HMMT 二月 2000 · ADV 赛 · 第 2 题
HMMT February 2000 — ADV Round — Problem 2
题目详情
英文原题
- Simplify ( ) + ( ) to the form a + bi .
2 2
1 1P
解析
英文解析
- Sin e os = and sin = , we an write the rst term as ( os + i sin ) .
3 2 3 2 3 3
6 p
4 1 4 3 4 4
Sin e os = and sin = , we an write the se ond term as ( os + i sin ) .
3 2 3 2 3 3
2 2
No w, we apply De moivre's Theorem to simplify the rst expression to ( os 6 + sin 6 ) =
3 3
4 4
( os 4 + sin 4 ) = 1+0 = 1. Similarly , we 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