2022高三·全国·专题练习
1 . (1)已知
,
求证:
;
(2)已知
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee43d3a6d69044d5070a80f16db9b658.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b417d14bf1fdac3a98e9bbf4ef61b1e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3d990ee8eb645976d9960e9b9f149cd.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2725462adcdba155727d15dddb87a500.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc78d1ba1b161f095e63a931442f1a2c.png)
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解题方法
2 . 在推导很多三角恒等变换公式时,我们可以利用平面向量的有关知识来研究,在一定程度上可以简化推理过程.如我们就可以利用平面向量来推导两角差的余弦公式:
.
具体过程如下:如图,在平面直角坐标系
内作单位圆
,以
为始边作角
.它们的终边与单位圆
的交点分别为
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/26/7b706348-a637-4720-a14e-ec7cff05289a.png?resizew=405)
则
,由向量数量积的坐标表示,有
.
设
的夹角为
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/685366b4bdd10c018e7d8a138eb57133.png)
,另一方面,由图(1)可知,
;
由图(2)可知
,于是
.
所以
,也有
;
所以,对于任意角
有:
.
此公式给出了任意角
的正弦、余弦值与其差角
的余弦值之间的关系,称为差角的余弦公式,简记作
.有了公式
以后,我们只要知道
的值,就可以求得
的值了.
阅读以上材料,利用图(3)单位圆及相关数据(图中
是
的中点),采取类似方法(用其他方法解答正确同等给分)解决下列问题:
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/26/2b6d08ff-a134-4b61-8ffc-63cf5a7df6da.png?resizew=389)
(1)判断
是否正确?(不需要证明)
(2)证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e6276ff5468f5aa9c6eaff479c26cc7.png)
具体过程如下:如图,在平面直角坐标系
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3e5af20b2f8c1fba4470f9650989e51.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4e288596fa3811dd2c17bded60e82e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/26/7b706348-a637-4720-a14e-ec7cff05289a.png?resizew=405)
则
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f87cd2e347fdb63ad7adb85c5d66915f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bce9cc4bfcdd7005d5db6c9276ef51d.png)
设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1625e24072a8fe6c277ecfca2c62cdaa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/685366b4bdd10c018e7d8a138eb57133.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cc0a497c53e96789db5bb3af445c955.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/655ee7e11f540619722504916419e009.png)
由图(2)可知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18eedcc65589e7529da85a578bd0ecb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0ee3a16cc8c88e1ad2a967a935ebb7d.png)
所以
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a689c643b92f5fafe77fb2c754b0184.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e6276ff5468f5aa9c6eaff479c26cc7.png)
所以,对于任意角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4e288596fa3811dd2c17bded60e82e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0fea9c7c16672335cd06f7d237e1495.png)
此公式给出了任意角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4e288596fa3811dd2c17bded60e82e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd927b4b5a7875528c1b54aa4bb8b2dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c5bcf44b6a1dd4daf8eca077ff72d4c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c5bcf44b6a1dd4daf8eca077ff72d4c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1455db71a4123b3317dcfce3e2005e0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22d521f8d021b20757d7a68107fcef1d.png)
阅读以上材料,利用图(3)单位圆及相关数据(图中
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/26/2b6d08ff-a134-4b61-8ffc-63cf5a7df6da.png?resizew=389)
(1)判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/faa56eac1ba22ed040fc34da3e114884.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/889623d5e61054f38a35aedd644c9ff5.png)
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3卷引用:宁夏石嘴山市第一中学2021-2022学年高一上学期期末考试数学试题
3 . 如图,在平面直角坐标系
中,以
轴为始边分别作角
,
,其终边分别与单位圆交于点
,
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/7/4c8c3063-1a5e-4629-9642-3313ab8ab9ff.png?resizew=166)
(1)证明:
;
(2)设
,
,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3e5af20b2f8c1fba4470f9650989e51.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/7/4c8c3063-1a5e-4629-9642-3313ab8ab9ff.png?resizew=166)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ff8eb79da2ae1202feebf45ba5e795c.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f02fede76ba43141bafdb8e40963235.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8d68cdb7adedb9367abd25689430d51.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f407134cad46827104d6654d92db1198.png)
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2卷引用:江苏省淮安市楚州中学、淮阴师范学院附属中学、新马高级中学2021-2022学年高一下学期期中联考数学试题
名校
解题方法
4 . (1)已知实数
,若函数
满足
,问:这样的函数
是否存在? 若存在,写出一个;若不存在,说明理由;
(2)写出三次函数
,使得
,对一切实数
成立,求
时,
的最大值和取最大值时
的值;
(3)设
,函数
,记M为
在区间[t,t+2]上的最大值,当
变化时,记m(t)为M的最小值.
①证明:m(t)的值是与t无关的常数(记为m)
②求m的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6b390d4f89c595551244f615b6856bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d889bdd690f84f91abd2c63dcc05139.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51e810d7540bf757d1bcdd62bea0f0fc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
(2)写出三次函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/932a4f4875c0d88716e36ac7f2eb3288.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04ed2e7ae36ecef5de68d8afd668d520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1591d4244dcf5539a4ae98f554e91e61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f28badcf9e6e095a9474b5d9fdad58b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9d51db103a0934d764e7f9da43fe6eb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ae06c488100e31570805778b1d322e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d09a2b7c019dae83e027830b82b3ee8d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76f0649064a085fb74c997fb507a9b6d.png)
①证明:m(t)的值是与t无关的常数(记为m)
②求m的值.
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20-21高一·全国·课后作业
5 . 证明:
(1)
;
(2)
.
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49aae968756928b30a40ac3775a56858.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c11eb6c04d2ce8d391994e41f1077292.png)
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5卷引用:广东省湛江市2021-2022学年高一上学期期末数学试题
广东省湛江市2021-2022学年高一上学期期末数学试题(已下线)第十章本章回顾(已下线)5.5.2简单的三角恒等变换(同步练习)-【一堂好课】2021-2022学年高一数学上学期同步精品课堂(人教A版2019必修第一册)(已下线)模块三 专题4 (三角函数)(拔高能力练)(北师大版)苏教版(2019)必修第二册课本习题第10章复习题
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6 . 对于定义域分别是
,
的函数
,
规定:函数![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24b59a616211f68934c64002449d9b63.png)
(I)若函数
,写出函数
的解析式并求函数
值域;
(II)若
,其中
是常数,且
,请设计一个定义域为
的函数
及一个
的值,使得
,并予以证明.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b5886cf72ed5a1073263eb9ff485c7c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48101d1755703877e99969012ddb4448.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbf5beca5f1a475dbf003bb2e27d51dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a1cfb60420ff7e72c1b9d64f69ae063.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24b59a616211f68934c64002449d9b63.png)
(I)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d6f83c8ab71cd4342a1381593a7bf09.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2598c9ca2183c4b0b8c1ac2ef979d712.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2598c9ca2183c4b0b8c1ac2ef979d712.png)
(II)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/424a743a9d5c65ec8976c5c041912d07.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/941647c1647511a05d56a58f0a21472d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aa0df7f1e45f9de29e802c7f19a4f64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1801eb64c822b33cfff1051cc8c5c96d.png)
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7 . 古希腊数学家普洛克拉斯曾说:“哪里有数学,哪里就有美,哪里就有发现……”,对称美是数学美的一个重要组成部分,比如圆,正多边形……,请解决以下问题:
![](https://img.xkw.com/dksih/QBM/2021/4/28/2709589299273728/2759660379553792/STEM/1eef0c92360245aa8e4c2533a2eebb6e.png?resizew=191)
(1)魏晋时期,我国古代数学家刘徽在《九章算术注》中提出了割圆术:“割之弥细,所失弥少,割之又割,以至于不可割,则与圆合体,而无所失矣”,割圆术可以视为将一个圆内接正n边形等分成n个等腰三角形(如图所示),当n变得很大时,等腰三角形的面积之和近似等于圆的面积,运用割圆术的思想,求
的近似值(结果保留
).
(2)正n边形的边长为a,内切圆的半径为r,外接圆的半径为R,求证:
.
![](https://img.xkw.com/dksih/QBM/2021/4/28/2709589299273728/2759660379553792/STEM/1eef0c92360245aa8e4c2533a2eebb6e.png?resizew=191)
(1)魏晋时期,我国古代数学家刘徽在《九章算术注》中提出了割圆术:“割之弥细,所失弥少,割之又割,以至于不可割,则与圆合体,而无所失矣”,割圆术可以视为将一个圆内接正n边形等分成n个等腰三角形(如图所示),当n变得很大时,等腰三角形的面积之和近似等于圆的面积,运用割圆术的思想,求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/440ce692fa6eef853b95f4c9ddba9294.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86ebba6ed1add0fe647c0226614b9290.png)
(2)正n边形的边长为a,内切圆的半径为r,外接圆的半径为R,求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d41b98a3d788ea1255c209653fb728d3.png)
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4卷引用:贵州省黔西南州金成实验学校2021-2022学年高一下学期4月质量监测数学试题
贵州省黔西南州金成实验学校2021-2022学年高一下学期4月质量监测数学试题江苏省镇江中学2020-2021学年高一下学期期中数学试题(已下线)数学与文学(已下线)压轴题三角函数新定义题(九省联考第19题模式)练
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解题方法
8 . 已知函数
满足:
,若
,且当
时,
.
(1)求a的值;
(2)当
时,求
的解析式;并判断
在
上的单调性(不需要证明);
(3)设
,
,若
,求实数m的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15afecc50a2c19fe2603ecb440dbab83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ed670b1f668778c6243f3f7470ee7d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27a003b586f8b63d0360bb3dfe15b176.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5eea59dfdb1381065682abad3006c28.png)
(1)求a的值;
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71bb7883ea87e6275472dbe14ee62357.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/953810dff2d248ff297b614947c0c7c5.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da4da56293412b83823ad7f803e16891.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84da8212603441cac973d7a3882f7188.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/241ece7ed9c29f97a6c930ab90f0652c.png)
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3卷引用:辽宁省名校联盟2022-2023学年高二上学期9月联合考试数学试题
名校
解题方法
9 . 对于数列
、
、
,若
对任意的
恒成立,则称数列
、
、
具有性质
.设
;
(1)证明:数列
、
、
具有性质
的一个充分条件为:
;
(2)若
,
、
、
满足(1)的充分条件,求
;
(3)若
、
、
的每一项均为有理数,但
每一项均为无理数,试给出数列
、
、
具有性质
的充要条件.若在此条件下令
,试探究数列
的一些性质(如单调性,极限,
的最大项等).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b70b67a5eb390a940febe9cff97a9579.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c01307616d815444fa236b25f7f96c39.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cea4ac187cbb465180e89f38250b3970.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b783cf91e34e692ce8e171f0965cb53f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c46ed0d01bf547620be3400c5d415ba.png)
(1)证明:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b783cf91e34e692ce8e171f0965cb53f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd5a99e65cc4c24c560a71efad4eab7a.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95d4f868a2ffb3650e3bd7f1bbc10353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b783cf91e34e692ce8e171f0965cb53f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa7152fb0ac4499ead440bd8fe4ba49e.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90af0a8f93cd6ee2d3f66aea5f9a3923.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b783cf91e34e692ce8e171f0965cb53f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b783cf91e34e692ce8e171f0965cb53f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2707474bbe935717bcd5f66aaa2f30f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
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您最近一年使用:0次
2020-02-04更新
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1132次组卷
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7卷引用:第八章 向量的数量积与三角恒等变换 8.2 三角恒等变换 8.2.4 三角恒等变换的应用
(已下线)第八章 向量的数量积与三角恒等变换 8.2 三角恒等变换 8.2.4 三角恒等变换的应用人教B版(2019) 必修第三册 逆袭之路 第八章 8.2 三角恒等变换 8.2.4 三角恒等变换的应用北师大版(2019) 必修第二册 金榜题名 第四章 三角恒等变换 §2 两角和与差的三角函数公式 2.4 积化和差与和差化积公式沪教版(2020) 必修第二册 堂堂清 第六章 6.2(5) 常用三角公式(已下线)专题5 三角函数人教B版(2019)必修第三册课本习题8.2.4 三角恒等变换的应用(已下线)大招11 积化和差公式