证明:(cos30º+i·sin30º)^n=cos(n·30º)+i·sin(n·30&
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证明:(cos30º+i·sin30º)^n=cos(n·30º)+i·sin(n·30º)
![证明:(cos30º+i·sin30º)^n=cos(n·30º)+i·sin(n·30&](/uploads/image/z/15394240-64-0.jpg?t=%E8%AF%81%E6%98%8E%EF%BC%9A%EF%BC%88cos30%26ordm%3B%2Bi%C2%B7sin30%26ordm%3B%EF%BC%89%5En%3Dcos%EF%BC%88n%C2%B730%26ordm%3B%EF%BC%89%2Bi%C2%B7sin%EF%BC%88n%C2%B730%26)
证明:
当n=1时,cos30°+i·sin30°=cos30°+sin30°,成立
假设当n=k-1时成立,即(cos30°+i·sin30°)^(k-1)=cos[(k-1)·30°]+i·sin[(k-1)·30°]
下面证明当n=k时成立:
(cos30°+i·sin30°)^k
=(cos30°+i·sin30°)^(k-1)×(cos30°+i·sin30°)
={cos[(k-1)·30°]+i·sin[(k-1)·30°]}×(cos30°+i·sin30°)
=cos[(k-1)·30°]cos30°-sin[(k-1)·30°]sin30°
+i·{sin[(k-1)·30°]cos30°+cos[(k-1)·30°]sin30°}
=cos[(k-1)·30°+30°]+i·sin[(k-1)·30°+30°]
=cos(k·30°)+i·sin(k·30°)
所以当n=k时成立.
∴(cos30º+i·sin30º)^n=cos(n·30º)+i·sin(n·30º)
当n=1时,cos30°+i·sin30°=cos30°+sin30°,成立
假设当n=k-1时成立,即(cos30°+i·sin30°)^(k-1)=cos[(k-1)·30°]+i·sin[(k-1)·30°]
下面证明当n=k时成立:
(cos30°+i·sin30°)^k
=(cos30°+i·sin30°)^(k-1)×(cos30°+i·sin30°)
={cos[(k-1)·30°]+i·sin[(k-1)·30°]}×(cos30°+i·sin30°)
=cos[(k-1)·30°]cos30°-sin[(k-1)·30°]sin30°
+i·{sin[(k-1)·30°]cos30°+cos[(k-1)·30°]sin30°}
=cos[(k-1)·30°+30°]+i·sin[(k-1)·30°+30°]
=cos(k·30°)+i·sin(k·30°)
所以当n=k时成立.
∴(cos30º+i·sin30º)^n=cos(n·30º)+i·sin(n·30º)
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