1
vote
1answer
44 views

How find this value of $\prod_{1\le i<j\le n}(w^i-w^j)^2$

give the positive integer number $n$, and $w=\cos{\dfrac{2\pi}{n}}+i\sin{\dfrac{2\pi}{n}}$ where $i^2=-1$ find the vaule $$\prod_{1\le i<j\le n}(w^i-w^j)^2$$ My try:note $$w^n=1$$ ...
1
vote
1answer
45 views

For what $k$ is $P(k):=\prod_{j=1}^{13}\cos\frac{\pi kj}{13}$ negative?

Let $k>0$ be an integer. For what $k$ is $P(k):=\prod_{j=1}^{13}\cos\frac{\pi kj}{13}$ negative? Since $13$ is prime, and for $\gcd(m,13)=1$, $P(2m)=P(2)=2^{-12}$ (can be shown by considering the ...
2
votes
1answer
215 views

A Trigonometric Sum Related to Gauss Sums

This is a problem given to me by fractals on Art of Problem Solving. I couldn't solve it so I'm posting it here for some thoughts on it. Let $$S = \sum_{j = 0}^{\lfloor n/2 \rfloor} ...
1
vote
0answers
153 views

How does trigonometry in a Galois field work?

This is a follow-up to this question. I'm interested in doing trigonometry in finite fields on a computer. I do not understand precisely how trigonometric functions are supposed to work in a finite ...
5
votes
4answers
592 views

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 ...
4
votes
2answers
943 views

If z is one of the fifth roots of unity, not 1…

If z is one of the fifth roots of unity, not 1, show that: $1+z+z^2+z^3+z^4=0$ Which wasn't too bad, but the next part is killing me: show that: $z-z^2+z^3-z^4=2i(sin(2\pi/5)-sin(\pi/5))$ Can ...
18
votes
2answers
3k views

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

Using n_th root of unity $$\Large\left(e^{\frac{2ki\pi}{n}}\right)^{n} = 1$$ Prove that $$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$