# The value of sin

Question:

The value of sin25° + sin210° + sin215° + ... + sin285° + sin290° is

(a) 7

(b) 8

(c) 9.5

(d) 10

Solution:

(c) 9.5

We have:

$\sin ^{2} 5^{\circ}+\sin ^{2} 10^{\circ}+\sin ^{2} 15^{\circ}+\ldots+\sin ^{2} 85^{\circ}+\sin ^{2} 90^{\circ}$

$=\sin ^{2} 5^{\circ}+\sin ^{2} 10^{\circ}+\sin ^{2} 15^{\circ}+\ldots+\sin ^{2}\left(90^{\circ}-10^{\circ}\right)+\sin ^{2}\left(90^{\circ}-5^{\circ}\right)+\sin ^{2} 90^{\circ}$

$=\sin ^{2} 5^{\circ}+\sin ^{2} 10^{\circ}+\sin ^{2} 15^{\circ}+\ldots+\cos ^{2} 10^{\circ}+\cos ^{2} 5^{\circ}+\sin ^{2} 90^{\circ}$

$=\left(\sin ^{2} 5^{\circ}+\cos ^{2} 5^{\circ}\right)+\left(\sin ^{2} 10^{\circ}+\cos ^{2} 10^{\circ}\right)++\left(\sin ^{2} 15^{\circ}+\cos ^{2} 15^{\circ}\right)$

$+\left(\sin ^{2} 20^{\circ}+\cos ^{2} 20^{\circ}\right)+\left(\sin ^{2} 25^{\circ}+\cos ^{2} 25^{\circ}\right)+\left(\sin ^{2} 30^{\circ}+\cos ^{2} 30^{\circ}\right)$

$+\left(\sin ^{2} 35^{\circ}+\cos ^{2} 35^{\circ}\right)+\left(\sin ^{2} 40^{\circ}+\cos ^{2} 40^{\circ}\right)+\sin ^{2} 45^{\circ}+\sin ^{2} 90^{\circ}$

$=1+1+1+1+1+1+1+1+\left(\frac{1}{\sqrt{2}}\right)^{2}+(1)^{2} \quad\left[\because \sin ^{2} \theta+\cos ^{2} \theta=1\right]$

$=8+\frac{1}{2}+1$

$=9.5$