HMMT 二月 2001 · ADV 赛 · 第 10 题

HMMT February 2001 — ADV Round — Problem 10

Alex picks his favorite point ( x, y ) in the first quadrant on the unit circle x + y = 1, such that a ray from the origin through ( x, y ) is θ radians counterclockwise from the positive ( ) 4 x +3 y − 1 x -axis. He then computes cos and is surprised to get θ . What is tan( θ )? 5

解析

Alex picks his favorite point ( x, y ) in the first quadrant on the unit circle x + y = 1, such that a ray from the origin through ( x, y ) is θ radians counterclockwise from the positive ( ) 4 x +3 y − 1 x -axis. He then computes cos and is surprised to get θ . What is tan( θ )? 5 Solution: x = cos( θ ), y = sin( θ ). By the trig identity you never thought you’d need, 4 x +3 y = cos( θ − φ ), where φ has sine 3/5 and cosine 4/5. Now θ − φ = θ is impossible, since 5 φ 6 = 0, so we must have θ − φ = − θ , hence θ = φ/ 2. Now use the trusty half-angle identities 1 to get tan( θ ) = . 3