Evaluate the sum: sin^2(3^{circ}) + sin^2(6^{ | vedclass

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(A) The given series is $S = \sin^2(3^{\circ}) + \sin^2(6^{\circ}) + \dots + \sin^2(87^{\circ}) + \sin^2(90^{\circ})$.This is a series of $30$ terms w

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$\frac{31}{2}$

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(A) The given series is $S = \sin^2(3^{\circ}) + \sin^2(6^{\circ}) + \dots + \sin^2(87^{\circ}) + \sin^2(90^{\circ})$.This is a series of $30$ terms where the angles are in an arithmetic progression: $3^{\circ}, 6^{\circ}, \dots, 90^{\circ}$.We can pair terms using the identity $\sin^2(	heta) + \sin^2(90^{\circ} - 	heta) = \sin^2(	heta) + \cos^2(	heta) = 1$.The pairs are $(3^{\circ}, 87^{\circ}), (6^{\circ}, 84^{\circ}), \dots, (42^{\circ}, 48^{\circ})$.There are $\frac{87-3}{3} + 1 = 29$ terms excluding $\sin^2(90^{\circ})$.Since $29$ is odd,there are $14$ pairs and one middle term $\sin^2(45^{\circ})$.Sum $= 14 	imes 1 + \sin^2(45^{\circ}) + \sin^2(90^{\circ}) = 14 + (\frac{1}{\sqrt{2}})^2 + 1^2 = 14 + 0.5 + 1 = 15.5 = \frac{31}{2}$.

Evaluate the sum: $\sin^2(3^{\circ}) + \sin^2(6^{\circ}) + \sin^2(9^{\circ}) + \dots + \sin^2(84^{\circ}) + \sin^2(87^{\circ}) + \sin^2(90^{\circ})$

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