Maple 2017.3 helps you. For example,
convert(cos((1/7)*Pi), RootOf);
RootOf(8*_Z^3-4*_Z^2-4*_Z+1, .9009688679)
convert(cos((1/27)*Pi), RootOf)
RootOf(512*_Z^9-1152*_Z^7+864*_Z^5-240*_Z^3+18*_Z-1, .9932383577)
See convert and RootOf for info.
Addition. Also
convert(sin(7*Pi*(1/22)), RootOf);
RootOf(32*_Z^5+16*_Z^4-32*_Z^3-12*_Z^2+6*_Z+1, .8412535328)
convert(sin(7*Pi*(1/22))^2, RootOf);
-(1/4)*(RootOf(_Z^20-_Z^18+_Z^16-_Z^14+_Z^12-_Z^10+_Z^8-_Z^6+_Z^4-_Z^2+1, index = 1)^7+
RootOf(_Z^20-_Z^18+_Z^16-_Z^14+_Z^12-_Z^10+_Z^8-_Z^6+_Z^4-_Z^2+1, index = 1)^15)^2