Is it possible get an algebraic expression for $\sin 1°$ that does not contain roots of negative values, so that it can be evaluated entirely using only real numbers?
For example, this paper gives some algebraic expressions for it, but they all involve complex numbers.
If this would be possible, you could also get an algebraic expression for $\sin(10^o)$ that doesn't contain complex numbers. That is one of three real roots of the cubic $8 z^3 - 6 z + 1$. Now see casus irreducibilis: given a cubic with rational coefficients that is irreducible over the rationals and has three real roots, it is impossible to express the roots using real radicals.