NDA 2015
Exam NDA
Year 2015
Subject Mathematics
Chapter Trigonometric Ratios & Identities NDA 2015 · Mathematics · Trigonometric Ratios & Identities
The value of ^2 5^ ^ + ^2 10^ ^ + ^2 15^ ^ + ^2 20^ ^ + . + ^2 90^...
The value of
sin 2 5 ∘ + sin 2 1 0 ∘ + sin 2 1 5 ∘ + sin 2 2 0 ∘ + … \sin ^2 5^{{}^{\circ}}+\sin ^2 10^{{}^{\circ}}+\sin ^2 15^{{}^{\circ}}+\sin ^2 20^{{}^{\circ}}+\ldots sin 2 5 ∘ + sin 2 1 0 ∘ + sin 2 1 5 ∘ + sin 2 2 0 ∘ + … .
+ sin 2 9 0 ∘ +\sin ^2 90^{{}^{\circ}} + sin 2 9 0 ∘ is
Step-by-step solution
sin 2 5 ∘ + sin 2 1 0 ∘ + sin 2 1 5 ∘ + … … … . . + … … … . . + sin 2 7 5 ∘ + sin 2 8 0 ∘ + sin 2 8 5 ∘ + sin 2 9 0 ∘ ⇒ sin 2 5 ∘ + sin 2 1 0 ∘ + sin 2 1 5 ∘ + … … . . sin 2 ( 90 − 1 5 ∘ ) + sin 2 ( 90 − 10 ) + sin 2 ( 90 − 5 ) + 1 ⇒ sin 2 5 + sin 2 10 + sin 2 1 5 ∘ + … … . . + cos 2 1 5 ∘ + cos 2 1 0 ∘ + cos 2 5 ∘ + 1 ⇒ ( 1 + 1 + 1 + … . 8 times ) + sin 2 4 5 ∘ + 1 ⇒ 8 + 1 2 + 1 = 19 2 \begin{aligned}
& \sin ^2 5^{{}^{\circ}}+\sin ^2 10^{{}^{\circ}}+\sin ^2 15^{{}^{\circ}}+\ldots \ldots \ldots..+\ldots \ldots \ldots..+\sin ^2 75^{{}^{\circ}}+ \\
& \sin ^2 80^{{}^{\circ}}+\sin ^2 85^{{}^{\circ}}+\sin ^2 90^{{}^{\circ}} \\
& \Rightarrow \sin ^2 5^{{}^{\circ}}+\sin ^2 10^{{}^{\circ}}+\sin ^2 15^{{}^{\circ}}+\ldots \ldots.. \sin ^2\left(90-15^{{}^{\circ}}\right)+ \\
& \sin ^2(90-10)+\sin ^2(90-5)+1 \\
& \Rightarrow \sin ^2 5+\sin ^2 10+\sin ^2 15^{{}^{\circ}}+\ldots \ldots..+\cos ^2 15^{{}^{\circ}}+ \\
& \cos ^2 10^{{}^{\circ}}+\cos ^2 5^{{}^{\circ}}+1 \\
& \Rightarrow(1+1+1+\ldots.8 \text{ times })+\sin ^2 45^{{}^{\circ}}+1 \\
& \Rightarrow 8+\frac{1}{2}+1=\frac{19}{2}
\end{aligned} sin 2 5 ∘ + sin 2 1 0 ∘ + sin 2 1 5 ∘ + ……… .. + ……… .. + sin 2 7 5 ∘ + sin 2 8 0 ∘ + sin 2 8 5 ∘ + sin 2 9 0 ∘ ⇒ sin 2 5 ∘ + sin 2 1 0 ∘ + sin 2 1 5 ∘ + …… .. sin 2 ( 90 − 1 5 ∘ ) + sin 2 ( 90 − 10 ) + sin 2 ( 90 − 5 ) + 1 ⇒ sin 2 5 + sin 2 10 + sin 2 1 5 ∘ + …… .. + cos 2 1 5 ∘ + cos 2 1 0 ∘ + cos 2 5 ∘ + 1 ⇒ ( 1 + 1 + 1 + … .8 times ) + sin 2 4 5 ∘ + 1 ⇒ 8 + 2 1 + 1 = 2 19
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