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HMMT 二月 2008 · 代数 · 第 8 题

HMMT February 2008 — Algebra — Problem 8

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 6 ] Compute arctan (tan 65 − 2 tan 40 ). (Express your answer in degrees.)
解析

英文解析

  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 .