Let's say that a real number has a simple trigonometric representation, if it can be represented as a product of zero or more rational powers of positive integers and zero or more (positive or negative) integer powers of $\sin(\cdot)$ at rational multiples of $\pi$ (the number of terms in the product is assumed to be finite, the empty product is taken to be $1$).
Examples:
- $\sqrt[3]{\sin\!\left(\frac\pi3\right)}$ has a simple trigonometric representation, because $\sqrt[3]{\sin\!\left(\frac\pi3\right)}=\frac{3^{1/6}}{2^{1/3}}$.
- $\sqrt{\sin\!\left(\frac\pi{10}\right)}$ has a simple trigonometric representation, because $\sqrt{\sin\!\left(\frac\pi{10}\right)}=\frac{2^{1/2}}{5^{1/4}}\,\sin\!\left(\frac\pi5\right)$.
- $\pi$ does not have a simple trigonometric representation, because it is not an algebraic number.
Questions:
- Do $\sqrt{\sin\!\left(\frac\pi5\right)},$ $\sqrt{\sin\!\left(\frac\pi8\right)},$ $\sqrt{\sin\!\left(\frac\pi{12}\right)},$ $\sqrt{\sin\!\left(\frac\pi{15}\right)},$ $\sqrt{\sin\!\left(\frac\pi{20}\right)},$ $\sqrt{\sin\!\left(\frac\pi{24}\right)}$ have simple trigonometric representations?
- Is there an algorithm that, given a rational power of $\sin(\cdot)$ at a rational multiple of $\pi$, would determine if it has a simple trigonometric representation? If so, could you give (or outline) a concrete example of such an algorithm (efficient, if possible)?
- More generally, is there an algorithm that, given a real algebraic number (in some explicit form, e.g. as its minimal polynomial and a rational isolating interval), would determine if it has a simple trigonometric representation? If so, could you give (or outline) a concrete example of such an algorithm (efficient, if possible)?
