302e6fba94998ee431e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
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10卷引用:北京市八一学校2019~2020学年第二学期高一期中考试数学试题
北京市八一学校2019~2020学年第二学期高一期中考试数学试题上海市青浦高级中学2018-2019学年高一下学期期中数学试题北京市景山学校远洋分校2020-2021学年高一下学期期中考试数学试题北京市北京理工大学附属中学2022-2023学年高一下学期数学期中练习试题北京市海淀区2017-2018学年高一上学期期末考试数学试题2017-2018北京市中国人民大学附属中学高一期末试题北京市平谷区2019-2020学年高一上学期期末数学试题(已下线)7.3 三角函数的图像和性质(难点)(课堂培优)-2021-2022学年高一数学课后培优练(苏教版2019必修第一册)(已下线)第7章《三角函数》 培优测试卷(一)-2021-2022学年高一数学上册同步培优训练系列(苏教版2019)北京市海淀区北京交通大学附属中学2021-2022学年高一下学期3月月考数学试题