103 Trigonometry Problems

5. Solutions to Advanced Problems 183
46. [USAMO 1995] Suppose a calculator is broken and 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 we can get q to
appear on the display panel of the calculator by pressing some finite sequence
of buttons. Assume that the calculator does real-number calculations with
infinite precision, and that all functions are in terms of radians.
Solution: Because cos −1 sin θ = π 2 − θ and tan ( π
2
− θ ) = 1
π
2
, we have for any x>0,
tan θ
for 0