If abs(a),abs(b) ≤ 100 there are some more simple possibilities:
φ∈{±atan(x<sup>±1</sup>):x∈{5/12, 7/24, 8/15, 9/40, 11/60, 12/35, 13/84, 16/63, 20/21, 20/99, 28/45, 33/56, 36/77, 39/80, 48/55, 60/91, 65/72}}
(of course these can lead to a, b way bigger than 100, but not every possibility has to be covered above 100) if φ is not an element of this set, either a or b are irrational (since you wrote in a previous answer, b isn't the irrational part, a is irrational), or you constructed the numbers some other way, but without knowing the algebraic curve you've chosen to do so, this wouldn't help anyhow, however as they are constructed to have this form in either way, it would be basically impossible to guess them...
φ∈{±atan(x<sup>±1</sup>):x∈{5/12, 7/24, 8/15, 9/40, 11/60, 12/35, 13/84, 16/63, 20/21, 20/99, 28/45, 33/56, 36/77, 39/80, 48/55, 60/91, 65/72}}
(of course these can lead to a, b way bigger than 100, but not every possibility has to be covered above 100) if φ is not an element of this set, either a or b are irrational (since you wrote in a previous answer, b isn't the irrational part, a is irrational), or you constructed the numbers some other way, but without knowing the algebraic curve you've chosen to do so, this wouldn't help anyhow, however as they are constructed to have this form in either way, it would be basically impossible to guess them...