### How to get sage to simplify sin(pi/16)?

I'm trying to look at numbers of the form sin(pi/2^n), but when I ask for a list of the first 10, I get [0,1, 1/2*sqrt(2), 1/2*sqrt(-sqrt(2) + 2), sin(1/16*pi), sin(1/32*pi)...]
Why are these not being reduced, even when I call sin(1/16*pi).simplify_full()? How can I make sage simplify them?