I am looking for an alternate expression for the cube root of two. The expression can include any integers, along with the constant pi (or tau). The following operations are allowed, and are the ONLY operations allowed, along with what can be derived from them:

- addition, subtraction, multiplication, division

- square roots

- trigonometric functions (like sin or cos^-1)

You are NOT allowed to use these:

- logarithms

- integrals or any calculus functions

- infinite series

- hyperbolic functions like sinh

For example, a real solution to 8x^3 - 6x - 1 = 0 can be best written in this form as cos(pi/9).

I have searched for a few days for a solution, with no luck, and a few exciting false starts that ended up quite sad. If you can provide a solution, or prove that there isn't one, that would be amazing.