Răspuns:
[tex] \frac{1}{4} [/tex]
Explicație pas cu pas:
Calculam [tex]\sin{\frac{11\pi}{12}} [/tex]:
[tex]\sin{\frac{11\pi}{12}}=\sin{\frac{12\pi-\pi}{12}}=\sin({\pi-\frac{\pi}{12}})=\sin\pi*\cos\frac{\pi}{12}-\cos\pi*\sin\frac{\pi}{12}=-\cos\pi*\sin\frac{\pi}{12}=-*(-1)*\sin\frac{\pi}{12}=\sin\frac{\pi}{12}[/tex]
Vedem cat face [tex]\sin\frac{\pi}{12}[/tex]:
[tex]\sin\frac{\pi}{12}=\sin({\frac{\pi}{3}-\frac{\pi}{4}}})=\sin{\frac{\pi}{3}}*\cos{\frac{\pi}{4}}-\sin{\frac{\pi}{4}}*\cos{\frac{\pi}{3}}=\frac{\sqrt3}{2}*\frac{\sqrt2}{2}-\frac{\sqrt2}{2}*\frac{1}{2}=\frac{\sqrt6-\sqrt2}{4}[/tex]
Procedam la fel cu [tex] \cos{\frac{23\pi}{12}} [/tex]:
[tex]\cos{\frac{23\pi}{12}}=\cos{\frac{24\pi-\pi}{12}}=\cos({\frac{24\pi}{12}-\frac{\pi}{12}})=\cos(2\pi-\frac{\pi}{12})=\cos2\pi*\cos{\frac{\pi}{12}}+\sin2\pi*\sin{\frac{\pi}{12}}=\cos{\frac{\pi}{12}}[/tex]
Vedem cat face [tex]\cos\frac{\pi}{12}[/tex]:
[tex]\cos\frac{\pi}{12}=\cos({\frac{\pi}{3}-\frac{\pi}{4}}})=\cos{\frac{\pi}{3}}*\cos{\frac{\pi}{4}}+\sin{\frac{\pi}{4}}*\sin{\frac{\pi}{3}}=\frac{1}{2}*\frac{\sqrt2}{2}+\frac{\sqrt2}{2}*\frac{\sqrt3}{2}=\frac{\sqrt6+\sqrt2}{4}[/tex]
Inmultim acum cele doua rezultate si avem:
[tex]\sin\frac{\pi}{12}*\cos\frac{\pi}{12}=\frac{\sqrt6-\sqrt2}{4}*\frac{\sqrt6+\sqrt2}{4}=\frac{6-2}{16}=\frac{4}{16}=\frac{1}{4}[/tex]
