Question

Aratati ca sin pi/12 * sin 5pi/12=1/4

Answer 1

   
[tex]\displaystyle\\ \sin\frac{\pi}{12}\times\sin\frac{5\pi}{12}=\frac{1}{4}\\\\ \text{Transformam unghiurile din radiani in grade.}\\\\ \frac{\pi}{12}=\frac{180^o}{12}=15^o\\\\ \frac{5\pi}{12}=\frac{5\times180^o}{12}=5\times15^o=75^o\\\\ \texttt{Rezolvare:}[/tex]


[tex]\displaystyle\\ \sin\frac{\pi}{12}\times\sin\frac{5\pi}{12}=\\\\ =\sin15^o\times\sin75^o=\\\\ =\sin(45^o-30^o)\times\sin(45^o+30^o)=\\\\ =\Big(\sin45^o\cos30^o-\cos45^o\sin30^o\Big)\Big(\sin45^o\cos30^o + \cos45^o\sin30^o\Big)=\\\\ =\Big(\frac{ \sqrt{2} }{2} \times \frac{ \sqrt{3} }{2} -\frac{ \sqrt{2} }{2}\times \frac{ 1}{2}\Big) \Big(\frac{ \sqrt{2} }{2} \times \frac{ \sqrt{3} }{2} +\frac{ \sqrt{2} }{2}\times \frac{ 1}{2}\Big)= [/tex]


[tex]\displaystyle\\ =\Big(\frac{ \sqrt{6}-\sqrt{2} }{4}\Big) \Big(\frac{ \sqrt{6}+\sqrt{2} }{4}\Big)=\\\\ =\frac{ (\sqrt{6}-\sqrt{2}) (\sqrt{6}+\sqrt{2}) }{4\times 4}= \\\\ =\frac{ ((\sqrt{6})^2-(\sqrt{2})^2 }{4\times 4}= \frac{6-2}{16}=\frac{4}{16}=\boxed{\bf \frac{1}{4}}[/tex]