Let's say that a number has a simple trigonometric representation, if it can be represented as a product of zero or more rational powers of prime numbers and zero or more integer powers of $\sin(\cdot)$ at rational multiples of $\pi$.

Examples:

• $\sqrt{\sin\!\left(\frac\pi{10}\right)}$ 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 $\sqrt{\sin\!\left(\frac\pi5\right)},\,\sqrt{\sin\!\left(\frac\pi7\right)},\,\sqrt{\sin\!\left(\frac\pi{12}\right)},\,\sqrt{\sin\!\left(\frac\pi{15}\right)},\,\sqrt{\sin\!\left(\frac\pi{20}\right)},\,\sqrt{\sin\!\left(\frac\pi{21}\right)}$ have a simple trigonometric representation?
• 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 a concrete example of such an algorithm?
• More generally, is there an algorithm that, given a real algebraic number (in a form of its minimal polynomial and a rational isolating interval), would determine if it has a simple trigonometric representation? If so, could you give a concrete example of such an algorithm?

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)?