8√3 cos 5П /12 cos п/12 sin П/6

Решите задачу:

$\pi =180$

$8 \sqrt{3} cos \frac{5 \pi }{12} cos \frac{ \pi }{12} sin \frac{ \pi }{6}=8 \sqrt{3} cos75cos15sin30$

$cos75=cos(45+30)=cos45cos30-sin45sin30= \frac{ \sqrt{2} }{2}* \frac{ \sqrt{3} }{2}- \frac{ \sqrt{2} }{2}* \frac{1}{2}= \frac{ \sqrt{2}* \sqrt{3}- \sqrt{2} }{4}= \frac{ \sqrt{3} }{4}$
$cos15=cos(45-30)=cos45cos30+sin45sin30=\frac{ \sqrt{2} }{2}* \frac{ \sqrt{3} }{2}+ \frac{ \sqrt{2} }{2}* \frac{1}{2}= \frac{ \sqrt{2}* \sqrt{3}+ \sqrt{2} }{4}= \frac{2 \sqrt{2} *\sqrt{3} }{4}$
$sin30= \frac{1}{2}$

$8 \sqrt{3} cos75cos15sin30=\frac{8 \sqrt{3} *\sqrt{3}*2 \sqrt{2}* \sqrt{3} }{4*2}=3*2 \sqrt{2*3} =6 \sqrt{6}$