Chứng Minh Cos Pi/15*cos 2pi/15*...*cos 7pi/15=1/128 - HOC247

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GIÚP VS AK:

CM: \(\cos\dfrac{\pi}{15}\cdot\cos\dfrac{2\pi}{15}\cdot\cdot\cdot\cos\dfrac{7\pi}{15}=\dfrac{1}{128}\)

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Trả lời (2)

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  \(\cos\dfrac{\pi}{15}.\cos\dfrac{2\pi}{15}...\cos\dfrac{7\pi}{15}=-\dfrac{1}{2}.\left(\cos\dfrac{\pi}{15}.\cos\dfrac{2\pi}{15}.\cos\dfrac{4\pi}{15}.\cos\dfrac{8\pi}{15}\right).\left(\cos\dfrac{3\pi}{15}.\cos\dfrac{6\pi}{15}\right)\)

  \(=-\dfrac{1}{2}.\left(\cos\dfrac{\pi}{15}.\cos\left(2.\dfrac{\pi}{15}\right).\cos\left(2^2.\dfrac{\pi}{15}\right).\cos\left(2^3\dfrac{\pi}{15}\right)\right).\left(\cos\dfrac{3\pi}{15}.\cos\left(2.\dfrac{3\pi}{15}\right)\right)\)

  \(=-\dfrac{1}{2}.\left(\dfrac{\sin\left(2^4.\dfrac{\pi}{15}\right)}{16.\sin\left(\dfrac{\pi}{15}\right)}\right).\left(\dfrac{\sin\left(2^2\dfrac{3\pi}{15}\right)}{4.\sin\left(\dfrac{3\pi}{15}\right)}\right)\)

  \(=-\dfrac{1}{2}.\left(\dfrac{\sin\left(\dfrac{16\pi}{15}\right)}{16.\sin\left(\dfrac{\pi}{15}\right)}\right).\left(\dfrac{\sin\left(\dfrac{12\pi}{15}\right)}{4.\sin\left(\dfrac{3\pi}{15}\right)}\right)\)

  \(=-\dfrac{1}{2}.\left(\dfrac{-\sin\left(\dfrac{\pi}{15}\right)}{16.\sin\left(\dfrac{\pi}{15}\right)}\right).\left(\dfrac{\sin\left(\dfrac{3\pi}{15}\right)}{4.\sin\left(\dfrac{3\pi}{15}\right)}\right)=\dfrac{1}{128}\)

  bởi Lương Linh 12/10/2018 Like (0) Báo cáo sai phạm
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  cosπ15.cos2π15...cos7π15=−12.(cosπ15.cos2π15.cos4π15.cos8π15).(cos3π15.cos6π15)cos⁡π15.cos⁡2π15...cos⁡7π15=−12.(cos⁡π15.cos⁡2π15.cos⁡4π15.cos⁡8π15).(cos⁡3π15.cos⁡6π15)

  =−12.(cosπ15.cos(2.π15).cos(22.π15).cos(23π15)).(cos3π15.cos(2.3π15))=−12.(cos⁡π15.cos⁡(2.π15).cos⁡(22.π15).cos⁡(23π15)).(cos⁡3π15.cos⁡(2.3π15))