HMMT 二月 2015 · 冲刺赛 · 第 8 题

HMMT February 2015 — Guts Round — Problem 8

[ 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# 2

解析

[ 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 . 9 1 2 2 Answer: 0 . 09 OR Use the difference of squares identity sin( a − b ) sin( a + b ) = sin( a ) − sin( b ) 100 9 2 2 2 to get 0 . 5 − 0 . 4 = 0 . 3 = 0 . 09 = . 100 2