HMMT 二月 2019 · 团队赛 · 第 3 题

HMMT February 2019 — Team Round — Problem 3

[ 25 ] For any angle 0 < θ < π/ 2, show that 0 < sin θ + cos θ + tan θ + cot θ − sec θ − csc θ < 1 .

解析

[ 25 ] For any angle 0 < θ < π/ 2, show that 0 < sin θ + cos θ + tan θ + cot θ − sec θ − csc θ < 1 . Proposed by: Yuan Yao We use the following geometric construction, which follows from the geometric definition of the trigono- metric functions: Let Z be a point on the unit circle in the coordinate plane with origin O . Let X , Y be 1 1 the projections of Z onto the x - and y -axis respectively, and let X , Y lie on x - and y -axis respectively 2 2 such that X Y is tangent to the unit circle at Z . Then we have 2 2 OZ = X Y = 1 , X Z = sin θ, Y Z = cos θ, X Z = tan θ, Y Z = cot θ, OX = sec θ, OY = csc θ. 1 1 1 1 2 2 2 2 It then suffices to show that 0 < X Y − X X − Y Y < 1 = X Y . The left inequality is true 2 2 1 2 1 2 1 1 because X X and Y Y are the projections of ZX and ZY onto x - and y -axis respectively. The right 1 2 1 2 2 2 inequality is true because X X + X Y + Y Y > X Y by triangle inequality. Therefore we are done. 1 2 1 1 1 2 2 2