Question

Evaluate the following :
$si{n}^2{30}^{\circ }+si{n}^2{45}^{\circ }+si{n}^2{60}^{\circ }+si{n}^2{90}^{\circ }$

Answer 1

Given ${\sin }^2{30}^{\circ }+{\sin }^2{45}^{\circ }+{\sin }^2{60}^{\circ }+{\sin }^2{90}^{\circ }$

$={\left(\frac{1}{2}\right)}^2+{\left(\frac{1}{\sqrt{2}}\right)}^2+{\left(\frac{\sqrt{3}}{2}\right)}^2+1^2$ $[\because \sin {45}^{\circ }=\frac{1}{\sqrt{2}},\sin {60}^{\circ }=\frac{\sqrt{3}}{2},\sin {30}^{\circ }=\frac{1}{2},\sin {90}^{\circ }=1]$

$=\frac{1}{4}+\frac{1}{2}+\frac{3}{4}+1$

$=\frac{1}{4}+\frac{2}{4}+\frac{3}{4}+\frac{4}{4}$

$=\frac{1+2+3+4}{4}$

$=\frac{10}{4}=\frac{5}{2}$