[Mastermander]

โจทย์ข้อ 6 พิมพ์ผิดครับ ตามที่คุณ nongtum ทำไว้แล้ว
\[
\cos \frac{\pi }{{2n + 1}}+\cos \frac{{3\pi }}{{2n + 1}} + \ldots + \cos \frac{{\left( {2n - 1} \right)\pi }}{{2n + 1}} = \varepsilon
\]\[
\left( {2\sin \frac{\pi }{{2n + 1}}} \right)\left( {\cos \frac{\pi }{{2n + 1}}+\cos \frac{{3\pi }}{{2n + 1}} + \ldots + \cos \frac{{\left( {2n - 1} \right)\pi }}{{2n + 1}}} \right) = \left( {2\sin \frac{\pi }{{2n + 1}}} \right)\varepsilon
\]\[
= 2\cos \frac{\pi }{{2n + 1}}\sin \frac{\pi }{{2n + 1}} + 2\cos \frac{{3\pi }}{{2n + 1}}\sin \frac{\pi }{{2n + 1}} + \ldots + 2\cos \frac{{\left( {2n - 1} \right)\pi }}{{2n + 1}}\sin \frac{\pi }{{2n + 1}}
\]\[\because 2\cos \alpha \sin \beta = \sin \left( {\alpha + \beta } \right) - \sin \left( {\alpha - \beta } \right)
\]\[
2\varepsilon \sin \frac{\pi }{{2n + 1}} = \sin \frac{{2\pi }}{{2n + 1}} + \sin \frac{{4\pi }}{{2n + 1}} - \sin \frac{{2n\pi }}{{2n + 1}} + \ldots + \sin \frac{{2n\pi }}{{2n + 1}} - \sin \frac{{\left( {2n - 2} \right)\pi }}{{2n + 1}}
\]\[
2\varepsilon \sin \frac{\pi }{{2n + 1}} = \sin \frac{{2n\pi }}{{2n + 1}}
\]\[\because \sin \left( {\pi - \theta } \right) = \sin \theta \quad \to \quad\sin \left( {\pi - \frac{{2n\pi }}{{2n + 1}}} \right) = \sin \frac{\pi }{{2n + 1}}
\]\[
1 = 2\varepsilon
\]\[
\varepsilon = \sum\limits_{k = 1}^n {\cos \frac{{\left( {2k - 1} \right)\pi }}{{2n + 1}}} = \frac{1}{2}
\]