If $a_k=\cos(2k\pi/n)-2+i\sin(2k\pi/n)$, evaluate $a_1a_2\cdots a_n$

Question

Define $$a_k=\cos\frac{2k\pi}{n}-2+i\sin\frac{2k\pi}{n}$$ How can I approach this product? $$a_1 a_2 \cdots a_n$$ I tried to investigate if one of terms is $0$, as it is a product, but no.

Any ideas?

I have already calculated this sum using the Euler formula. $$a_1+a_2+\cdots +a_n,\;n>1$$

(original problem images: $a_k$, product, sum)

$e^{ix} = cos(x)+isin(x)$ and the sum of a geometric progression. — Paul, Apr 30 '20 at 06:49
Thank you @Paul. I resolved the sum, but I have some problems on some product of factors ;( — Akhtubir, Apr 30 '20 at 07:03
https://math.stackexchange.com/questions/1909362/product-of-one-minus-the-tenth-roots-of-unity — lab bhattacharjee, Apr 30 '20 at 07:20

score 2 · Accepted Answer · edited Apr 30 '20 at 07:41

2

So, $a_k;1\le k \le n$

are the roots of $$(x+2)^n=1\iff x^n+\cdots+2^n-1=0$$

Using Vieta's formula $$\prod_{k=1}^n a_k=(-1)^n\cdot\dfrac{2^n-1}1$$

edited Apr 30 '20 at 07:41

Jean-Claude Arbaut

23,277

answered Apr 30 '20 at 07:19

lab bhattacharjee

274,582

Great, I understand why you used the Veita's formula. Can you explain me how you knew the exact equation (x+2)^n by given roots? – Akhtubir Apr 30 '20 at 07:22
@Akhtubir, $$a_k+2=?$$ – lab bhattacharjee Apr 30 '20 at 07:35
You are great. I understand now, sorry for my language and for my little stupid question. – Akhtubir Apr 30 '20 at 07:38

If $a_k=\cos(2k\pi/n)-2+i\sin(2k\pi/n)$, evaluate $a_1a_2\cdots a_n$

1 Answers1