Numbers of the form:

$\backslash \; cos\; (\backslash \; frac\; \{K\; \backslash \; pi\}\; \{N\})\; \backslash \; qquad\; \backslash \; sin\; (\backslash \; frac\; \{K\; \backslash \; pi\}\; \{N\})\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{K\; \backslash \; pi\}\; \{N\})$

are algebraic numbers and for this reason, they admit a minimal Polynôme on $\backslash \; mathbb\; \{Q\}$. One obtains even polynomials of least degrees by accepting that the coefficients of the polynomial belong to a quadratic Corps $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{D\})$. D being a divider of N.

In this article, one will thus name $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{D\})$ the whole of the polynomials with coefficients in the quadratic body $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{D\})$.

Minimal polynomial of number of the form 2cos (k$\backslash \; pi$/n) or 2sin (k$\backslash \; pi$/n).

Below are, in the order of degree crescents, the first polynomials minimal of the numbers of the form 2.cos (k$\backslash \; pi$/n) or of the form 2.sin (k$\backslash \; pi$/n). Factor 2 is there only to simplify the coefficients of the polynomial.

Polynomials of the first degree

$X\; -\; 1\; \backslash \; qquad\; X\; +\; 1\; \backslash \; qquad\; X\; -\; 2\; \backslash \; qquad\; X\; +\; 2$

are respectively the minimal polynomials of the numbers:

$2\; \backslash \; cos\; (\backslash \; frac\; \{\backslash \; pi\}\; \{3\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{2\; \backslash \; pi\}\; \{3\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (0),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; pi).\; ~$

Who can also put themselves in the form:

$2\; \backslash \; sin\; (\backslash \; frac\; \{\backslash \; pi\}\; \{6\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; \backslash \; pi\}\; \{6\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{\backslash \; pi\}\; \{2\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; \backslash \; pi\}\; \{2\}).\; ~$

$X\; -\; \backslash \; sqrt\; \{2\}\; \backslash \; qquad\; X\; +\; \backslash \; sqrt\; \{2\}\; \backslash \; qquad\; X\; -\; \backslash \; sqrt\; \{3\}\; \backslash \; qquad\; X\; +\; \backslash \; sqrt\; \{3\}$

are respectively the minimal polynomials in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{2\})$ and $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{3\})$ of the numbers:

$2\; \backslash \; cos\; (\backslash \; frac\; \{\backslash \; pi\}\; \{4\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{3\; \backslash \; pi\}\; \{4\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{\backslash \; pi\}\; \{6\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{5\; \backslash \; pi\}\; \{6\}).\; ~$

Who can also put themselves in the form:

$2\; \backslash \; sin\; (\backslash \; frac\; \{\backslash \; pi\}\; \{4\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; \backslash \; pi\}\; \{4\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{\backslash \; pi\}\; \{3\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; \backslash \; pi\}\; \{3\}).\; ~$

Polynomials of second degree

$x^2\; -\; X\; \backslash \; sqrt\; \{2\}\; -\; 1\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{2\})$ of the numbers:

$2\; \backslash \; cos\; (\backslash \; frac\; \{\backslash \; pi\}\; \{12\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{7\; \backslash \; pi\}\; \{12\}).\; ~$

Who can also put themselves in the form:

$2\; \backslash \; sin\; (\backslash \; frac\; \{-\; \backslash \; pi\}\; \{12\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{5\; \backslash \; pi\}\; \{12\}).\; ~$

$x^2\; +\; X\; \backslash \; sqrt\; \{2\}\; -\; 1\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{2\})$ of the numbers:

$2\; \backslash \; cos\; (\backslash \; frac\; \{5\; \backslash \; pi\}\; \{12\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{11\; \backslash \; pi\}\; \{12\}).\; ~$

Who can also put themselves in the form:

$2\; \backslash \; sin\; (\backslash \; frac\; \{\backslash \; pi\}\; \{12\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 5\; \backslash \; pi\}\; \{12\}).\; ~$

$x^2\; -\; X\; \backslash \; sqrt\; \{5\}\; +\; 1\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{5\})$ of the numbers:

$2\; \backslash \; cos\; (\backslash \; frac\; \{\backslash \; pi\}\; \{5\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{2\; \backslash \; pi\}\; \{5\}).\; ~$

Who can also put themselves in the form:

$2\; \backslash \; sin\; (\backslash \; frac\; \{\backslash \; pi\}\; \{10\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{3\; \backslash \; pi\}\; \{10\}).\; ~$

$x^2\; +\; X\; \backslash \; sqrt\; \{5\}\; +\; 1\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{5\})$ of the numbers:

$2\; \backslash \; cos\; (\backslash \; frac\; \{3\; \backslash \; pi\}\; \{5\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{4\; \backslash \; pi\}\; \{5\}).\; ~$

Who can also put themselves in the form:

$2\; \backslash \; sin\; (\backslash \; frac\; \{-\; \backslash \; pi\}\; \{10\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 3\; \backslash \; pi\}\; \{10\}).\; ~$

Polynomials of the third degree

$x^3\; -\; x^2\; -\; 2x\; +\; 1\; ~$

Is the minimal polynomial of the numbers:

$2\; \backslash \; cos\; (\backslash \; frac\; \{\backslash \; pi\}\; \{7\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{3\; \backslash \; pi\}\; \{7\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{5\; \backslash \; pi\}\; \{7\}).\; ~$

Who can also put themselves in the form:

$2\; \backslash \; sin\; (\backslash \; frac\; \{\backslash \; pi\}\; \{14\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 3\; \backslash \; pi\}\; \{14\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{5\; \backslash \; pi\}\; \{14\}).\; ~$

$x^3\; +\; x^2\; -\; 2x\; -\; 1\; ~$

Is the minimal polynomial of the numbers:

$2\; \backslash \; cos\; (\backslash \; frac\; \{2\; \backslash \; pi\}\; \{7\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{4\; \backslash \; pi\}\; \{7\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{6\; \backslash \; pi\}\; \{7\}).\; ~$

Who can also put themselves in the form:

$2\; \backslash \; sin\; (\backslash \; frac\; \{-\; \backslash \; pi\}\; \{14\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{3\; \backslash \; pi\}\; \{14\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 5\; \backslash \; pi\}\; \{14\}).\; ~$

$x^3\; -\; 3x\; -\; 1\; ~$

Is the minimal polynomial of the numbers:

$2\; \backslash \; cos\; (\backslash \; frac\; \{\backslash \; pi\}\; \{9\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{5\; \backslash \; pi\}\; \{9\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{7\; \backslash \; pi\}\; \{9\}).\; ~$

Who can also put themselves in the form:

$2\; \backslash \; sin\; (\backslash \; frac\; \{-\; \backslash \; pi\}\; \{18\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 5\; \backslash \; pi\}\; \{18\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{7\; \backslash \; pi\}\; \{18\}).\; ~$

$x^3\; -\; 3x\; +\; 1\; ~$

Is the minimal polynomial of the numbers:

$2\; \backslash \; cos\; (\backslash \; frac\; \{2\; \backslash \; pi\}\; \{9\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{4\; \backslash \; pi\}\; \{9\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{8\; \backslash \; pi\}\; \{9\}).\; ~$

Who can also put themselves in the form:

$2\; \backslash \; sin\; (\backslash \; frac\; \{\backslash \; pi\}\; \{18\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{5\; \backslash \; pi\}\; \{18\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 7\; \backslash \; pi\}\; \{18\}).\; ~$

$x^3\; -\; x^2\; \backslash \; sqrt\; \{7\}\; +\; \backslash \; sqrt\; \{7\}\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{7\})$ of the numbers:

$2\; \backslash \; cos\; (\backslash \; frac\; \{\backslash \; pi\}\; \{14\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{3\; \backslash \; pi\}\; \{14\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{9\; \backslash \; pi\}\; \{14\}).\; ~$

Who can also put themselves in the form:

$2\; \backslash \; sin\; (\backslash \; frac\; \{-\; \backslash \; pi\}\; \{7\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{2\; \backslash \; pi\}\; \{7\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{3\; \backslash \; pi\}\; \{7\}).\; ~$

$x^3\; +\; x^2\; \backslash \; sqrt\; \{7\}\; -\; \backslash \; sqrt\; \{7\}\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{7\})$ of the numbers:

$2\; \backslash \; cos\; (\backslash \; frac\; \{5\; \backslash \; pi\}\; \{14\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{11\; \backslash \; pi\}\; \{14\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{13\; \backslash \; pi\}\; \{14\}).\; ~$

Who can also put themselves in the form:

$2\; \backslash \; sin\; (\backslash \; frac\; \{\backslash \; pi\}\; \{7\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 2\; \backslash \; pi\}\; \{7\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 3\; \backslash \; pi\}\; \{7\}).\; ~$

$x^3\; -\; 3x\; -\; \backslash \; sqrt\; \{3\}\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{3\})$ of the numbers:

$2\; \backslash \; cos\; (\backslash \; frac\; \{\backslash \; pi\}\; \{18\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{11\; \backslash \; pi\}\; \{18\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{13\; \backslash \; pi\}\; \{18\}).\; ~$

Who can also put themselves in the form:

$2\; \backslash \; sin\; (\backslash \; frac\; \{-\; \backslash \; pi\}\; \{9\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 2\; \backslash \; pi\}\; \{9\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{4\; \backslash \; pi\}\; \{9\}).\; ~$

$x^3\; -\; 3x\; +\; \backslash \; sqrt\; \{3\}\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{3\})$ of the numbers:

$2\; \backslash \; cos\; (\backslash \; frac\; \{5\; \backslash \; pi\}\; \{18\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{7\; \backslash \; pi\}\; \{18\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{17\; \backslash \; pi\}\; \{18\}).\; ~$

Who can also put themselves in the form:

$2\; \backslash \; sin\; (\backslash \; frac\; \{\backslash \; pi\}\; \{9\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{2\; \backslash \; pi\}\; \{9\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 4\; \backslash \; pi\}\; \{9\}).\; ~$

$2x^3\; +\; (1\; +\; \backslash \; sqrt\; \{13\})\; x^2\; -\; 2x\; -\; 3\; -\; \backslash \; sqrt\; \{13\}\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{13\})$ of the numbers:

$2\; \backslash \; cos\; (\backslash \; frac\; \{4\; \backslash \; pi\}\; \{13\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{10\; \backslash \; pi\}\; \{13\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{12\; \backslash \; pi\}\; \{13\}).\; ~$

Who can also put themselves in the form:

$2\; \backslash \; sin\; (\backslash \; frac\; \{5\; \backslash \; pi\}\; \{26\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 7\; \backslash \; pi\}\; \{26\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 11\; \backslash \; pi\}\; \{26\}).\; ~$

$2x^3\; +\; (1\; -\; \backslash \; sqrt\; \{13\})\; x^2\; -\; 2x\; -\; 3\; +\; \backslash \; sqrt\; \{13\}\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{13\})$ of the numbers:

$2\; \backslash \; cos\; (\backslash \; frac\; \{2\; \backslash \; pi\}\; \{13\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{6\; \backslash \; pi\}\; \{13\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{8\; \backslash \; pi\}\; \{13\}).\; ~$

Who can also put themselves in the form:

$2\; \backslash \; sin\; (\backslash \; frac\; \{\backslash \; pi\}\; \{26\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{9\; \backslash \; pi\}\; \{26\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 3\; \backslash \; pi\}\; \{26\}).\; ~$

$2x^3\; -\; (1\; -\; \backslash \; sqrt\; \{13\})\; x^2\; -\; 2x\; +\; 3\; -\; \backslash \; sqrt\; \{13\}\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{13\})$ of the numbers:

$2\; \backslash \; cos\; (\backslash \; frac\; \{5\; \backslash \; pi\}\; \{13\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{7\; \backslash \; pi\}\; \{13\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{11\; \backslash \; pi\}\; \{13\}).\; ~$

Who can also put themselves in the form:

$2\; \backslash \; sin\; (\backslash \; frac\; \{3\; \backslash \; pi\}\; \{26\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; \backslash \; pi\}\; \{26\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 9\; \backslash \; pi\}\; \{26\}).\; ~$

$2x^3\; -\; (1\; +\; \backslash \; sqrt\; \{13\})\; x^2\; -\; 2x\; +\; 3\; +\; \backslash \; sqrt\; \{13\}\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{13\})$ of the numbers:

$2\; \backslash \; cos\; (\backslash \; frac\; \{\backslash \; pi\}\; \{13\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{3\; \backslash \; pi\}\; \{13\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{9\; \backslash \; pi\}\; \{13\}).\; ~$

Who can also put themselves in the form:

$2\; \backslash \; sin\; (\backslash \; frac\; \{7\; \backslash \; pi\}\; \{26\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{11\; \backslash \; pi\}\; \{26\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 5\; \backslash \; pi\}\; \{26\}).\; ~$

Polynomials of the fourth degree

$x^4\; -\; 4x^2\; +\; 2\; ~$

Is the minimal polynomial of the numbers:

$2\; \backslash \; cos\; (\backslash \; frac\; \{\backslash \; pi\}\; \{8\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{3\; \backslash \; pi\}\; \{8\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{5\; \backslash \; pi\}\; \{8\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{7\; \backslash \; pi\}\; \{8\}).\; ~$

Who can also put themselves in the form:

$2\; \backslash \; sin\; (\backslash \; frac\; \{\backslash \; pi\}\; \{8\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; \backslash \; pi\}\; \{8\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{3\; \backslash \; pi\}\; \{8\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 3\; \backslash \; pi\}\; \{8\}).\; ~$

$x^4\; -\; 4x^2\; +\; 2\; -\; \backslash \; sqrt\; \{2\}\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{2\})$ of the numbers:

$2\; \backslash \; cos\; (\backslash \; frac\; \{\backslash \; pi\}\; \{16\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{7\; \backslash \; pi\}\; \{16\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{9\; \backslash \; pi\}\; \{16\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{15\; \backslash \; pi\}\; \{16\}).\; ~$

Who can also put themselves in the form:

$2\; \backslash \; sin\; (\backslash \; frac\; \{\backslash \; pi\}\; \{16\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{7\; \backslash \; pi\}\; \{16\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; \backslash \; pi\}\; \{16\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 7\; \backslash \; pi\}\; \{16\}).\; ~$

$x^4\; -\; 4x^2\; +\; 2\; +\; \backslash \; sqrt\; \{2\}\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{2\})$ of the numbers:

$2\; \backslash \; cos\; (\backslash \; frac\; \{3\; \backslash \; pi\}\; \{16\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{5\; \backslash \; pi\}\; \{16\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{11\; \backslash \; pi\}\; \{16\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{13\; \backslash \; pi\}\; \{16\}).\; ~$

Who can also put themselves in the form:

$2\; \backslash \; sin\; (\backslash \; frac\; \{3\; \backslash \; pi\}\; \{16\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{5\; \backslash \; pi\}\; \{16\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 3\; \backslash \; pi\}\; \{16\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 5\; \backslash \; pi\}\; \{16\}).\; ~$

$x^4\; -\; 4x^2\; +\; 2\; -\; \backslash \; sqrt\; \{3\}\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{3\})$ of the numbers:

$2\; \backslash \; cos\; (\backslash \; frac\; \{\backslash \; pi\}\; \{24\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{11\; \backslash \; pi\}\; \{24\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{13\; \backslash \; pi\}\; \{24\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{23\; \backslash \; pi\}\; \{24\}).\; ~$

Who can also put themselves in the form:

$2\; \backslash \; sin\; (\backslash \; frac\; \{\backslash \; pi\}\; \{24\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{11\; \backslash \; pi\}\; \{24\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; \backslash \; pi\}\; \{24\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 11\; \backslash \; pi\}\; \{24\}).\; ~$

$x^4\; -\; 4x^2\; +\; 2\; +\; \backslash \; sqrt\; \{3\}\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{3\})$ of the numbers:

$2\; \backslash \; cos\; (\backslash \; frac\; \{5\; \backslash \; pi\}\; \{24\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{7\; \backslash \; pi\}\; \{24\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{17\; \backslash \; pi\}\; \{24\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{19\; \backslash \; pi\}\; \{24\}).\; ~$

Who can also put themselves in the form:

$2\; \backslash \; sin\; (\backslash \; frac\; \{5\; \backslash \; pi\}\; \{24\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{7\; \backslash \; pi\}\; \{24\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 5\; \backslash \; pi\}\; \{24\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 7\; \backslash \; pi\}\; \{24\}).\; ~$

$x^4\; -\; 5x^2\; +\; 5\; ~$

Is the minimal polynomial of the numbers:

$2\; \backslash \; cos\; (\backslash \; frac\; \{\backslash \; pi\}\; \{10\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{3\; \backslash \; pi\}\; \{10\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{7\; \backslash \; pi\}\; \{10\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{9\; \backslash \; pi\}\; \{10\}).\; ~$

Who can also put themselves in the form:

$2\; \backslash \; sin\; (\backslash \; frac\; \{\backslash \; pi\}\; \{5\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; \backslash \; pi\}\; \{5\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{2\; \backslash \; pi\}\; \{5\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 2\; \backslash \; pi\}\; \{5\}).\; ~$

$x^4\; -\; x^3\; \backslash \; sqrt\; \{5\}\; -\; 2x^2\; +\; 2x\; \backslash \; sqrt\; \{5\}\; +\; 5\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{5\})$ of the numbers:

$2\; \backslash \; cos\; (\backslash \; frac\; \{\backslash \; pi\}\; \{15\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{2\; \backslash \; pi\}\; \{15\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{8\; \backslash \; pi\}\; \{15\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{11\; \backslash \; pi\}\; \{15\}).\; ~$

Who can also put themselves in the form:

$2\; \backslash \; sin\; (\backslash \; frac\; \{-\; \backslash \; pi\}\; \{30\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 7\; \backslash \; pi\}\; \{30\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{11\; \backslash \; pi\}\; \{30\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{13\; \backslash \; pi\}\; \{30\}).\; ~$

$2x^4\; -\; (1\; -\; \backslash \; sqrt\; \{17\})\; x^3\; -\; (3\; +\; \backslash \; sqrt\; \{17\})\; x^2\; -\; 2\; (2\; +\; \backslash \; sqrt\; \{17\})\; X\; -\; 2\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{17\})$ of the numbers:

$2\; \backslash \; cos\; (\backslash \; frac\; \{\backslash \; pi\}\; \{17\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{9\; \backslash \; pi\}\; \{17\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{13\; \backslash \; pi\}\; \{17\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{15\; \backslash \; pi\}\; \{17\}).\; ~$

Who can also put themselves in the form:

$2\; \backslash \; sin\; (\backslash \; frac\; \{15\; \backslash \; pi\}\; \{34\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 1\; \backslash \; pi\}\; \{34\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 9\; \backslash \; pi\}\; \{34\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 13\; \backslash \; pi\}\; \{34\}).\; ~$

$2x^4\; -\; (1\; +\; \backslash \; sqrt\; \{17\})\; x^3\; -\; (3\; -\; \backslash \; sqrt\; \{17\})\; x^2\; -\; 2\; (2\; -\; \backslash \; sqrt\; \{17\})\; X\; -\; 2\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{17\})$ of the numbers:

$2\; \backslash \; cos\; (\backslash \; frac\; \{3\; \backslash \; pi\}\; \{17\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{5\; \backslash \; pi\}\; \{17\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{7\; \backslash \; pi\}\; \{17\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{11\; \backslash \; pi\}\; \{17\}).\; ~$

Who can also put themselves in the form:

$2\; \backslash \; sin\; (\backslash \; frac\; \{3\; \backslash \; pi\}\; \{34\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{7\; \backslash \; pi\}\; \{34\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{11\; \backslash \; pi\}\; \{34\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 5\; \backslash \; pi\}\; \{34\}).\; ~$

$2x^4\; +\; (1\; +\; \backslash \; sqrt\; \{17\})\; x^3\; -\; (3\; -\; \backslash \; sqrt\; \{17\})\; x^2\; +\; 2\; (2\; -\; \backslash \; sqrt\; \{17\})\; X\; -\; 2\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{17\})$ of the numbers:

$2\; \backslash \; cos\; (\backslash \; frac\; \{6\; \backslash \; pi\}\; \{17\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{10\; \backslash \; pi\}\; \{17\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{12\; \backslash \; pi\}\; \{17\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{14\; \backslash \; pi\}\; \{17\}).\; ~$

Who can also put themselves in the form:

$2\; \backslash \; sin\; (\backslash \; frac\; \{5\; \backslash \; pi\}\; \{34\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 3\; \backslash \; pi\}\; \{34\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 7\; \backslash \; pi\}\; \{34\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 11\; \backslash \; pi\}\; \{34\}).\; ~$

$2x^4\; +\; (1\; -\; \backslash \; sqrt\; \{17\})\; x^3\; -\; (3\; +\; \backslash \; sqrt\; \{17\})\; x^2\; +\; 2\; (2\; +\; \backslash \; sqrt\; \{17\})\; X\; -\; 2\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{17\})$ of the numbers:

$2\; \backslash \; cos\; (\backslash \; frac\; \{2\; \backslash \; pi\}\; \{17\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{4\; \backslash \; pi\}\; \{17\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{8\; \backslash \; pi\}\; \{17\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{16\; \backslash \; pi\}\; \{17\}).\; ~$

Who can also put themselves in the form:

$2\; \backslash \; sin\; (\backslash \; frac\; \{\backslash \; pi\}\; \{34\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{9\; \backslash \; pi\}\; \{34\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{13\; \backslash \; pi\}\; \{34\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 15\; \backslash \; pi\}\; \{34\}).\; ~$

$x^4\; +\; x^3\; \backslash \; sqrt\; \{5\}\; -\; 2x^2\; -\; 2x\; \backslash \; sqrt\; \{5\}\; +\; 5\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{5\})$ of the numbers:

$2\; \backslash \; cos\; (\backslash \; frac\; \{4\; \backslash \; pi\}\; \{15\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{7\; \backslash \; pi\}\; \{15\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{13\; \backslash \; pi\}\; \{15\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{14\; \backslash \; pi\}\; \{15\}).\; ~$

Who can also put themselves in the form:

$2\; \backslash \; sin\; (\backslash \; frac\; \{\backslash \; pi\}\; \{30\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{7\; \backslash \; pi\}\; \{30\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 11\; \backslash \; pi\}\; \{30\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 13\; \backslash \; pi\}\; \{30\}).\; ~$

$x^4\; -\; x^3\; \backslash \; sqrt\; \{2\}\; -\; 3x^2\; +\; 3x\; \backslash \; sqrt\; \{2\}\; -\; 1\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{2\})$ of the numbers:

$2\; \backslash \; cos\; (\backslash \; frac\; \{\backslash \; pi\}\; \{20\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{7\; \backslash \; pi\}\; \{20\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{9\; \backslash \; pi\}\; \{20\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{17\; \backslash \; pi\}\; \{20\}).\; ~$

Who can also put themselves in the form:

$2\; \backslash \; sin\; (\backslash \; frac\; \{\backslash \; pi\}\; \{20\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{3\; \backslash \; pi\}\; \{20\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 7\; \backslash \; pi\}\; \{20\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{9\; \backslash \; pi\}\; \{20\}).\; ~$

$x^4\; +\; x^3\; \backslash \; sqrt\; \{2\}\; -\; 3x^2\; -\; 3x\; \backslash \; sqrt\; \{2\}\; -\; 1\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{2\})$ of the numbers:

$2\; \backslash \; cos\; (\backslash \; frac\; \{3\; \backslash \; pi\}\; \{20\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{11\; \backslash \; pi\}\; \{20\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{13\; \backslash \; pi\}\; \{20\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{19\; \backslash \; pi\}\; \{20\}).\; ~$

Who can also put themselves in the form:

$2\; \backslash \; sin\; (\backslash \; frac\; \{-\; \backslash \; pi\}\; \{20\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 3\; \backslash \; pi\}\; \{20\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{7\; \backslash \; pi\}\; \{20\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 9\; \backslash \; pi\}\; \{20\}).\; ~$

$x^4\; -\; x^3\; \backslash \; sqrt\; \{15\}\; +\; 4x^2\; -\; 1\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{15\})$ of the numbers:

$2\; \backslash \; cos\; (\backslash \; frac\; \{\backslash \; pi\}\; \{30\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{7\; \backslash \; pi\}\; \{30\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{11\; \backslash \; pi\}\; \{30\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{17\; \backslash \; pi\}\; \{30\}).\; ~$

Who can also put themselves in the form:

$2\; \backslash \; sin\; (\backslash \; frac\; \{-\; \backslash \; pi\}\; \{15\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{2\; \backslash \; pi\}\; \{15\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{4\; \backslash \; pi\}\; \{15\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{7\; \backslash \; pi\}\; \{15\}).\; ~$

$x^4\; +\; x^3\; \backslash \; sqrt\; \{15\}\; +\; 4x^2\; -\; 1\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{15\})$ of the numbers:

$2\; \backslash \; cos\; (\backslash \; frac\; \{13\; \backslash \; pi\}\; \{30\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{19\; \backslash \; pi\}\; \{30\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{23\; \backslash \; pi\}\; \{30\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{29\; \backslash \; pi\}\; \{30\}).\; ~$

Who can also put themselves in the form:

$2\; \backslash \; sin\; (\backslash \; frac\; \{\backslash \; pi\}\; \{15\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 2\; \backslash \; pi\}\; \{15\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 4\; \backslash \; pi\}\; \{15\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 7\; \backslash \; pi\}\; \{15\}).\; ~$

Polynomials of the fifth degree

$x^5\; +\; x^4\; -\; 4x^3\; -\; 3x^2\; +\; 3x\; +\; 1\; ~$

Is the minimal polynomial of the numbers:

$2\; \backslash \; cos\; (\backslash \; frac\; \{2\; \backslash \; pi\}\; \{11\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{4\; \backslash \; pi\}\; \{11\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{6\; \backslash \; pi\}\; \{11\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{8\; \backslash \; pi\}\; \{11\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{10\; \backslash \; pi\}\; \{11\}).\; ~$

Who can also put themselves in the form:

$2\; \backslash \; sin\; (\backslash \; frac\; \{-\; \backslash \; pi\}\; \{22\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{3\; \backslash \; pi\}\; \{22\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 5\; \backslash \; pi\}\; \{22\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{7\; \backslash \; pi\}\; \{22\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 9\; \backslash \; pi\}\; \{22\}).\; ~$

$x^5\; -\; x^4\; -\; 4x^3\; +\; 3x^2\; +\; 3x\; -\; 1\; ~$

Is the minimal polynomial of the numbers:

$2\; \backslash \; cos\; (\backslash \; frac\; \{\backslash \; pi\}\; \{11\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{3\; \backslash \; pi\}\; \{11\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{5\; \backslash \; pi\}\; \{11\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{7\; \backslash \; pi\}\; \{11\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{9\; \backslash \; pi\}\; \{11\}).\; ~$

Who can also put themselves in the form:

$2\; \backslash \; sin\; (\backslash \; frac\; \{\backslash \; pi\}\; \{22\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 3\; \backslash \; pi\}\; \{22\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{5\; \backslash \; pi\}\; \{22\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 7\; \backslash \; pi\}\; \{22\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{9\; \backslash \; pi\}\; \{22\}).\; ~$

$x^5\; +\; x^4\; \backslash \; sqrt\; \{11\}\; -\; 3x^2\; \backslash \; sqrt\; \{11\}\; -\; 11x\; -\; \backslash \; sqrt\; \{11\}\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{11\})$ of the numbers:

$2\; \backslash \; cos\; (\backslash \; frac\; \{3\; \backslash \; pi\}\; \{22\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{13\; \backslash \; pi\}\; \{22\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{15\; \backslash \; pi\}\; \{22\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{17\; \backslash \; pi\}\; \{22\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{21\; \backslash \; pi\}\; \{22\}).\; ~$

Who can also put themselves in the form:

$2\; \backslash \; sin\; (\backslash \; frac\; \{-\; \backslash \; pi\}\; \{11\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 2\; \backslash \; pi\}\; \{11\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 3\; \backslash \; pi\}\; \{11\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{4\; \backslash \; pi\}\; \{11\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 5\; \backslash \; pi\}\; \{11\}).\; ~$

$x^5\; -\; x^4\; \backslash \; sqrt\; \{11\}\; +\; 3x^2\; \backslash \; sqrt\; \{11\}\; -\; 11x\; +\; \backslash \; sqrt\; \{11\}\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{11\})$ of the numbers:

$2\; \backslash \; cos\; (\backslash \; frac\; \{\backslash \; pi\}\; \{22\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{5\; \backslash \; pi\}\; \{22\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{7\; \backslash \; pi\}\; \{22\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{9\; \backslash \; pi\}\; \{22\}),\; \backslash \; qquad\; 2\; \backslash \; cos\; (\backslash \; frac\; \{19\; \backslash \; pi\}\; \{22\}).\; ~$

Who can also put themselves in the form:

$2\; \backslash \; sin\; (\backslash \; frac\; \{\backslash \; pi\}\; \{11\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{2\; \backslash \; pi\}\; \{11\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{3\; \backslash \; pi\}\; \{11\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{-\; 4\; \backslash \; pi\}\; \{11\}),\; \backslash \; qquad\; 2\; \backslash \; sin\; (\backslash \; frac\; \{5\; \backslash \; pi\}\; \{11\}).\; ~$

Etc…

Polynomials minimals of number of the tan form (k$\backslash \; pi$/n).

Below are, in the order of degree crescents, the first polynomials minimal of the numbers of the tan form (kπ/n).

Polynomials of the first degree

$X\; -\; 1\; \backslash \; qquad\; X\; +\; 1$

are respectively the minimal polynomials of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{\backslash \; pi\}\; \{4\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{3\; \backslash \; pi\}\; \{4\}).\; ~$

$X\; -\; \backslash \; sqrt\; \{3\}\; \backslash \; qquad\; X\; +\; \backslash \; sqrt\; \{3\}\; \backslash \; qquad\; X\; -\; \backslash \; frac\; \{1\}\; \{\backslash \; sqrt\; \{3\}\}\; \backslash \; qquad\; X\; +\; \backslash \; frac\; \{1\}\; \{\backslash \; sqrt\; \{3\}\}$

are respectively the minimal polynomials in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{3\})$ of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{\backslash \; pi\}\; \{3\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{2\; \backslash \; pi\}\; \{3\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{\backslash \; pi\}\; \{6\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{5\; \backslash \; pi\}\; \{6\}).\; ~$

Polynomials of the second degree

$x^2\; -\; 4x\; +\; 1\; ~$

Is the minimal polynomial of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{\backslash \; pi\}\; \{12\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{5\; \backslash \; pi\}\; \{12\}).\; ~$

$x^2\; +\; 4x\; +\; 1\; ~$

Is the minimal polynomial of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{7\; \backslash \; pi\}\; \{12\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{11\; \backslash \; pi\}\; \{12\}).\; ~$

$x^2\; -\; 2x\; \backslash \; sqrt\; \{2\}\; +\; 1\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{2\})$ of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{\backslash \; pi\}\; \{8\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{3\; \backslash \; pi\}\; \{8\}).\; ~$

$x^2\; +\; 2x\; \backslash \; sqrt\; \{2\}\; +\; 1\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{2\})$ of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{5\; \backslash \; pi\}\; \{8\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{7\; \backslash \; pi\}\; \{8\}).\; ~$

Polynomials of the third degree

$x^3\; +\; x^2\; \backslash \; sqrt\; \{7\}\; -\; 7x\; +\; \backslash \; sqrt\; \{7\}\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{7\})$ of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{\backslash \; pi\}\; \{7\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{2\; \backslash \; pi\}\; \{7\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{4\; \backslash \; pi\}\; \{7\}).\; ~$

$x^3\; -\; x^2\; \backslash \; sqrt\; \{7\}\; -\; 7x\; -\; \backslash \; sqrt\; \{7\}\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{7\})$ of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{3\; \backslash \; pi\}\; \{7\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{5\; \backslash \; pi\}\; \{7\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{6\; \backslash \; pi\}\; \{7\}).\; ~$

$x^3\; -\; 3x^2\; \backslash \; sqrt\; \{3\}\; -\; 3x\; +\; \backslash \; sqrt\; \{3\}\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{3\})$ of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{\backslash \; pi\}\; \{9\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{4\; \backslash \; pi\}\; \{9\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{7\; \backslash \; pi\}\; \{7\}).\; ~$

$x^3\; +\; 3x^2\; \backslash \; sqrt\; \{3\}\; -\; 3x\; -\; \backslash \; sqrt\; \{3\}\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{3\})$ of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{2\; \backslash \; pi\}\; \{9\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{5\; \backslash \; pi\}\; \{9\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{8\; \backslash \; pi\}\; \{7\}).\; ~$

$x^3\; +\; x^2\; \backslash \; sqrt\; \{7\}\; +\; X\; -\; \backslash \; frac\; \{1\}\; \{\backslash \; sqrt\; \{7\}\}\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{7\})$ of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{\backslash \; pi\}\; \{14\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{9\; \backslash \; pi\}\; \{14\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{11\; \backslash \; pi\}\; \{14\}).\; ~$

$x^3\; -\; x^2\; \backslash \; sqrt\; \{7\}\; +\; X\; +\; \backslash \; frac\; \{1\}\; \{\backslash \; sqrt\; \{7\}\}\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{7\})$ of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{3\; \backslash \; pi\}\; \{14\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{5\; \backslash \; pi\}\; \{14\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{13\; \backslash \; pi\}\; \{14\}).\; ~$

$x^3\; -\; x^2\; \backslash \; sqrt\; \{3\}\; -\; 3x\; +\; \backslash \; frac\; \{1\}\; \{\backslash \; sqrt\; \{3\}\}\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{3\})$ of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{\backslash \; pi\}\; \{18\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{13\; \backslash \; pi\}\; \{18\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{7\; \backslash \; pi\}\; \{18\}).\; ~$

$x^3\; +\; x^2\; \backslash \; sqrt\; \{3\}\; -\; 3x\; -\; \backslash \; frac\; \{1\}\; \{\backslash \; sqrt\; \{3\}\}\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{3\})$ of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{5\; \backslash \; pi\}\; \{18\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{11\; \backslash \; pi\}\; \{18\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{17\; \backslash \; pi\}\; \{18\}).\; ~$

$(2\; \backslash \; sqrt\; \{7\}\; +\; 3\; \backslash \; sqrt\; \{3\})\; x^3\; +\; x^2\; -\; (2\; \backslash \; sqrt\; \{7\}\; +\; \backslash \; sqrt\; \{3\})\; X\; +\; 1\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{3\})\; (\backslash \; sqrt\; \{7\})$ of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{\backslash \; pi\}\; \{21\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{4\; \backslash \; pi\}\; \{21\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{16\; \backslash \; pi\}\; \{21\}).\; ~$

$(2\; \backslash \; sqrt\; \{7\}\; -\; 3\; \backslash \; sqrt\; \{3\})\; x^3\; +\; x^2\; -\; (2\; \backslash \; sqrt\; \{7\}\; -\; \backslash \; sqrt\; \{3\})\; X\; +\; 1\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{3\})\; (\backslash \; sqrt\; \{7\})$ of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{2\; \backslash \; pi\}\; \{21\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{8\; \backslash \; pi\}\; \{21\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{11\; \backslash \; pi\}\; \{21\}).\; ~$

$(2\; \backslash \; sqrt\; \{7\}\; -\; 3\; \backslash \; sqrt\; \{3\})\; x^3\; -\; x^2\; -\; (2\; \backslash \; sqrt\; \{7\}\; -\; \backslash \; sqrt\; \{3\})\; X\; -\; 1\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{3\})\; (\backslash \; sqrt\; \{7\})$ of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{10\; \backslash \; pi\}\; \{21\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{13\; \backslash \; pi\}\; \{21\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{19\; \backslash \; pi\}\; \{21\}).\; ~$

$(2\; \backslash \; sqrt\; \{7\}\; +\; 3\; \backslash \; sqrt\; \{3\})\; x^3\; -\; x^2\; -\; (2\; \backslash \; sqrt\; \{7\}\; +\; \backslash \; sqrt\; \{3\})\; X\; -\; 1\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{3\})\; (\backslash \; sqrt\; \{7\})$ of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{5\; \backslash \; pi\}\; \{21\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{17\; \backslash \; pi\}\; \{21\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{26\; \backslash \; pi\}\; \{21\}).\; ~$

$x^3\; -\; (4\; -\; \backslash \; sqrt\; \{7\})\; x^2\; -\; (11\; -\; 4\; \backslash \; sqrt\; \{7\})\; X\; +\; 8\; -\; 3\; \backslash \; sqrt\; \{7\}\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{7\})$ of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{\backslash \; pi\}\; \{28\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{9\; \backslash \; pi\}\; \{28\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{25\; \backslash \; pi\}\; \{28\}).\; ~$

$x^3\; -\; (4\; +\; \backslash \; sqrt\; \{7\})\; x^2\; -\; (11\; +\; 4\; \backslash \; sqrt\; \{7\})\; X\; +\; 8\; +\; 3\; \backslash \; sqrt\; \{7\}\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{7\})$ of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{5\; \backslash \; pi\}\; \{28\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{13\; \backslash \; pi\}\; \{28\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{17\; \backslash \; pi\}\; \{28\}).\; ~$

$x^3\; +\; (4\; +\; \backslash \; sqrt\; \{7\})\; x^2\; -\; (11\; +\; 4\; \backslash \; sqrt\; \{7\})\; X\; -\; 8\; -\; 3\; \backslash \; sqrt\; \{7\}\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{7\})$ of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{11\; \backslash \; pi\}\; \{28\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{15\; \backslash \; pi\}\; \{28\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{23\; \backslash \; pi\}\; \{28\}).\; ~$

$x^3\; +\; (4\; -\; \backslash \; sqrt\; \{7\})\; x^2\; -\; (11\; -\; 4\; \backslash \; sqrt\; \{7\})\; X\; -\; 8\; +\; 3\; \backslash \; sqrt\; \{7\}\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{7\})$ of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{3\; \backslash \; pi\}\; \{28\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{19\; \backslash \; pi\}\; \{28\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{27\; \backslash \; pi\}\; \{28\}).\; ~$

$x^3\; -\; 3\; (2\; -\; \backslash \; sqrt\; \{3\})\; x^2\; -\; 3x\; +\; 2\; -\; \backslash \; sqrt\; \{3\}\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{3\})$ of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{\backslash \; pi\}\; \{36\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{13\; \backslash \; pi\}\; \{36\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{25\; \backslash \; pi\}\; \{36\}).\; ~$

$x^3\; -\; 3\; (2\; +\; \backslash \; sqrt\; \{3\})\; x^2\; -\; 3x\; +\; 2\; +\; \backslash \; sqrt\; \{3\}\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{3\})$ of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{5\; \backslash \; pi\}\; \{36\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{17\; \backslash \; pi\}\; \{36\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{29\; \backslash \; pi\}\; \{36\}).\; ~$

$x^3\; +\; 3\; (2\; +\; \backslash \; sqrt\; \{3\})\; x^2\; -\; 3x\; -\; 2\; -\; \backslash \; sqrt\; \{3\}\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{3\})$ of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{7\; \backslash \; pi\}\; \{36\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{19\; \backslash \; pi\}\; \{36\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{31\; \backslash \; pi\}\; \{36\}).\; ~$

$x^3\; +\; 3\; (2\; -\; \backslash \; sqrt\; \{3\})\; x^2\; -\; 3x\; -\; 2\; +\; \backslash \; sqrt\; \{3\}\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{3\})$ of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{11\; \backslash \; pi\}\; \{36\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{23\; \backslash \; pi\}\; \{36\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{35\; \backslash \; pi\}\; \{36\}).\; ~$

Polynomials of the fourth degree

$x^4\; -\; 10x^2\; +\; 5\; ~$

Is the minimal polynomial of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{\backslash \; pi\}\; \{5\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{2\; \backslash \; pi\}\; \{5\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{3\; \backslash \; pi\}\; \{5\})\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{4\; \backslash \; pi\}\; \{5\}).\; ~$

$x^4\; -\; 2x^2\; +\; 2\; ~$

Is the minimal polynomial of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{\backslash \; pi\}\; \{10\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{3\; \backslash \; pi\}\; \{10\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{5\; \backslash \; pi\}\; \{10\})\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{7\; \backslash \; pi\}\; \{10\}).\; ~$

$x^4\; -\; 4x^3\; -\; 14x^2\; -\; 4x\; +\; 1\; ~$

Is the minimal polynomial of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{\backslash \; pi\}\; \{20\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{9\; \backslash \; pi\}\; \{20\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{13\; \backslash \; pi\}\; \{20\})\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{17\; \backslash \; pi\}\; \{20\}).\; ~$

$x^4\; +\; 4x^3\; -\; 14x^2\; +\; 4x\; +\; 1\; ~$

Is the minimal polynomial of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{3\; \backslash \; pi\}\; \{20\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{7\; \backslash \; pi\}\; \{20\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{11\; \backslash \; pi\}\; \{20\})\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{19\; \backslash \; pi\}\; \{20\}).\; ~$

$x^4\; +\; 8x^3\; +\; 2x^2\; -\; 8x\; +\; 1\; ~$

Is the minimal polynomial of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{\backslash \; pi\}\; \{24\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{5\; \backslash \; pi\}\; \{24\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{13\; \backslash \; pi\}\; \{24\})\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{17\; \backslash \; pi\}\; \{24\}).\; ~$

$x^4\; -\; 8x^3\; +\; 2x^2\; +\; 8x\; +\; 1\; ~$

Is the minimal polynomial of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{7\; \backslash \; pi\}\; \{24\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{11\; \backslash \; pi\}\; \{24\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{19\; \backslash \; pi\}\; \{24\})\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{23\; \backslash \; pi\}\; \{24\}).\; ~$

$x^4\; +\; 4x^3\; \backslash \; sqrt\; \{2\}\; +\; 2x^2\; -\; 4x\; \backslash \; sqrt\; \{2\}\; +\; 1\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{2\})$ of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{\backslash \; pi\}\; \{16\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{3\; \backslash \; pi\}\; \{16\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{9\; \backslash \; pi\}\; \{16\})\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{11\; \backslash \; pi\}\; \{16\}).\; ~$

$x^4\; -\; 4x^3\; \backslash \; sqrt\; \{2\}\; +\; 2x^2\; +\; 4x\; \backslash \; sqrt\; \{2\}\; +\; 1\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{2\})$ of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{5\; \backslash \; pi\}\; \{16\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{7\; \backslash \; pi\}\; \{16\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{13\; \backslash \; pi\}\; \{16\})\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{15\; \backslash \; pi\}\; \{16\}).\; ~$

$x^4\; +\; 6x^3\; \backslash \; sqrt\; \{3\}\; +\; 8x^2\; -\; 2x\; \backslash \; sqrt\; \{3\}\; -\; 1\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{3\})$ of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{2\; \backslash \; pi\}\; \{15\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{8\; \backslash \; pi\}\; \{15\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{11\; \backslash \; pi\}\; \{15\})\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{14\; \backslash \; pi\}\; \{15\}).\; ~$

$x^4\; -\; 6x^3\; \backslash \; sqrt\; \{3\}\; +\; 8x^2\; +\; 2x\; \backslash \; sqrt\; \{3\}\; -\; 1\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{3\})$ of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{\backslash \; pi\}\; \{15\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{4\; \backslash \; pi\}\; \{15\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{7\; \backslash \; pi\}\; \{15\})\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{13\; \backslash \; pi\}\; \{15\}).\; ~$

$x^4\; -\; 2x^3\; \backslash \; sqrt\; \{3\}\; -\; 8x^2\; +\; 6x\; \backslash \; sqrt\; \{3\}\; -\; 1\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{3\})$ of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{\backslash \; pi\}\; \{30\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{7\; \backslash \; pi\}\; \{30\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{13\; \backslash \; pi\}\; \{30\})\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{19\; \backslash \; pi\}\; \{30\}).\; ~$

$x^4\; +\; 2x^3\; \backslash \; sqrt\; \{3\}\; -\; 8x^2\; -\; 6x\; \backslash \; sqrt\; \{3\}\; -\; 1\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{3\})$ of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{11\; \backslash \; pi\}\; \{30\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{17\; \backslash \; pi\}\; \{30\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{23\; \backslash \; pi\}\; \{30\})\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{29\; \backslash \; pi\}\; \{30\}).\; ~$

Polynomials of the fifth degree

$x^5\; +\; 3x^4\; \backslash \; sqrt\; \{11\}\; +\; 22x^3\; +\; 2x^2\; \backslash \; sqrt\; \{11\}\; -\; 11x\; -\; \backslash \; sqrt\; \{11\}\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{11\})$ of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{2\; \backslash \; pi\}\; \{11\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{6\; \backslash \; pi\}\; \{11\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{7\; \backslash \; pi\}\; \{11\})\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{8\; \backslash \; pi\}\; \{11\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{10\; \backslash \; pi\}\; \{11\}).\; ~$

$x^5\; -\; 3x^4\; \backslash \; sqrt\; \{11\}\; +\; 22x^3\; -\; 2x^2\; \backslash \; sqrt\; \{11\}\; -\; 11x\; +\; \backslash \; sqrt\; \{11\}\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{11\})$ of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{\backslash \; pi\}\; \{11\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{3\; \backslash \; pi\}\; \{11\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{4\; \backslash \; pi\}\; \{11\})\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{5\; \backslash \; pi\}\; \{11\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{9\; \backslash \; pi\}\; \{11\}).\; ~$

$x^5\; -\; x^4\; \backslash \; sqrt\; \{11\}\; -\; 2x^3\; +\; 2x^2\; \backslash \; sqrt\; \{11\}\; -\; 3x\; +\; \backslash \; frac\; \{1\}\; \{\backslash \; sqrt\; \{11\}\}\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{11\})$ of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{\backslash \; pi\}\; \{22\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{3\; \backslash \; pi\}\; \{22\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{5\; \backslash \; pi\}\; \{22\})\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{9\; \backslash \; pi\}\; \{22\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{15\; \backslash \; pi\}\; \{22\}).\; ~$

$x^5\; +\; x^4\; \backslash \; sqrt\; \{11\}\; -\; 2x^3\; -\; 2x^2\; \backslash \; sqrt\; \{11\}\; -\; 3x\; -\; \backslash \; frac\; \{1\}\; \{\backslash \; sqrt\; \{11\}\}\; ~$

Is the minimal polynomial in $\backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{11\})$ of the numbers:

$\backslash \; tan\; (\backslash \; frac\; \{7\; \backslash \; pi\}\; \{22\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{13\; \backslash \; pi\}\; \{22\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{17\; \backslash \; pi\}\; \{22\})\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{19\; \backslash \; pi\}\; \{22\}),\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; frac\; \{21\; \backslash \; pi\}\; \{22\}).\; ~$

Etc…

General formulas in relation to the coefficients of the trigonometrical minimal polynomials.

One will find below some formulas relating to the Relations between coefficients and roots of a polynomial canceler of the trigonometrical numbers treated in this article. Some of these formulas reveal to the second member the root of a divider of the denominator being in the argument of the goniometrical functions to the first member. What makes it possible to foresee the reasons for which, one finds in the coefficients of the minimal polynomial of such roots.

Formulas of summations

$\backslash \; forall\; N\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\; ^*\; \backslash \; qquad\; \backslash \; sum\_\; \{k=1\}\; ^n\; \backslash \; cos\; \backslash \; left\; (\backslash \; frac\; \{2k\; \backslash \; pi\}\; \{2n+1\}\; \backslash \; right)\; =\; -\; \backslash \; frac\; \{1\}\; \{2\}\; ~$

$\backslash \; forall\; N\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\; ^*\; \backslash \; qquad\; \backslash \; sum\_\; \{k=1\}\; ^n\; \backslash \; cos\; \backslash \; left\; (\backslash \; frac\; \{(2k-1)\; \backslash \; pi\}\; \{2n\}\; \backslash \; right)\; =\; 0\; ~$

Formulas of products

(In what follows Ent (X) indicates the whole part of X)

$\backslash \; forall\; N\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\; ^*\; \backslash \; qquad\; \backslash \; prod\_\; \{k=1\}\; ^n\; \backslash \; cos\; \backslash \; left\; (\backslash \; frac\; \{K\; \backslash \; pi\}\; \{2n+1\}\; \backslash \; right)\; =\; \backslash \; frac\; \{1\}\; \{2^n\}\; ~$

$\backslash \; forall\; N\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\; ^*\; \backslash \; qquad\; \backslash \; prod\_\; \{k=1\}\; ^n\; \backslash \; sin\; \backslash \; left\; (\backslash \; frac\; \{K\; \backslash \; pi\}\; \{2n+1\}\; \backslash \; right)\; =\; \backslash \; frac\; \{\backslash \; sqrt\; \{2n+1\}\}\; \{2^n\}\; ~$

$\backslash \; forall\; N\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\; ^*\; \backslash \; qquad\; \backslash \; prod\_\; \{k=1\}\; ^n\; \backslash \; tan\; \backslash \; left\; (\backslash \; frac\; \{K\; \backslash \; pi\}\; \{2n+1\}\; \backslash \; right)\; =\; \backslash \; sqrt\; \{2n+1\}\; ~$

$\backslash \; forall\; N\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\; ^*\; \backslash \; qquad\; \backslash \; prod\_\; \{k=1\}\; ^n\; \backslash \; cos\; \backslash \; left\; (\backslash \; frac\; \{2k\; \backslash \; pi\}\; \{2n+1\}\; \backslash \; right)\; =\; \backslash \; frac\; \{(-\; 1)\; ^\; \{Ent\; (\backslash \; frac\; \{n-1\}\; \{2\})\}\}\{2^n\}\; ~$

$\backslash \; forall\; N\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\; ^*\; \backslash \; qquad\; \backslash \; prod\_\; \{k=1\}\; ^n\; \backslash \; sin\; \backslash \; left\; (\backslash \; frac\; \{2k\; \backslash \; pi\}\; \{2n+1\}\; \backslash \; right)\; =\; \backslash \; frac\; \{\backslash \; sqrt\; \{2n+1\}\}\; \{2^n\}\; ~$

$\backslash \; forall\; N\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\; ^*\; \backslash \; qquad\; \backslash \; prod\_\; \{k=1\}\; ^n\; \backslash \; tan\; \backslash \; left\; (\backslash \; frac\; \{2k\; \backslash \; pi\}\; \{2n+1\}\; \backslash \; right)\; =\; (-\; 1)\; ^\; \{Ent\; (\backslash \; frac\; \{n-1\}\; \{2\})\}\backslash sqrt\{2n+1\}\; ~$

$\backslash \; forall\; N\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\; -\; \backslash \; left\; \backslash \; \{0,1\; \backslash \; right\; \backslash \}\; \backslash \; qquad\; \backslash \; prod\_\; \{k=1\}\; ^\; \{n-1\}\; \backslash \; cos\; \backslash \; left\; (\backslash \; frac\; \{K\; \backslash \; pi\}\; \{2n\}\; \backslash \; right)\; =\; \backslash \; frac\; \{\backslash \; sqrt\; \{N\}\}\; \{2^\; \{n-1\}\}\; ~$

$\backslash \; forall\; N\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\; -\; \backslash \; left\; \backslash \; \{0,1\; \backslash \; right\; \backslash \}\; \backslash \; qquad\; \backslash \; prod\_\; \{k=1\}\; ^\; \{n-1\}\; \backslash \; sin\; \backslash \; left\; (\backslash \; frac\; \{K\; \backslash \; pi\}\; \{2n\}\; \backslash \; right)\; =\; \backslash \; frac\; \{\backslash \; sqrt\; \{N\}\}\; \{2^\; \{n-1\}\}\; ~$

$\backslash \; forall\; N\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\; -\; \backslash \; left\; \backslash \; \{0,1\; \backslash \; right\; \backslash \}\; \backslash \; qquad\; \backslash \; prod\_\; \{k=1\}\; ^\; \{n-1\}\; \backslash \; tan\; \backslash \; left\; (\backslash \; frac\; \{K\; \backslash \; pi\}\; \{2n\}\; \backslash \; right)\; =\; 1\; ~$

$\backslash \; forall\; N\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\; ^*\; \backslash \; qquad\; \backslash \; prod\_\; \{k=1\}\; ^\; \{2n\}\; \backslash \; cos\; \backslash \; left\; (\backslash \; frac\; \{K\; \backslash \; pi\}\; \{2n+1\}\; \backslash \; right)\; =\; \backslash \; frac\; \{1\}\; \{(-\; 4)\; ^n\}\; ~$

$\backslash \; forall\; N\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\; ^*\; \backslash \; qquad\; \backslash \; prod\_\; \{k=1\}\; ^\; \{2n\}\; \backslash \; sin\; \backslash \; left\; (\backslash \; frac\; \{K\; \backslash \; pi\}\; \{2n+1\}\; \backslash \; right)\; =\; \backslash \; frac\; \{N\}\; \{4^\; \{n-1\}\}\; ~$

$\backslash \; forall\; N\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\; ^*\; \backslash \; qquad\; \backslash \; prod\_\; \{k=1\}\; ^\; \{2n\}\; \backslash \; tan\; \backslash \; left\; (\backslash \; frac\; \{K\; \backslash \; pi\}\; \{2n+1\}\; \backslash \; right)\; =\; (-\; 1)\; ^n\; (2n+1)\; ~$

$\backslash \; forall\; N\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\; ^*\; \backslash \; qquad\; \backslash \; prod\_\; \{k=1\}\; ^\; \{N\}\; \backslash \; cos\; \backslash \; left\; (\backslash \; frac\; \{(2k-1)\; \backslash \; pi\}\; \{4n\}\; \backslash \; right)\; =\; \backslash \; frac\; \{\backslash \; sqrt\; \{2\}\}\; \{2^n\}\; ~$

$\backslash \; forall\; N\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\; ^*\; \backslash \; qquad\; \backslash \; prod\_\; \{k=1\}\; ^\; \{N\}\; \backslash \; sin\; \backslash \; left\; (\backslash \; frac\; \{(2k-1)\; \backslash \; pi\}\; \{4n\}\; \backslash \; right)\; =\; \backslash \; frac\; \{\backslash \; sqrt\; \{2\}\}\; \{2^n\}\; ~$

$\backslash \; forall\; N\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\; ^*\; \backslash \; qquad\; \backslash \; prod\_\; \{k=1\}\; ^\; \{N\}\; \backslash \; tan\; \backslash \; left\; (\backslash \; frac\; \{(2k-1)\; \backslash \; pi\}\; \{4n\}\; \backslash \; right)\; =\; 1\; ~$

$\backslash \; forall\; N\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\; ^*\; \backslash \; qquad\; \backslash \; prod\_\; \{k=1\}\; ^\; \{N\}\; \backslash \; cos\; \backslash \; left\; (\backslash \; frac\; \{(2k-1)\; \backslash \; pi\}\; \{4n+2\}\; \backslash \; right)\; =\; \backslash \; frac\; \{\backslash \; sqrt\; \{2n+1\}\}\; \{2^n\}\; ~$

$\backslash \; forall\; N\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\; ^*\; \backslash \; qquad\; \backslash \; prod\_\; \{k=1\}\; ^\; \{N\}\; \backslash \; sin\; \backslash \; left\; (\backslash \; frac\; \{(2k-1)\; \backslash \; pi\}\; \{4n+2\}\; \backslash \; right)\; =\; \backslash \; frac\; \{1\}\; \{2^n\}\; ~$

$\backslash \; forall\; N\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\; ^*\; \backslash \; qquad\; \backslash \; prod\_\; \{k=1\}\; ^\; \{N\}\; \backslash \; tan\; \backslash \; left\; (\backslash \; frac\; \{(2k-1)\; \backslash \; pi\}\; \{4n+2\}\; \backslash \; right)\; =\; \backslash \; frac\; \{1\}\; \{\backslash \; sqrt\; \{2n+1\}\}\; ~$

In the continuation, one will indicate by P (N), the whole of the integers lower than N and first with N. For example:

$P\; (15)\; =\; \backslash \; left\; \backslash \; \{1,2,4,7,8,11,13,14\; \backslash \; right\; \backslash \}$

One will also indicate by $\backslash \; mathbb\; \{P\}$, the whole of the prime numbers.

One has then:

$\backslash \; forall\; N\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\; -\; \backslash \; left\; \backslash \; \{0,1\; \backslash \; right\; \backslash \}\; \backslash \; qquad\; \backslash \; prod\_\; \{K\; \backslash \; in\; P\; (4n)\}\; \backslash \; tan\; \backslash \; left\; (\backslash \; frac\; \{K\; \backslash \; pi\}\; \{4n\}\; \backslash \; right)\; =\; 1\; ~$

$\backslash \; forall\; p\; \backslash \; in\; \backslash \; mathbb\; \{P\}\; -\; \backslash \; left\; \backslash \; \{2\; \backslash \; right\; \backslash \},\; \backslash \; forall\; N\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\; ^*\; \backslash \; qquad\; \backslash \; prod\_\; \{K\; \backslash \; in\; P\; (p^n)\}\; \backslash \; tan\; \backslash \; left\; (\backslash \; frac\; \{K\; \backslash \; pi\}\; \{p^n\}\; \backslash \; right)\; =\; (-\; 1)\; ^\; \backslash \; frac\; \{p-1\}\; \{2\}\; p\; ~$

$\backslash \; forall\; p\; \backslash \; in\; \backslash \; mathbb\; \{P\}\; -\; \backslash \; left\; \backslash \; \{2\; \backslash \; right\; \backslash \},\; \backslash \; forall\; N\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\; ^*\; \backslash \; qquad\; \backslash \; prod\_\; \{K\; \backslash \; in\; P\; (2p^n)\}\; \backslash \; tan\; \backslash \; left\; (\backslash \; frac\; \{K\; \backslash \; pi\}\; \{2p^n\}\; \backslash \; right)\; =\; \backslash \; frac\; \{(-\; 1)\; ^\; \backslash \; frac\; \{p-1\}\; \{2\}\}\; \{p\}\; ~$

For any N having at least 2 prime numbers odd in its decomposition in factors first, one a:

$\backslash \; prod\_\; \{K\; \backslash \; in\; P\; (N)\}\; \backslash \; tan\; \backslash \; left\; (\backslash \; frac\; \{K\; \backslash \; pi\}\; \{N\}\; \backslash \; right)\; =\; 1\; ~$

For any N which is not a power of 2, one a:

$\backslash \; prod\_\; \{K\; \backslash \; in\; P\; (4n)\}\; 2\; \backslash \; cos\; \backslash \; left\; (\backslash \; frac\; \{K\; \backslash \; pi\}\; \{4n\}\; \backslash \; right)\; =\; 1\; ~$

$\backslash \; forall\; N\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\; -\; \backslash \; left\; \backslash \; \{0,1,2\; \backslash \; right\; \backslash \}\; \backslash \; qquad\; \backslash \; prod\_\; \{K\; \backslash \; in\; P\; (2^n)\}\; 2\; \backslash \; cos\; \backslash \; left\; (\backslash \; frac\; \{K\; \backslash \; pi\}\; \{2^n\}\; \backslash \; right)\; =\; 2\; ~$

$\backslash \; forall\; p\; \backslash \; in\; \backslash \; mathbb\; \{P\}\; -\; \backslash \; left\; \backslash \; \{2\; \backslash \; right\; \backslash \},\; \backslash \; forall\; N\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\; ^*\; \backslash \; qquad\; \backslash \; prod\_\; \{K\; \backslash \; in\; P\; (p^n)\}\; 2\; \backslash \; cos\; \backslash \; left\; (\backslash \; frac\; \{K\; \backslash \; pi\}\; \{p^n\}\; \backslash \; right)\; =\; (-\; 1)\; ^\; \backslash \; frac\; \{p-1\}\; \{2\}\; ~$

$\backslash \; forall\; p\; \backslash \; in\; \backslash \; mathbb\; \{P\}\; -\; \backslash \; left\; \backslash \; \{2\; \backslash \; right\; \backslash \},\; \backslash \; forall\; N\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\; ^*\; \backslash \; qquad\; \backslash \; prod\_\; \{K\; \backslash \; in\; P\; (2p^n)\}\; 2\; \backslash \; cos\; \backslash \; left\; (\backslash \; frac\; \{K\; \backslash \; pi\}\; \{2p^n\}\; \backslash \; right)\; =\; (-\; 1)\; ^\; \backslash \; frac\; \{p-1\}\; \{2\}\; p\; ~$

For any N having at least 2 prime numbers odd in its decomposition in factors first, one a:

$\backslash \; prod\_\; \{K\; \backslash \; in\; P\; (N)\}\; 2\; \backslash \; cos\; \backslash \; left\; (\backslash \; frac\; \{K\; \backslash \; pi\}\; \{N\}\; \backslash \; right)\; =\; 1\; ~$

If 2n+1 is a power of a prime number p,

$\backslash \; prod\_\; \{K\; \backslash \; N,\; K\; \backslash \; in\; P\; (2n+1)\}\; \backslash \; tan\; \backslash \; left\; (\backslash \; frac\; \{K\; \backslash \; pi\}\; \{2n+1\}\; \backslash \; right)\; =\; \backslash \; sqrt\; \{p\}\; ~$

If 2n+1 contains at least two prime numbers distinct in its decomposition in factors first,

$\backslash \; prod\_\; \{K\; \backslash \; N,\; K\; \backslash \; in\; P\; (2n+1)\}\; \backslash \; tan\; \backslash \; left\; (\backslash \; frac\; \{K\; \backslash \; pi\}\; \{2n+1\}\; \backslash \; right)\; =\; 1\; ~$

Etc…

Category: Trigonometry Category: Remarkable polynomial Category: List in connection with mathematics