# Questions tagged [roots-of-unity]

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

89 questions
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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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### Concurrent lines proof for a regular 18-gon

Let $X_1 X_2 \dotsb X_{18}$ be a regular 18-gon. Show that $X_1 X_{10}$, $X_2 X_{13}$, and $X_3 X_{15}$ are concurrent. What would be the best way to prove this? I am actually struggling ...
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### Complex numbers and Roots of unity

I have no clue how to begin these problems. How do I start? I don't think I should pound em out...Thanks. Let P be the set of $42^{\text{nd}}$ roots of unity, and let Q be the set of $70^{\text{th}}$...
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### A Polygon is inscribed in a circle $\Gamma$

A regular polygon P is inscribed in a circle $\Gamma$. Let A, B, and C, be three consecutive vertices on the polygon P, and let M be a point on the arc AC of $\Gamma$ that does not contain B. Prove ...
Let $\omega=e^{\frac{2\pi i}{n}}$. Prove that $\Pi_{k=1}^{n-1} (1-\omega^k)=n.$ So far, I've tried brute-forcing it by expanding out the product, but it ended up getting too messy--and now I'm ...