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प्रश्न
The value of sin25° + sin210° + sin215° + ... + sin285° + sin290° is
विकल्प
7
8
9.5
10
उत्तर
9.5
We have:
\[ \sin^2 5^\circ + \sin^2 10^\circ + \sin^2 15^\circ + . . . + \sin^2 85^\circ + \sin^2 90^\circ\]
\[ = \sin^2 5^\circ + \sin^2 10^\circ + \sin^2 15^\circ + . . . + \sin^2 \left( 90^\circ - 10^\circ \right) + \sin^2 \left( 90^\circ - 5^\circ \right) + \sin^2 90^\circ\]
\[ = \sin^2 5^\circ + \sin^2 10^\circ + \sin^2 15^\circ + . . . + \cos^2 10^\circ + \cos^2 5^\circ + \sin^2 90^\circ\]
\[ = \left( \sin^2 5^\circ + \cos^2 5^\circ \right) + \left( \sin^2 10^\circ + \cos^2 10^\circ \right) + + \left( \sin^2 15^\circ + \cos^2 15^\circ \right)\]
\[ + \left( \sin^2 20^\circ + \cos^2 20^\circ \right) + \left( \sin^2 25^\circ + \cos^2 25^\circ \right) + \left( \sin^2 30^\circ + \cos^2 30^\circ \right) \]
\[ + \left( \sin^2 35^\circ + \cos^2 35^\circ \right) + \left( \sin^2 40^\circ + \cos^2 40^\circ \right) + \sin^2 45^\circ + \sin^2 90^\circ\]
\[ = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + \left( \frac{1}{\sqrt{2}} \right)^2 + \left( 1 \right)^2 \left[ \because \sin^2 \theta + \cos^2 \theta = 1 \right]\]
\[ = 8 + \frac{1}{2} + 1\]
\[ = 9 . 5\]
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