$\sin(\frac{\pi}{p}) $ not expressible by positive radicals and $\sin(\frac{\pi}{q_i})$ ??

Question

We have the following identities:

$\sin(\frac{\pi}{1})=0$

$\sin(\frac{\pi}{2})=1$

$\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}$

$\sin(\frac{\pi}{4})=\sqrt{\frac{1}{2}}$

Lets start with a definition.

Rules for construction :

We start with the set of positive integers and we extend with a finite set $A$ of algebraic positive real numbers $a_i$, the set may even be empty.

further these rules apply;

1. if $x$ is in the set then so is $-x$.
2. if $x,y$ are in the set $x+y$ is in the set.
3. if $x,y$ are in the set then $xy$ is in the set.
4. if $x>0$ is in the set than the positive real $n$ th root is in the set.
5. if $x$ is in the set and $n$ is a strict positive integer then $\frac{x}{n}$ is in the set.

so for instance

$\sqrt{\frac{2}{5} + 3 \sqrt7-\frac{1}{13}}$

is in the set and so is $2^{1/3}$ while

$\sqrt{\frac{2}{5} - 3 \sqrt7-\frac{1}{13}}$

is not ( rule 4)

$1/\sqrt2$ is in the set in a sense , because it is equal to $\sqrt{1/2}$ but it is not allowed to write it like that, because it does not use the laws for construction.

$\sin(\pi/17)$ can be expressed or constructed in that way, but again it not an acceptable form as such.

So what algebraic numbers can be written like that ?

this is close to galois theory. ( expressions by radicals as galois theory usually studies also always divisions and roots of non-positive reals or complex numbers )

But that is not the main question.

Notice I did not mention any elements of A yet.

What I want is to know if a given number can be given constructed with a given set $A$ and the construction rules.

In particular this :

Let $p$ be a given odd prime but not a fermat prime.

Let $q_i$ be odd primes but not fermat primes all smaller than $p$.

Let the set A contain only $\sin(\pi/q_1),\sin(\pi/q_2),...$.

When can we construct the number

$$\sin(\frac{\pi}{p})=..$$

for a given value $p$ ?

see also :

Algebraic numbers expressible in terms of real-valued radicals

For which angles we know the $\sin$ value algebraically (exact)?

Closed form of $\cos(\frac{\pi}{7})$

which are related or similar but different.

edit

comment :

In my construction I did not allow divisions apart from by integers.

A related question is if we also allow division

6. if $x,y$ are in the set and both nonzero than $x/y$ is in the set.

This makes a nice follow-up question.

However I did not make this the main question because I believe this is not more powerful. In other words if $\sin(\frac{\pi}{p})$ cannot be expressed without rule 6 than I do not believe it can be expressed including rule 6.

This is because making a denominator rootfree is usually possible.

On the other hand if the denominator contains a sine that does not reduce to root form , it usually is not removable from the denominator.

For those 2 reasons I believe what I believe.

More insight in it would be welcome ofcourse.

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