1-factorizse-p-x-x-n-1-inside-C-x-2-find-the-value-of-k-1-n-1-sin-kpi-n-3-find-also-the-value-of-k-0-n-1-sin-kpi-n-

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) \dots . b e c o n t i n u e d \dots . .