Question

计算(con7π/4+isin7π/4)(cos3π/4+isin3π/4) (con7π/4+isin7π/4)(cos3π/4+isin3π/4)

Answer 1

可以用Euler公式计算,较为简单：
cos7π/4 + isin7π/4 = e^(i*7π/4),cos3π/4 + isin3π/4 = e^(i*3π/4).
所以原式
=e^(i*7π/4)*e^(i*3π/4)
=e^(i*5π/2)
=cos(5π/2) + isin(5π/2)
=i
也可以直接按复数四则运算来算：
(con7π/4+isin7π/4)(cos3π/4+isin3π/4)
=(cos7π/4cos3/4-sin7π/4sin3π/4)+i(cos7π/4sin3π/4+sin7π/4cos3π/4) (利用sin,cos的和角公式)
=cos(7π/4+3π/4) + isin(7π/4+3π/4)
=cos5π/2 + isin5π/2
=i
即原式 = i.