1) factorizse p(x) =x^n −1 inside C[x] 2) find the value of Π_(k=1) ^(n−1) sin(((kπ)/n)) 3)find also the value of Π


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Question Number 28370 by abdo imad last updated on 24/Jan/18

$1) f a c t o r i z s e p (x) = x^{n} - 1 i n s i d e C [x]$ $2) f i n d t h e v a l u e o f \prod_{k = 1}^{n - 1} s i n (\frac{k π}{n})$ $3) f i n d a l s o t h e v a l u e o f \prod_{k = 0}^{n - 1} s i n (\frac{k π}{n} + θ) .$
Commented by abdo imad last updated on 26/Jan/18

$1) t h e r o o t s o f p (x) a r e t h e c o m p l e x z_{k} = e^{i \frac{2 k π}{n}} a n d k \in [[0, n - 1]]$ $a n d p (x) = λ \prod_{k = 1}^{n - 1} (x - z_{k}) . i t s c l e a r t h a t λ = 1$ $2) p (x) = (x - 1) \prod_{k = 1}^{n - 1} (x - e^{i \frac{2 k π}{n}}) f o r x \neq 1$ $\frac{p (x)}{x - 1} = \prod_{k = 1}^{n - 1} (x - e^{i \frac{2 k π}{n}})$ $\Rightarrow {l i m}_{x \to 1} \frac{x^{n} - 1}{x - 1} = \prod_{k = 1}^{n - 1} (1 - c o s (\frac{2 k π}{n}) - i s i n (\frac{2 k π}{n}))$ $\Rightarrow n = \prod_{k = 1}^{n - 1} (2 {s i n}^{} (\frac{k π}{n}) - 2 i s i n (\frac{k π}{n}) c o s (\frac{k π}{n}))$ $\Rightarrow n = {(- 2 i)}^{n - 1} \prod_{k = 1}^{n - 1} (s i n (\frac{k π}{n}) (e^{i \frac{k π}{n}}))$ $\Rightarrow n = {(- 2 i)}^{n - 1} (\prod_{k = 1}^{n - 1} s i n (\frac{k π}{n})) e^{i \frac{π}{n} \sum_{k = 1}^{n - 1} k}$ $\Rightarrow n = {(- 2 i)}^{n - 1} e^{i \frac{π}{n} \frac{n (n - 1)}{2}} \prod_{k = 1}^{n - 1} s i n (\frac{k π}{n})$ $= {(- i)}^{n - 1} i^{n - 1} 2^{n - 1} \prod_{k = 1}^{n - 1} s i n (\frac{k π}{n}) = 2^{n - 1} \prod_{k = 1}^{n - 1} s i n (\frac{k π}{n})$ $\Rightarrow \prod_{k = 1}^{n - 1} s i n (\frac{k π}{n}) = \frac{n}{2^{n - 1}} . (n ⩾ 2) . . . . b e c o n t i n u e d . . . . .$
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