Question

$\sin ^{2}\left(3^{\circ}\right)+\sin ^{2}\left(6^{\circ}\right)+\sin ^{2}\left(9^{\circ}\right)+\ldots+\sin ^{2}\left(84^{\circ}\right)$ $+\sin ^{2}\left(87^{\circ}\right)+\sin ^{2}\left(90^{\circ}\right)=$

• A. $\frac{31}{2}$
• B. $\frac{39}{2}$
• C. $\frac{59}{2}$
• D. 36

Answer 1

Answer: (A)

We have,
$\sin ^{2}\left(3^{\circ}\right)+\sin ^{2}\left(6^{\circ}\right)+\sin ^{2}\left(9^{\circ}\right)+\ldots+\sin ^{2}\left(84^{\circ}\right)+\sin ^{2}\left(87^{\circ}\right)+\sin ^{2} 90^{\circ}$
$=\sin ^{2}\left(3^{\circ}\right)+\sin ^{2}\left(87^{\circ}\right)+\sin ^{2}\left(6^{\circ}\right)+\sin ^{2}\left(84^{\circ}\right)+ \ldots \sin ^{2} 90^{\circ}$
$=1+1+1+\ldots 15$ times $+\left(\frac{1}{\sqrt{2}}\right)^{2}=15+\frac{1}{2}=\frac{31}{2}$