名校
1 . 任意一个复数z的代数形式都可写成复数三角形式,即
,其中i为虚数单位,
,
.棣莫弗定理由法国数学家棣莫弗(1667~1754)创立.设两个复数用三角函数形式表示为:
,
,则:
.如果令
,则能导出复数乘方公式:
.请用以上知识解决以下问题.
(1)试将
写成三角形式;
(2)试应用复数乘方公式推导三倍角公式:
;![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1266e4d6e189cbd788785b44eb4491d6.png)
(3)记
,由棣莫弗定理得
,从而得
,复数
,我们称其为1在复数域内的三次方根. 若
为64在复数域内的6次方根.求
取值构成的集合,其中
,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1810abd6348f8d3863be355fdce70c78.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7fea9021362c5e232929a37a05225cf6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f687593cb4ecef31667bf2320fdfe000.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45eef4221f949bbea8498b39ac1c136a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c825b7acba8f9997d38806be7b3b87eb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5137aa9fb53b43fd558b2f1c26b0951.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed43030ca376eb5e3331d75f103fc762.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1c6bdabdb3bfa767e0cb2f73eec6270.png)
(1)试将
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb232df15bbcb2addccf8d5e7adc4d1f.png)
(2)试应用复数乘方公式推导三倍角公式:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a47bdf9c678020d1d50082f7bb208557.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1266e4d6e189cbd788785b44eb4491d6.png)
(3)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd0c30155ec5bc576f72e97afc42abaf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2443c796f97e4b9b209a207abb3bf1f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3eabab9c270c5390e9930a1376e6906.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c88c2ca3f32231770665622da3ba4be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/930009e5e260660214817c4eaae0c712.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/75cd58d17916b906defc4d6817514272.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b71af6590f0f369c164a054a8b63bf6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11b70cd6a9f071d3a89f3c1c65b609b2.png)
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