1 . 复平面是人类漫漫数学历史中的一副佳作,他以虚无缥缈的数字展示了人类数学最纯粹的浪漫.欧拉公式可以说是这座数学王座上最璀璨的明珠,相关的内容是,欧拉公式:
,其中
表示虚数单位,
是自然对数的底数.数学家泰勒对此也提出了相关公式:
其中的感叹号!表示阶乘
,试回答下列问题:
(1)试证明欧拉公式.
(2)利用欧拉公式,求出以下方程的所有复数解.
①
;②
;
(3)求出角度
的
倍角公式(用
表示,
).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5aa584db159b0f9bfae801d0134393b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a7035cd4adda5d72a9fc9f9fda75995.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/041a7c8fc017f596542c5e6ec7d1c40b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/574f94ac7dfd3477b58799e0251bb6a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a260aee25664815506d2720174b03829.png)
(1)试证明欧拉公式.
(2)利用欧拉公式,求出以下方程的所有复数解.
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bde2a8df1f0418c41a6e077c7f5de21.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1150e58bbcb15a349fb5b9b5ef708d41.png)
(3)求出角度
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f9d7bbcbeb05fbbb06463120f9a6811.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aefd06c239145a2b6ae87a955aa51414.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8cd112c1cb203187e3c9554617c45b8.png)
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解题方法
2 . 一般地,任意给定一个角
,它的终边
与单位圆的交点
的坐标,无论是横坐标
还是纵坐标
,都是唯一确定的,所以点
的横坐标
、纵坐标
都是角
的函数.下面给出这些函数的定义:
的纵坐标
叫作
的正弦函数,记作
,即
;
②把点
的横坐标
叫作
的余弦函数,记作
,即
;
③把点
的纵坐标
的倒数叫作
的余割,记作
,即
;
④把点
的横坐标
的倒数叫作
的正割,记作
,即
.
下列结论正确的有( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8bd1adf2f585d7c3f22e6755c2a545ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abd13974aebe38eb2a1d744a01ea5aa5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8f798a9af75a091a8be0b71f2038260.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74b5644d2be8b74f1bc61bf90efc87a8.png)
②把点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3a282d19bf2c827a98d4443330f7ca1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9320c4f0e8a937495d22f2ddb50f93f3.png)
③把点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eeb7866904d35a4e6d7ff6e2639270a0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2692ac8eb7a6934c2ba4b8ff30735c7.png)
④把点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9085195f5d04d5e224f4c8df470a9c7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33476d63f1cab19df8fc0ff420eaa636.png)
下列结论正确的有( )
A.![]() |
B.当![]() ![]() |
C.函数![]() ![]() |
D.当![]() ![]() ![]() |
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3 . 正五角星是一个非常优美的几何图形,其与黄金分割有着密切的联系,在如图所示的五角星中,以
为顶点的多边形为正边边形,设
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ae0c3a889da94fd660aa8dc57e25b38.png)
________ ,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13285b04c6606d0af14d148e344ff93e.png)
________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1d75df9d80ce1e0b7cb50464e293864.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ae1b2a249841cb942fea24811feda9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ae0c3a889da94fd660aa8dc57e25b38.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13285b04c6606d0af14d148e344ff93e.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/4/22/2e3852b0-2d69-4bda-994f-6edb4bd2b582.png?resizew=128)
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|
768次组卷
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4卷引用:河北省邯郸市2024届高三下学期学业水平选择性模拟考试数学试题
河北省邯郸市2024届高三下学期学业水平选择性模拟考试数学试题河北省邯郸市2024届高三第四次调研监测数学试题辽宁省辽阳市2023-2024学年高三下学期二模数学试卷(已下线)第五套 艺体生新高考全真模拟 (二模重组卷)
名校
4 . 已知
,求
( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccfbb9f2fb125298744d7bdf814cfac8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4e304cf018473bb54edb166fcd6502b.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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解题方法
5 . 在平面直角坐标系中,若角
的顶点为原点,始边为
轴非负半轴,终边经过点
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30fb45e6002c63ec6836ad7237dc35e0.png)
___________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/132c30ea68aa40630eb1b86e4f607b2c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/111fc953684b51df437ef2bc02290ad5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30fb45e6002c63ec6836ad7237dc35e0.png)
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解题方法
6 . 设
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ae3c401925fc74138be84e19b895cc5.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
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解题方法
7 . 若
,则
( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a59c46d0c5d753fae933b06b9ebeb8b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ab73ca961dbb5cc2c7b8bec69b95e26.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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解题方法
8 . 已知角
的终边过点
,角
的终边与角
的终边关于
轴对称,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/994d6e7f47b3b98aa848c05d0af8f65e.png)
_________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d00d5d8cc31aaca7fc2eafb99eac7629.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/994d6e7f47b3b98aa848c05d0af8f65e.png)
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名校
解题方法
9 . 已知函数
,若
,则
( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c31f72684eab187b945a32800c3498db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f364a0e06a0c751425d4e864d1e5b54.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cbd78b7cb1f5cfdd36c44f7e1b86b72b.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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解题方法
10 . 始边与
轴的正半轴重合的角
的终边过点
,则
=_________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f4d63efe3b074b8644890d2c4902f26.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a12182d69090968701ca7c3c410bdceb.png)
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上海市闵行区2024届高三下学期学业质量调研(二模)数学试卷江苏省?邮市第?中学2023-2024学年高一下学期4月阶段测试数学试卷(已下线)专题07 一轮复习三角函数(1)--高二期末考点大串讲(人教A版2019)