Question

Arătați ca,dacă n * sin(pi/12) * cos(pi/12) = 1 ,atunci n=4

Answer 1

 

$\displaystyle\bf\\\frac{\pi}{12}=15^o\\\\sin15^o=sin(45^o-30^o)=sin45^ocos30^o-sin30^ocos45^o=\\\\=\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}-\frac{1}{2}\times\frac{\sqrt{2}}{2}= \frac{\sqrt{6}}{4}-\frac{\sqrt{2}}{4}=\boxed{\bf\frac{\sqrt{6}-\sqrt{2}}{4}}\\\\\\cos15^o=cos(45^o-30^o)=cos45^ocos30^o+sin45^osin30^o=\\\\=\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\times\frac{1}{2}= \frac{\sqrt{6}}{4}+\frac{\sqrt{2}}{4}=\boxed{\frac{\bf\sqrt{6}+\sqrt{2}}{4}}$

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$\displaystyle\bf\\n\times sin15^o\times cos15^o=1\\\\n\times\frac{\sqrt{6}-\sqrt{2}}{4}\times\frac{\sqrt{6}+\sqrt{2}}{4}=1\\\\n\times\frac{\Big(\sqrt{6}-\sqrt{2}\Big)\times\Big(\sqrt{6}+\sqrt{2}\Big)}{4\times4} =1\\\\n\times\frac{6-2}{16}=1\\\\\frac{4n}{16}=1\\\\\frac{n}{4}=1 \\\\n=4\times1\\\\\boxed{\bf n=4}\\\\cctd$