HMMT 二月 2004 · 代数 · 第 5 题

英文解析

5. There exists a positive real number x such that cos(tan ( x )) = x . Find the value ofx .2
   √
   Solution: ( − 1 + 5) / 2 − 1
   Draw a right triangle with legs 1 , x ; then the angle θ opposite x is tan x , and we can
   √ √
   2 2
   compute cos( θ ) = 1 / x + 1. Thus, we only need to solve x = 1 / x + 1. This is
   √
   4 2 4 2
   equivalent to x x + 1 = 1. Square both sides to get x + x = 1 ⇒ x + x − 1 = 0.2
   √
   2 2
   Use the quadratic formula to get the solution x = ( − 1 + 5) / 2 (unique since x must be positive).
   4 4