Question

Evaluate $\cos {20}^{\circ }+\cos {100}^{\circ }+\cos {140}^{\circ }.$

Answer 1

$\cos {20}^{\circ }+\cos {100}^{\circ }+\cos {140}^{\circ }$
$=(\cos {20}^{\circ }+\cos {140}^{\circ })+\cos ({90}^{\circ }+{10}^{\circ })$
$=2\cos {80}^{\circ }\cos {60}^{\circ }-\sin {10}^{\circ }=\cos {80}^{\circ }-sin{10}^{\circ }$
$=\sin {10}^{\circ }-\sin {10}^{\circ }=0$