#1
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µÃÕ⡳¤ÃѺ
cos(2pi/7) + cos(4pi/7) + cos(6pi/7) = ?
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#2
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Åͧ´Ù¡ÃзÙéà¡èÒ¹Õé¤ÃѺ
http://www.mathcenter.net/forum/showthread.php?t=10874 |
#3
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⨷ÂìµèÒ§¡Ñ¹¹Ð¤ÃѺ
¡ÃзÙéà¡èÒà»ç¹ $cos\frac{\pi }{7} - cos\frac{2\pi }{7} + cos\frac{3\pi }{7}$ Êèǹ¡ÃзÙé¹Õéà»ç¹ $cos\frac{2\pi }{7} + cos\frac{4\pi }{7} + cos\frac{6\pi }{7}$ ..¡ÓÅѧ¹Ñè§ÁÖ¹ËÑÇÍÂÙèàŤÃѺ...à´ÕëÂǤԴä´é¤è͵ͺ¤ÃѺ
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#4
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àÊÃÔÁ»ÃÐʺ¡Òó줳ԵÈÒʵÃìªØ´·Õè1...͹ءÃÁµÃÕ⡳ÁÔµÔ..à¢Õ¹â´Â¤Ø³GON
¤ÓµÍº ¤×Í$-\frac{1}{2} $ $cos\frac{2\pi }{7} + cos\frac{4\pi }{7} + cos\frac{6\pi }{7}$ $\Delta =\frac{4\pi }{7}-\frac{2\pi }{7}=\frac{6\pi }{7}-\frac{4\pi }{7}=\frac{2\pi }{7}$ $\frac{\Delta}{2} =\frac{\pi }{7}$ àÍÒ$2sin\frac{\Delta}{2} $¤Ù³à¢éÒä» ¨Ðä´éÇèÒ $2sin\frac{\pi }{7}cos\frac{2\pi }{7} =sin(\frac{\pi }{7}+\frac{2\pi }{7})+sin(\frac{\pi }{7}-\frac{2\pi }{7})=sin\frac{3\pi }{7}+sin(-\frac{\pi }{7}) = sin\frac{3\pi }{7}-sin\frac{\pi }{7}$....(1) ¨Ò¡ÊÙµÃ$2sinAcosB=sin(A+B)+sin(A-B)$ áÅÐ $sin(-A) = -sinA$ $2sin\frac{\pi }{7}cos\frac{4\pi }{7} = sin\frac{5\pi }{7}-sin\frac{3\pi }{7}$.....(2) $2sin\frac{\pi }{7}cos\frac{6\pi }{7} = sin\pi -sin\frac{5\pi }{7}$.....(3) (1)+(2)+(3); $2sin\frac{\pi }{7}(cos\frac{2\pi }{7} + cos\frac{4\pi }{7} + cos\frac{6\pi }{7}) = sin\pi -sin\frac{\pi }{7} =-sin\frac{\pi }{7}$ $\therefore cos\frac{2\pi }{7} + cos\frac{4\pi }{7} + cos\frac{6\pi }{7} = -\frac{1}{2}$ ¢Íº¤Ø³¤Ø³GON ¼Ùéà¢Õ¹º·¤ÇÒÁ´éǤÃѺ
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#5
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¢Íº¤Ø³ÁÒ¡¤ÃѺ ¾Í´Õ¼Á¤Ô´ä´éÍÕ¡ÇÔ¸Õà»ç¹á¹ÇáËÇ¡ÁØÁ¡ÇèÒ¹Õé¹Ô´Ë¹èÍÂÍèФÃѺ àÍÒäÇé¨Ðà¢Õ¹áÅéÇÊ᡹ÁÒãËé´Ù¹Ð¤ÃѺ (à¢Õ¹ Latex ·ÕäÃãªéàÇÅÒ¹Ò¹ÁÒ¡¤ÃѺ T T)
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