Trig Identity for sin(x/3)

What is the identity for sin(x/3), similar to sin(x/2) = +/- sqrt[(1 - cos x) / 2]?

Let t = x/3

sin(x) =>
sin(3t) =>
sin(2t)cos(t) + sin(t)cos(2t) =>
2sin(t)cos(t)^2 + sin(t)cos(t)^2 - sin(t)^3 =>
3sin(t)cos(t)^2 - sin(t)^3 =>
3sin(t) * (1 - sin(t)^2) - sin(t)^3 =>
3sin(t) - 3sin(t)^3 - sin(t)^3 =>
3sin(t) - 4sin(t)^3

sin(x) = 3sin(x/3) - 4sin(x/3)^3
4sin(x/3)^3 - 3sin(x/3) + sin(x) = 0

sin(x/3) = k

4k^3 - 3k + sin(x) = 0

Solve the cubic equation for k and you have your relationship