formula for $\sin{\frac{\theta}{3}}$

Question

Using $\sin{3\theta}=3\sin{\theta}-4\sin^3{\theta}$, we may calculate $\sin{\frac{\theta}{3}}$.

But the formula for root of a cubic equation is very complicated can you help me to find a formula for $\sin{\frac{\theta}{3}}$ as it will help to compute $\sin20^\circ$, $\sin10^\circ$ etc. which will be useful.

Answer 1

Using Cardano's method to solve $$3s-4s^3=\sin t$$ leads to finding the cube roots of $$\pm\sqrt{1-\sin^2t}+i\sin t=\pm\cos t+i\sin t=\pm e^{\mp it}.$$ These are non-real number in general, so you end up using polar form, and so go back to square one.

Answer 2

No, there are no such simple formulas. Suppose, for instance, that $\theta=\frac\pi6$. Then, to compute $\sin\left(\frac\pi3\right)$, you would have to solve the equation $3s-4s^3=\frac12$. Problem: this is an irreducible cubic (in $\mathbb{Q}[x]$) with $3$ real roots. Therefore, the roots cannot be expressed algebraically (starting from rational numbers) without the use of complex numbers.