<s>[INST] Can we write $\sin(\frac{\pi}{14})$ as an finite expression using only basic operations? First, let us recall some trigonometric values
$\sin(0)=0$
$\sin(\pi/6)=\frac{1}{2}$
$\sin(\pi/3)=\frac{\sqrt{3}}{2}$
$\sin(\pi/2)=1$
$\sin(\pi/10)=\frac{\sqrt{5}-1}{4}$
$\sin(\pi/12)=\frac{\sqrt{3}-1}{2\sqrt{2}}$
Here, we can observe that for some values of $\theta$, $\sin(\theta)$ can be expressed as a sum of finite radical terms. It is also easy to see that, there exists infinite such $\theta$'s. Say, I have given $\sin(\pi/14)$, can we determine whether it can also be expressed as sum of finite radical terms? What about the general case? Are there any criteria that $\theta$ must obey in order to do that?
 [/INST] It is not so bad !
As @Doug M commented, it is one of the roots of
$$8x^3 - 4x^2 - 4x + 1 = 0$$
Using the trigonometric method for three real roots,we have
$$\sin \left(\frac{\pi }{14}\right)=\frac{1}{6} \left(1+2 \sqrt{7} \sin \left(\frac{1}{3} \csc ^{-1}\left(2
   \sqrt{7}\right)\right)\right)$$
Otherwise, using radicals
$$\sin \left(\frac{\pi }{14}\right)=\frac{1}{6}-\frac{7^{2/3} \left(1-i \sqrt{3}\right)}{6\ 2^{2/3} \sqrt[3]{-1+3 i
   \sqrt{3}}}-\frac{1}{12} \left(1+i \sqrt{3}\right) \sqrt[3]{\frac{7}{2} \left(-1+3
   i \sqrt{3}\right)}$$
The second and third terms are two of the roots of
$$46656 x^6-1512 x^3+343=0$$ which is a quadratic in $x^3$.
The solutions of
$$46656 X^2-1512 X+343=0$$ are
$$X_{1,2}=\frac{7}{432} \left(1\pm3 i \sqrt{3}\right)$$
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