返回题库

HMMT 二月 2001 · 团队赛 · 第 5 题

HMMT February 2001 — Team Round — Problem 5

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. Find all x between − and such that 1 − sin x − cos x = .
    2 2 16
解析

英文解析

  1. Find all x between − and such that 1 − sin x − cos x = .
    2 2 16
    4 4 4
    2 1 2
    Solution: 1 − sin x − cos x = ⇒ (16 − 16 cos x ) − sin x − 1 = 0 ⇒ 16 sin x −
    √16
    2 2 1 3
    16 sin x + 1 = 0. Use the quadratic formula in sin x to obtain sin x = ± . Since
    2 4
    √
    3 π 5 π2
    cos 2 x = 1 − 2 sin x = ± , we get x = ± , ± .
    2 12 12