Question

Find the value of $\sin\, 12^°\, \sin\, 48^°\, \sin \,54^° .$

• A. $\frac{1}{2}$
• B. $\frac{1}{4}$
• C. $\frac{1}{6}$
• D. $\frac{1}{8}$

Answer 1

Answer: (D)

$\sin 12^{\circ} \sin 48^{\circ} \sin 54^{\circ}$
$=\sin 12^{\circ} \sin \left(60^{\circ}-12^{\circ}\right) \sin \left(90^{\circ}-36^{\circ}\right)$
$=\frac{\sin 12^{\circ} \sin \left(60^{\circ}-12^{\circ}\right) \sin 72^{\circ} \cos 36^{\circ}}{\sin 72^{\circ}}$
$=\frac{\sin 12^{\circ}\left(60^{\circ}-\sin ^{2} 12^{\circ}\right) \cos 36^{\circ}}{\sin 72^{\circ}}$
$=\frac{\sin 12^{\circ}\left(\frac{3}{4}-\sin ^{2} 12^{\circ}\right) \cos 36^{\circ}}{\sin 72^{\circ}}$
$=\frac{\sin 36^{\circ} \cos 36^{\circ}}{4 \sin 72^{\circ}}$
$=\frac{1}{2} \frac{\sin 72^{\circ}}{4 \sin 72^{\circ}}=\frac{1}{8}$