Question

The value of ${\sin }^2{0}^{\circ }+{\sin }^2{5}^{\circ }+{\sin }^{2}10^{\circ }+\dots +$ ${\sin }^{2}85^{\circ }$ is ___

Answer 1

Answer: 8.5

${\sin }^2{0}^{\circ }+{\sin }^2{5}^{\circ }+{\sin }^{2}10^{\circ }+\dots +{\sin }^{2}85^{\circ }$
$=\left({\sin }^2{5}^{\circ }+{\sin }^{2}85^{\circ }\right)+\left({\sin }^{2}10^{\circ }+{\sin }^{2}80^{\circ }\right)\dots +{\sin }^2{0}^{\circ }+{\sin }^{2}45^{\circ }$
$\left(\sin 85^{\circ }=\cos 5^{\circ }\right)=8+0+{\left(\frac{1}{\sqrt{2}}\right)}^2=\frac{17}{2}=8.5$