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HMMT 二月 2015 · 冲刺赛 · 第 8 题

HMMT February 2015 — Guts Round — Problem 8

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

  1. [ 5 ] Evaluatesin(arcsin(0 . 4) + arcsin(0 . 5)) · sin(arcsin(0 . 5) − arcsin(0 . 4)) ,
    where for x ∈ [ − 1 , 1], arcsin( x ) denotes the unique real number y ∈ [ − π, π ] such that sin( y ) = x .
    HMMT FEBRUARY 2015, 21 FEBRUARY 2015 — GUTS ROUND
    Organization Team Team ID# 2

英文原题

[ 5 ] Evaluate
sin(arcsin(0 . 4) + arcsin(0 . 5)) · sin(arcsin(0 . 5) − arcsin(0 . 4)) ,
where for x ∈ [ − 1 , 1], arcsin( x ) denotes the unique real number y ∈ [ − π, π ] such that sin( y ) = x .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
HMMT FEBRUARY 2015, 21 FEBRUARY 2015 — GUTS ROUND
Organization Team Team ID#

解析

英文解析

  1. [ 5 ] Evaluatesin(arcsin(0 . 4) + arcsin(0 . 5)) · sin(arcsin(0 . 5) − arcsin(0 . 4)) ,
    where for x ∈ [ − 1 , 1], arcsin( x ) denotes the unique real number y ∈ [ − π, π ] such that sin( y ) = x .
    9 1 2 2
    Answer: 0 . 09 OR Use the difference of squares identity sin( a − b ) sin( a + b ) = sin( a ) − sin( b )
    9100
    2 2 2
    to get 0 . 5 − 0 . 4 = 0 . 3 = 0 . 09 = .
    2100