Integer Trig Hunt

A mathematical hunt:

This Demonstration considers expressions of the form where , , , , and are positive rational numbers or integers and is one of the six trigonometric functions, sin, cos, tan, cot, sec, or csc. For any set of such parameters, the value will never be an integer (nor even an algebraic number).

The goal of the hunt is to find values of the parameters such that is as near as possible to an integer. Easy integers are excluded, like the sine or cosine of a small number, which can easily be made very close to 0 or 1. Using all possible slider combinations, more than 10³⁰ possible values of can be tested, allowing the hunter to be busy for weeks. How small can you make the difference between the value and an integer?