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][1] and [RootOf][2] for info.


  [1]: https://www.maplesoft.com/support/help/Maple/view.aspx?path=convert&term=convert
  [2]: https://www.maplesoft.com/support/help/Maple/view.aspx?path=RootOf&term=RootOf

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)