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HMMT 二月 2008 · 代数 · 第 8 题

HMMT February 2008 — Algebra — Problem 8

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 6 ] Compute arctan (tan 65 − 2 tan 40 ). (Express your answer in degrees.)
解析
  1. [ 6 ] Compute arctan (tan 65 − 2 tan 40 ). (Express your answer in degrees as an angle between 0 and ◦ 180 .) ◦ Answer: 25 First Solution: We have 2 ◦ cot 25 − 1 1 ◦ ◦ ◦ ◦ ◦ ◦ tan 65 − 2 tan 40 = cot 25 − 2 cot 50 = cot 25 − = = tan 25 . ◦ ◦ cot 25 cot 25 ◦ Therefore, the answer is 25 . Second Solution: We have ◦ ◦ ◦ 2 1 + tan 20 4 tan 20 (1 − tan 20 ) ◦ ◦ ◦ ◦ ◦ tan 65 − 2 tan 40 = − = = tan(45 − 20 ) = tan 25 . 2 ◦ ◦ ◦ ◦ 1 − tan 20 1 − tan 20 (1 − tan 20 )(1 + tan 20 ) ◦ Again, the answer is 25 .