1995 USAMO 第 2 题

A calculator is broken so that the only keys that still work are the $\, \sin, \; \cos,$ $\tan, \; \sin^{-1}, \; \cos^{-1}, \,$ and $\, \tan^{-1} \,$ buttons. The display initially shows 0. Given any positive rational number $\, q, \,$ show that pressing some finite sequence of buttons will yield $\, q$. Assume that the calculator does real number calculations with infinite precision. All functions are in terms of radians.

计算器坏了，唯一还能用的键是 $\、\sin、\; \cos,$$\tan,\; \sin^{-1}, \; \cos^{-1}、\,$和$\、\tan^{-1} \,$按钮。显示屏最初显示 0。给定任何正有理数$\, q, \,$表明按下某些有限的按钮序列将产生$\, q$。假设计算器以无限精度进行实数计算。所有函数均以弧度表示。