Question

Calculate without resorting to table.
$\frac{96\sin {80}^{\circ }\sin {65}^{\circ }\sin {35}^{\circ }}{\sin {20}^{\circ }+\sin {50}^{\circ }+\sin {110}^{\circ }}$

Answer 1

$\sin {20}^{\circ }+\sin {50}^{\circ }+\sin {110}^{\circ }=2\sin {10}^{\circ }\cos {10}^{\circ }+2\sin {80}^{\circ }\cos {30}^{\circ }=2\sin {10}^{\circ }\cos {10}^{\circ }+2\cos {10}^{\circ }\sin {60}^{\circ }$

$=2\cos {10}^{\circ }(\sin {10}^{\circ }+\sin {60}^{\circ })=2\cos {10}^{\circ }\left(2\sin {35}^{\circ }\cos {25}^{\circ }\right)=4\sin {80}^{\circ }\sin {35}^{\circ }\sin {65}^{\circ }$

$\frac{96\sin {80}^{\circ }\sin {65}^{\circ }\sin {35}^{\circ }}{\sin {20}^{\circ }+\sin {50}^{\circ }+\sin {110}^{\circ }}=\frac{96}{4}=24$