Question

$96\sqrt{3}\sin \frac{\pi }{48}\cos \frac{\pi }{48}\cos \frac{\pi }{24}\cos \frac{\pi }{12}\cos \frac{\pi }{6}$ has the value $=$ ....................

• A. $9$
• B. $18$
• C. $-9$
• D. $-18$

Answer 1

Answer: (A)

$9$

$96\sqrt{3}\sin \frac{\pi }{48}\cos \frac{\pi }{48}\cos \frac{\pi }{24}\cos \frac{\pi }{12}\cos \frac{\pi }{6}$

$=96\sqrt{3}\cdot \frac{1}{2}\left(2\sin \frac{\pi }{48}\cos \frac{\pi }{48}\right)\cos \frac{\pi }{24}\cos \frac{\pi }{12}\cos \frac{\pi }{6}$

$=96\sqrt{3}\cdot \frac{1}{2}\sin \frac{\pi }{24}\cos \frac{\pi }{24}\cos \frac{\pi }{12}\cos \frac{\pi }{6}$

$=96\sqrt{3}\cdot \frac{1}{4}\left(2\sin \frac{\pi }{24}\cos \frac{\pi }{24}\right)\cos \frac{\pi }{12}\cos \frac{\pi }{6}$

$=96\sqrt{3}\cdot \frac{1}{4}\sin \frac{\pi }{12}\cos \frac{\pi }{12}\cos \frac{\pi }{6}$

$=96\sqrt{3}\cdot \frac{1}{8}\left(2\sin \frac{\pi }{12}\cos \frac{\pi }{12}\right)\cos \frac{\pi }{6}$

$=96\sqrt{3}\cdot \frac{1}{8}\sin \frac{\pi }{6}\cos \frac{\pi }{6}$

$=96\sqrt{3}\cdot \frac{1}{16}\left(2\sin \frac{\pi }{6}\cos \frac{\pi }{6}\right)$

$=96\sqrt{3}\cdot \frac{1}{16}\sin \frac{\pi }{3}$

$=96\sqrt{3}\cdot \frac{1}{16}\cdot \frac{\sqrt{3}}{2}=9$