Question

$log (\sin \,1^\circ ) \times \log(\sin\, 2^\circ) \times \log(\sin \,3^\circ)\dots log (sin \,179^{\circ})$

• A. is positive
• B. is negative
• C. lies between 1 and 180
• D. is zero

Answer 1

Answer: (D)

$\log \left(\sin 1^{\circ}\right) \cdot \log \left(\sin 2^{\circ}\right) \cdot \log \left(\sin 3^{\circ}\right) \ldots \log \left(\sin 179^{\circ}\right)$
$=\log \sin 1^{\circ} \cdot \log \sin 2^{\circ} \ldots \log \sin 90^{\circ} \ldots \log \sin 179^{\circ}$
$= \log \sin 1^{\circ} \cdot \log \sin 2^{\circ} \ldots \log (1) \ldots \log \sin 179^{\circ} \,\,\,\,\,\left(\because \sin 90^{\circ}=1\right) $
$= \log \sin 1^{\circ} \cdot \log \sin 2 \ldots 0 \ldots \log \sin 179^{\circ}$
$(\because \,log \,1=0)$
$= 0$