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HMMT 二月 2000 · ADV 赛 · 第 2 题

HMMT February 2000 — ADV Round — Problem 2

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. Simplify ( ) + ( ) to the form a + bi . 2 2 P 1 1
解析
  1. Sin e os = and sin = , w e an write the rst term as ( os + i sin ) . 3 2 3 2 3 3 p 6 4 1 4 3 4 4 Sin e os = and sin = , w e an write the se ond term as ( os + i sin ) . 3 2 3 2 3 3 2 2 No w, w e apply DeMoivre's Theorem to simplify the rst expression to ( os 6 + sin 6 ) = 3 3 4 4 ( os 4 + sin 4 ) = 1+0 = 1. Similarly , w e simplify the se ond expression to ( os 6 + sin 6 ) = 3 3 ( os 8 + sin 8 ) = 1 + 0 = 1. Th us, the total sum is 1 + 1 = 2 . 1 1 1 1 n n +2