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#1
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͹ءÃÁ ¢Í§ sin áÅÐ cos
ËÇÑ´´Õ¤Ñº ¼ÙéÃÙé·Ø¡·èÒ¹ ËÒÂ仹ҹàÅ ªèǧ¹Õé§Ò¹àÂÍШѧ äÁè¤èÍÂÁÕàÇÅÒ¤Ô´àÅ¢àÅÂ
ËÅѧ¨Ò¡¤Ô´ËÒÇÔ¸Õ¡ÒÃËÒÃÒ¡ÊÁ¡ÒáÓÅѧ 5 ÍÂÙè¹Ò¹ ã¹·ÕèÊØ´¡çËÒäÁèä´é¤Ñº ·Äɯբͧ Abel à»ç¹¨ÃÔ§ áÎÐ áÅÐÂѧ¤Ô´ÇÔ¸Õ¡ÒÃÍÔ¹·Ôà¡Ã·áººãËÁè ¡ç¤Ô´ÁÒáÅéÇ áµèÁÒÃÙé·ÕËÅѧÇèÒ¼Ô´ àÅÂàÅÔ¡ÅéÁ 令ԴÍÂèÒ§Í×è¹á·¹¤ÃѺ ¨¹µÍ¹¹Õé¼ÁÊÒÁÒö ËÒ͹ءÃÁ ¢Í§ sin áÅÐ cos ä´éáÅéÇ ¤ÃÒǹÕé¶Ù¡ªÑÇÃìæ¤ÃѺ à¾ÃÒÐÅͧ¡´à¤Ã×èͧ¤Ô´àÅ¢µÃǨÊͺ´ÙáÅéÇ áµè¼Á¡ÅÑÇÇèÒÍÒ¨¨ÐÁÕ¤¹¤Ô´ÁÒä´é¡è͹ áÅéǹèФÃѺ ÍÂèÒ§àªè¹ ¨§ËÒ¤èҢͧ cos 1 + cos 2 + cos 3 + ....... + cos 90 = ? sin 1 + sin 2 + sin 3 + ....... + sin 90 = ? àÅÂÍÂÒ¡¶ÒÁ·èÒ¹¼ÙéÃÙéã¹·Õè¹ÕéÇèÒà¤ÂÁÕã¤Ã¤Ô´ËÒ͹ءÃÁ sin áÅÐ cos ä´éÁÒ¡è͹¼ÁÃÖÂѧ¤ÃѺ ¶éÒÁÕªèÇ ÁҺ͡˹èÍ ËÃ×Íã¤ÃÍÂÒ¡¾ÔÊÙ¨¹ìÇèÒÊٵüÁ¶Ù¡µéͧËÃ×ÍäÁè ¡çµÑé§â¨·Âì͹ءÃÁ ÁÒÅͧ´Ù¡çä´é¹Ð¤ÃѺ
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¤ÇÒÁ¾ÂÒÂÒÁáÅÐÍ´·¹ à»ç¹¤Ø³ÊÁºÑµÔ˹Ö觢ͧ¹Ñ¡¤³ÔµÈÒʵÃì...... ´Ù¼Á¨Ô ¼Á¾ÂÒÂÒÁ ËÒÊٵ÷ÑèÇä»ã¹¡ÒÃËÒÃÒ¡¢Í§ÊÁ¡ÒáÓÅѧ 5 ÍÂÙè¤ÃÖè§»Õ ÂѧäÁèä´éàŤѺ Î×Íææææ |
#2
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͹ءÃÁ·Õè¤Ø³¡Ó˹´ÁÒ ÁÕº·¤ÇÒÁÍÂÙèáÅéÇã¹àÇ纹Õé
áµè ¶éҤسªèÇ·Ӣé͹Õé¨Ð¢Íº¤Ø³ÁÒ¡¤ÃѺ Evaluate $$ \cos1+\frac{\cos2}{2}+\frac{\cos3}{3}+... $$
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âÅ¡¹ÕéÁÕ¤¹ÍÂÙè 10 »ÃÐàÀ· ¤×Í ¤¹·Õèà¢éÒã¨àÅ¢°Ò¹Êͧ áÅФ¹·ÕèäÁèà¢éÒ㨠|
#3
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ËÒ!!! ÁÕ¤¹¤Ô´ä´é¡è͹«ÐáÃéÇàËÃͤѺ
¶éÒ§Ñ鹢ͼÁ´Ùº·¤ÇÒÁ¹Ñé¹ä´éÁÑê¤ÃѺ ¨Ð´ÙÇèÒ¤Ô´àËÁ×͹¡Ñ¹ÃÖ»èÒÇ Êèǹ⨷Âì¢Í§¤Ø³ Mastermander µÍ¹àÅÔ¡§Ò¹¼Á¨Ðà¡çºä»¤Ô´¤ÃѺ ( ⨷Âì ¤Ø³ Mastermander ÂѧÂÒ¡àËÁ×͹à´ÔÁ¹Ð ¤Ñº )
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¤ÇÒÁ¾ÂÒÂÒÁáÅÐÍ´·¹ à»ç¹¤Ø³ÊÁºÑµÔ˹Ö觢ͧ¹Ñ¡¤³ÔµÈÒʵÃì...... ´Ù¼Á¨Ô ¼Á¾ÂÒÂÒÁ ËÒÊٵ÷ÑèÇä»ã¹¡ÒÃËÒÃÒ¡¢Í§ÊÁ¡ÒáÓÅѧ 5 ÍÂÙè¤ÃÖè§»Õ ÂѧäÁèä´éàŤѺ Î×Íææææ |
#4
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º·¤ÇÒÁâ´Â : ¹Ò ¾§Èì·Í§ á«èàÎé§
á¶Á⨷ÂìÍÕ¡ 2 ¢éͤÃѺ 1. ¨§ËÒ¤èÒ x ¨Ò¡ÊÁ¡Òà $$ \sum_{\theta=1}^{999}\sin(x\theta)=\sum_{\theta=1}^{999}\cos(x\theta) $$ 2.
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âÅ¡¹ÕéÁÕ¤¹ÍÂÙè 10 »ÃÐàÀ· ¤×Í ¤¹·Õèà¢éÒã¨àÅ¢°Ò¹Êͧ áÅФ¹·ÕèäÁèà¢éÒ㨠02 µØÅÒ¤Á 2006 16:11 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 1 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ Mastermander |
#5
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âÅ¡¹ÕéÁÕ¤¹ÍÂÙè 10 »ÃÐàÀ· ¤×Í ¤¹·Õèà¢éÒã¨àÅ¢°Ò¹Êͧ áÅФ¹·ÕèäÁèà¢éÒ㨠|
#6
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ÍéÒÇ ¾Õè¡Ã¤Ô´ä´é¡è͹áÅéÇàËÃͤѺ ÇÔ¸Õ¤Ô´µèÒ§¡Ñ¹àÅ硹éÍÂáµè¤ÓµÍºàËÁ×͹¡Ñ¹àŤѺ
µÍ¹¹Õé¼Á¾ÂÒÂÒÁËÒ͹ءÃÁ¢Í§ tan ÍÂÙè ÂѧËÒäÁèä´éàŤѺ àÎéÍ ¢Íº¤Ø³ ¤Ø³ Mastermander ·ÕèãËé¢éÍÁÙŹФѺ ⨷Âì 2 ¢éÍËÅѧ ¾ÍÃÙéá¹Ç㹡Ò÷ÓáÅéǤѺ ¶éÒÃÙé ¡ÒÃËÒ͹ءÃÁ sin áÅÐ cos ¡ç¤§·Óä´é áµè¢éÍáá·ÕèãËéÁÒ cos1 + (cos2)/2 + (cos3)/3 + .... ¹Õè·ÓÂÑ§ä§ ¤Ñº à©ÅÂ˹èͤѺ ¤Ø³ Mastermander ¹Õè¢éÍÁÙŤÇÒÁÃÙéá¹è¹»Öé¡áºº¹Õé ÁÕ˹ѧÊ×Í´Õæá¹Ð¹ÓºéÒ§ÁÑê¤Ѻ µÍ¹¹ÕéÍÂÒ¡ ÈÖ¡ÉÒ¤³ÔµÈÒʵÃìà¾ÔèÁàµÔÁ¹èФѺ ẺÇèÒÃéÒ§ÃÒ仹ҹ
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¤ÇÒÁ¾ÂÒÂÒÁáÅÐÍ´·¹ à»ç¹¤Ø³ÊÁºÑµÔ˹Ö觢ͧ¹Ñ¡¤³ÔµÈÒʵÃì...... ´Ù¼Á¨Ô ¼Á¾ÂÒÂÒÁ ËÒÊٵ÷ÑèÇä»ã¹¡ÒÃËÒÃÒ¡¢Í§ÊÁ¡ÒáÓÅѧ 5 ÍÂÙè¤ÃÖè§»Õ ÂѧäÁèä´éàŤѺ Î×Íææææ |
#7
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I know only the answer is $-\frac{\ln(2-2\cos1)}{2}=0.0420195$...(by program)
Solution in next reply
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âÅ¡¹ÕéÁÕ¤¹ÍÂÙè 10 »ÃÐàÀ· ¤×Í ¤¹·Õèà¢éÒã¨àÅ¢°Ò¹Êͧ áÅФ¹·ÕèäÁèà¢éÒ㨠08 µØÅÒ¤Á 2006 22:17 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 2 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ Mastermander |
#8
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·Õè¨ÃÔ§ÁÕáËÅè§ÃÇÁ E-book Ẻ˹ѧÊ×Íà»ç¹àÅèÁ æ ÁËÒÈÒÅÍÂÙè·Õè 2 ·Õè¹Ð¤ÃѺ à»ç¹áºº·ÕèÁѹËÁ´ÍÒÂØ·Ò§ÅÔ¢ÊÔ·¸Ôì»ÃÐÁÒ³¹Ñé¹ à¤ÂµÍºäÇéã¹ÊÑ¡¡ÃзÙé˹Öè§
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#9
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Short solution
(Editied)
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âÅ¡¹ÕéÁÕ¤¹ÍÂÙè 10 »ÃÐàÀ· ¤×Í ¤¹·Õèà¢éÒã¨àÅ¢°Ò¹Êͧ áÅФ¹·ÕèäÁèà¢éÒ㨠09 µØÅÒ¤Á 2006 13:44 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 1 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ Mastermander |
#10
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à»ç¹¡Òà solve ·ÕèàÂÕèÂÁ¤Ñº ÊÑé¹æ ä´é㨤ÇÒÁ
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¤ÇÒÁ¾ÂÒÂÒÁáÅÐÍ´·¹ à»ç¹¤Ø³ÊÁºÑµÔ˹Ö觢ͧ¹Ñ¡¤³ÔµÈÒʵÃì...... ´Ù¼Á¨Ô ¼Á¾ÂÒÂÒÁ ËÒÊٵ÷ÑèÇä»ã¹¡ÒÃËÒÃÒ¡¢Í§ÊÁ¡ÒáÓÅѧ 5 ÍÂÙè¤ÃÖè§»Õ ÂѧäÁèä´éàŤѺ Î×Íææææ |
#11
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ÊÃØ»ä´éÇèÒ
$$\sin z+\frac{\sin 2z}{2}+\frac{\sin 3z}{3}+...=\arctan(\frac{\sin z}{1-\cos z})$$ $$\cos z+\frac{\cos 2z}{2}+\frac{\cos 3z}{3}+...=-\frac{\ln(2-2\cos z)}{2}$$ áµè $\frac{\sin z}{1-\cos z} = \cot\frac{z}{2}=\tan(\frac{\pi}{2}-\frac{z}{2})$ ´Ñ§¹Ñ鹨֧ä´éÊÙµÃÊÇÂæ§ÒÁ·ÕèàË繴ѧ¹Õé $$ \sum_{n=1}^{\infty} \frac{\sin nz}{n}=\arctan\big(\cot\frac{z}{2}\big)=\arctan\big(\tan(\frac{\pi}{2}-\frac{z}{2})\big)$$
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âÅ¡¹ÕéÁÕ¤¹ÍÂÙè 10 »ÃÐàÀ· ¤×Í ¤¹·Õèà¢éÒã¨àÅ¢°Ò¹Êͧ áÅФ¹·ÕèäÁèà¢éÒ㨠09 µØÅÒ¤Á 2006 20:20 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 2 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ Mastermander |
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