Question

The value of $\sin {10}^{\circ }\sin {30}^{\circ }\sin {50}^{\circ }\sin {70}^{\circ }$ is :

• A. $\frac{1}{32}$
• B. $\frac{1}{16}$
• C. $\frac{1}{18}$
• D. $\frac{1}{36}$

Answer 1

The correct option is B $\frac{1}{16}$

$\sin {10}^{\circ }\sin {30}^{\circ }\sin {50}^{\circ }\sin {70}^{\circ }$
$=\frac{1}{4}\sin {10}^{\circ }\left(2\sin {50}^{\circ }\sin {70}^{\circ }\right)$
$=\frac{1}{4}\sin {10}^{\circ }(\cos {20}^{\circ }-\cos {120}^{\circ })$
$=\frac{1}{8}\left(2\sin {10}^{\circ }\cos {20}^{\circ }\right)+\frac{1}{8}\sin {10}^{\circ }$
$=\frac{1}{8}\cos {30}^{\circ }-\frac{1}{8}\sin {10}^{\circ }+\frac{1}{8}\sin {10}^{\circ }$
$=\frac{1}{16}$