ISI2016-DCG-65

Question

The value of $\sin^{2}5^{\circ}+\sin^{2}10^{\circ}+\sin^{2}15^{\circ}+\cdots+\sin^{2}90^{\circ}$ is

• A. $8$
• B. $9$
• C. $9.5$
• D. None of these

Answer 1

Correct Answer - Option (C)

Formulae required for finding out the answer -

• $\sin^{2}{\theta} + \cos^2 {\theta} = 1$
• $\sin \theta = \cos{(90^{\circ} - \theta)}$

Now, we have to find

$\sin^2{5^\circ} + \sin^2{10^\circ} + \sin^2{15^\circ} +\dots + \sin^2{80^\circ} + \sin^2{85^\circ} +\sin^2{90^\circ} $

Now, we have

$\sin^2{85^\circ} = \sin85^\circ \cdot \sin85^\circ = \cos(90^\circ - 85^\circ) \cdot \cos(90^\circ - 85^\circ) = \cos^2 5^\circ $

similarly,

$\sin^280 = \cos^{2} 10$

$\sin^275 = \cos^{2} 15$

$\sin^270 = \cos^{2} 20$

$\sin^265 = \cos^{2} 25$

$\sin^260 = \cos^{2} 30$

$\sin^255 = \cos^{2} 35$

$\sin^250 = \cos^{2} 40$

$\sin^{2} 45 = ?$

we have no choice for $45^\circ$ since there is only one term of $45^\circ$

replacing the terms in above equation 

$\underbrace{\sin^2{5^\circ} + \cos^2{5^\circ} + \sin^2{10^\circ} + \cos^2{10^\circ} + \dots + \sin^2{5^\circ} + \cos^2{40^\circ}}_\text{8 Pairs} + \sin^2{45^\circ} +\sin^2{90^\circ} $

Thus the answer will be $= 8 + 0.5 + 1 = 9.5$

Answer 2

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

$\sin^{2}5^{\circ}+\sin^{2}10^{\circ}+\sin^{2}15^{\circ}+\cdots+\sin^{2}85^{\circ}+1$

Number of terms in $5,10,15,20,\dots 85$

$\implies l = a + (n-1)d,\:$Where $l = $ last terms, $a = $ first term, $n = $ total number of terms

$\implies 85 = 5 + (n-1)5$

$\implies n = 17$

We know that  $\sin^{2}\theta + \:cos^{2}\theta = 1$

$\Large \star$ Every pairs gives value $'1'.$

$\:\:[\because \sin^{2} 5^{\circ} + \sin^{2} 85^{\circ} = \sin^{2} 5 ^{\circ} + \sin^{2} (90^{\circ}-5^{\circ}) = \sin^{2} 5 ^{\circ}  + \cos^{2} 5 ^{\circ} = 1].$

$\therefore\text{Number of pairs} = \dfrac{\text{Number of terms}}{2} = \dfrac{n}{2} = \dfrac{17}{2} = 8.5$

Now, $\underbrace{\sin^{2}5^{\circ}+\sin^{2}10^{\circ}+\sin^{2}15^{\circ}+\cdots+\sin^{2}85^{\circ}}_{8.5}+1 = 8.5 + 1 = 9.5$

So, the correct answer is $(C).$