I'm going to look into what's the case for 'tan'. For 'tan',
tan(pi/4)=tan(45°) is rational as well - I'm going to look into
whether the 'S' of 'tan' is only the integer multiples pi/4, or
whether there are more.
It's only those:
For which θ is θ/π and tan(θ) rational?
Assume t=tan(θ)∈ℚ. Note that sin (2θ)=(2t)/(1+t^2)∈ℚ. Since sin(2θ) is
rational and 2θ/π is rational, then 2θ is an integer multiple of π/2.
So, θ is an integer multiple of π/4.
(For the other direction, this can be verified with a lookup table.)
Best regards,
Maxime Devos
