Question

$log(\sin 1^{\circ })\times \log (\sin 2^{\circ })\times \log (\sin 3^{\circ })\dots log(sin179^{\circ })$

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

Answer 1

Answer: (D)

$\log \left(\sin 1^{\circ }\right)\cdot \log \left(\sin 2^{\circ }\right)\cdot \log \left(\sin 3^{\circ }\right)\dots \log \left(\sin 179^{\circ }\right)$
$=\log \sin 1^{\circ }\cdot \log \sin 2^{\circ }\dots \log \sin 90^{\circ }\dots \log \sin 179^{\circ }$
$=\log \sin 1^{\circ }\cdot \log \sin 2^{\circ }\dots \log (1)\dots \log \sin 179^{\circ }\left(\because \sin 90^{\circ }=1\right)$
$=\log \sin 1^{\circ }\cdot \log \sin 2\dots 0\dots \log \sin 179^{\circ }$
$(\because log1=0)$
$=0$