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$$(\frac{1}{sin(20)})^2+(\frac{1}{sin(40)})^2+(\frac{1}{sin(80) })^2 $$ »Å.ÁØÁà»ç¹Í§ÈÒ
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Hope is what makes us strong. It's why we are here. It is what we fight with when all else is lost. |
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Íѹ¹ÕéÇÔ¸Õ¼Á¹Ð¤ÃѺ ÁÕÇÔ¸Õä˹ÊÑ鹡ÇèÒ¹Õéáºè§»Ñ¹´éǤÃѺ
¨Ò¡ $sin(x)^2 = \frac{1-cos(2x)}{2}$ $$(\frac{1}{sin(20)})^2+(\frac{1}{sin(40})^2+(\frac{1}{sin(80) })^2 $$ $$ = \frac{2}{1-cos(40)}+\frac{2}{1-cos(80)}+\frac{2}{1-cos(160)}$$ $$ = 2*\frac{(1-cos(40))(1-cos(80))+(1-cos(40))(1-cos(160))+(1-cos(160))(1-cos(80))}{(1-cos(40))(1-cos(80)(1-cos(160)}$$ $$ =2*\frac{3-2(cos(40)+cos(80)+(cos(160))+cos(40)cos(80)+cos(40)cos(160)+cos(80)cos(160)}{1-(cos(40)+cos(80)+cos(160))+cos(40)cos(80)+cos(40)cos(160)+cos(80)cos(160)-cos(40)cos(80)cos(160)} $$ $cos(40)+cos(80)+cos(160)=0$ $$ =2*\frac{3+cos(40)cos(80)+cos(40)cos(160)+cos(80)cos(160)}{1+cos(40)cos(80)+cos(40)cos(160)+cos(80)cos(160)-cos(40)cos(80)cos(160)} $$ $cos(40)cos(80)+cos(40)cos(160)+cos(80)cos(160)=\frac{-3}{4}$ $cos(40)cos(80)cos(160)=\frac{-1}{8}$ ¨Ðä´éÇèÒ $$ =2*\frac{3+cos(40)cos(80)+cos(40)cos(160)+cos(80)cos(160)}{1+cos(40)cos(80)+cos(40)cos(160)+cos(80)cos(160)-cos(40)cos(80)cos(160)}$$ $$= 2*\frac{3-\frac{3}{4}}{1-\frac{3}{4}+\frac{1}{8}} $$ $$=12$$
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Hope is what makes us strong. It's why we are here. It is what we fight with when all else is lost. 26 µØÅÒ¤Á 2014 19:57 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 3 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ FranceZii Siriseth |
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Hope is what makes us strong. It's why we are here. It is what we fight with when all else is lost. |
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Hope is what makes us strong. It's why we are here. It is what we fight with when all else is lost. |
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$=\frac{(sin 40cos 10)^2+(sin 20cos 10)^2+(sin20sin40)^2}{(sin 20sin 40cos 10)^2}$ $=\frac{({\frac{sin 50+sin30}{2}})^2 +({\frac{sin 30+sin10}{2}})^2+({\frac{cos 20+cos 60}{2}})^2}{(\frac{\sqrt{3}}{8})^2}$ $=\frac{\frac{1}{4}[(cos 40+\frac{1}{2})^2 +(\frac{1}{2}+sin 10)^2+(cos 20-\frac{1}{2})^2]}{(\frac{\sqrt{3}}{8})^2}$ $=\frac{\frac{1}{4}[(cos^2 40+cos 40+\frac{1}{4})+(sin^2 10+sin 10+\frac{1}{4})+(cos^2 20-cos 20+\frac{1}{4})]}{(\frac{\sqrt{3}}{8})^2}$ $=\frac{\frac{1}{4}(cos^2 40+sin^2 10+cos^2 20+cos 40+sin 10-cos 20+\frac{3}{4})}{(\frac{\sqrt{3}}{8})^2}$ $=\frac{\frac{1}{4}(\frac{cos 80+1}{2} +\frac{1-cos 20}{2}+\frac{cos 40+1}{2}+cos 40+sin 10-cos 20+\frac{3}{4})}{(\frac{\sqrt{3}}{8})^2}$ $=\frac{\frac{1}{4}(\frac{sin 10+1}{2} +\frac{1-cos 20}{2}+\frac{cos 40+1}{2}+cos 40+sin 10-cos 20+\frac{3}{4})}{(\frac{\sqrt{3}}{8})^2}$ $=\frac{\frac{3}{8}(cos 40+sin 10-cos 20 +\frac{3}{2})}{(\frac{\sqrt{3}}{8})^2}$ $=\frac{\frac{3}{8}(0 +\frac{3}{2})}{\frac{3}{64}}$ $=12$ |
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