code: def _minpoly_sin(ex, x):
      """
      Returns the minimal polynomial of ``sin(ex)``
      see https://mathworld.wolfram.com/TrigonometryAngles.html
      """
      c, a = ex.args[0].as_coeff_Mul()
      if a is pi:
          if c.is_rational:
              n = c.q
              q = sympify(n)
              if q.is_prime:
                  # for a = pi*p/q with q odd prime, using chebyshevt
                  # write sin(q*a) = mp(sin(a))*sin(a);
                  # the roots of mp(x) are sin(pi*p/q) for p = 1,..., q - 1
                  a = dup_chebyshevt(n, ZZ)
                  return Add(
                      *[x ** (n - i - 1) * a[i] for i in range(n)]
                  )
              if c.p == 1:
                  if q == 9:
                      return 64 * x**6 - 96 * x**4 + 36 * x**2 - 3
  
              if n % 2 == 1:
                  # for a = pi*p/q with q odd, use
                  # sin(q*a) = 0 to see that the minimal polynomial must be
                  # a factor of dup_chebyshevt(n, ZZ)
                  a = dup_chebyshevt(n, ZZ)
                  a = [x ** (n - i) * a[i] for i in range(n + 1)]
                  r = Add(*a)
                  _, factors = factor_list(r)
                  res = _choose_factor(factors, x, ex)
                  return res
  
              expr = ((1 - cos(2 * c * pi)) / 2) ** S.Half
              res = _minpoly_compose(expr, x, QQ)
              return res
  
      raise NotAlgebraic(
          "%s does not seem to be an algebraic element" % ex
      )
  
imports: from functools import reduce
  from sympy.core.add import Add
  from sympy.core.exprtools import Factors
  from sympy.core.function import expand_mul, expand_multinomial, _mexpand
  from sympy.core.mul import Mul
  from sympy.core.numbers import I, Rational, pi, _illegal
  from sympy.core.singleton import S
  from sympy.core.symbol import Dummy
  from sympy.core.sympify import sympify
  from sympy.core.traversal import preorder_traversal
  from sympy.functions.elementary.exponential import exp
  from sympy.functions.elementary.miscellaneous import sqrt, cbrt
  from sympy.functions.elementary.trigonometric import cos, sin, tan
  from sympy.ntheory.factor_ import divisors
  from sympy.utilities.iterables import subsets
  from sympy.polys.domains import ZZ, QQ, FractionField
  from sympy.polys.orthopolys import dup_chebyshevt
  from sympy.polys.polyerrors import NotAlgebraic, GeneratorsError
  from sympy.polys.polytools import Poly, PurePoly, invert, factor_list, groebner, resultant, degree, poly_from_expr, parallel_poly_from_expr, lcm
  from sympy.polys.polyutils import dict_from_expr, expr_from_dict
  from sympy.polys.ring_series import rs_compose_add
  from sympy.polys.rings import ring
  from sympy.polys.rootoftools import CRootOf
  from sympy.polys.specialpolys import cyclotomic_poly
  from sympy.utilities import numbered_symbols, public, sift
label_desc: Use image node_id 11 for calling a global function with example usage: _minpoly_sin(ex, x) and returns: res, Add, res, unknown