# Tagged Questions

numbers $z$ such that $z^n=1$ for some natural number $n$; here usually $z$ is in $\mathbb C$ or some other field

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### Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$

Using $\text{n}^{\text{th}}$ root of unity $$\large\left(e^{\frac{2ki\pi}{n}}\right)^{n} = 1$$ Prove that $$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$
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### Is $\sqrt 7$ the sum of roots of unity?

Let $a_n$ and $b_n$ be 2 sequences of $n$ rationals. Is it possible that $\sqrt 7 = \sum_{m=1}^{n} a_m (-1)^{b_m}$ ? Is it possible that $\sqrt{17}$ = $\sum_{m=1}^{n} a_m (-1)^{b_m}$ ? How to ...
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### Prove that $\cot^2{(\pi/7)} + \cot^2{(2\pi/7)} + \cot^2{(3\pi/7)} = 5$

Prove that $\cot^2{(\pi/7)} + \cot^2{(2\pi/7)} + \cot^2{(3\pi/7)} = 5$ . I am sure this is derived from using roots of unity and Euler's complex number function, but I am very uncomfortable in these ...
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### A little more on $\sqrt[3]{\cos\bigl(\tfrac{2\pi}7\bigr)}+\sqrt[3]{\cos\bigl(\tfrac{4\pi}7\bigr)}+\sqrt[3]{\cos\bigl(\tfrac{8\pi}7\bigr)}$

Using a special case of an identity by Ramanujan, we find that given the roots $x_i$ of $$x^3 + x^2 - (3 n^2 + n)x + n^3=0\tag1$$ which, since its discriminant is negative, always has three real ...
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### Conjecture about some group semiring representations ( and roots of unity ).

Let $\Bbb R_+=[0,\infty)$ be a semiring. $\Bbb R_+[C_n]$ is the group semiring formed by the semiring $\Bbb R_+$ and the cylic group $C_n$. Let $\Bbb R_+[X_n]$ be the polynomial semiring. (...
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### Finding the Roots of Unity

I have the following equation, $$z^5 = -16 + (16\sqrt 3)i$$I am asked to write down the 5th roots of unity and find all the roots for the above equation expressing each root in the form $re^{i\theta}$....
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### Can $\sin(\pi/25)$ be expressed in radicals, revisited

This was inspired by this post. Let, $$q = e^{2\pi\, i/m}$$ D. Speyer's answer can be generalized as, $$\sin\Big(\frac{\pi}{m^2}\Big) = \frac{i}{2}\Big(-q^{1/(2m)}+q^{-1/(2m)} \Big)\tag1$$ while R....
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### Nth root of Unity

Hi all I am in higher level mathematics and I am taking the IB. We started doing problems associated with nth root of unity. I understand how to find the roots of for example: $$Z^3 - 1 =0$$ and ...
### Proof that $\mathbb G_n \bigcap \mathbb G_m = \mathbb G_{(m:n)}$
Being $\mathbb G_n$ the roots of unity for $n \in \mathbb N$, prove that $\mathbb G_n \bigcap \mathbb G_m = \mathbb G_{(m:n)}.$