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Question Number 165314 by mathlove last updated on 29/Jan/22

sin^2 1+sin^2 2+sin^2 3+.....+sin^2 90=?

$$\mathrm{sin}\:^{\mathrm{2}} \mathrm{1}+\mathrm{sin}\:^{\mathrm{2}} \mathrm{2}+\mathrm{sin}\:^{\mathrm{2}} \mathrm{3}+.....+{sin}^{\mathrm{2}} \mathrm{90}=? \\ $$

Commented by cortano1 last updated on 29/Jan/22

45.5

$$\mathrm{45}.\mathrm{5}\: \\ $$

Commented by shunmisaki007 last updated on 29/Jan/22

sin^2 (1°)+sin^2 (2°)+sin^2 (3°)+...+sin^2 (90°) =(sin^2 (1°)+sin^2 (89°))+(sin^2 (2°)+sin^2 (88°))+(sin^2 (3°)+sin^2 (87°))+...+(sin^2 (44°)+sin^2 (46°))+sin^2 (45°)+sin^2 (90°) =(sin^2 (1°)+cos^2 (1°))+(sin^2 (2°)+cos^2 (2°))+(sin^2 (3°)+cos^2 (3°))+...+(sin^2 (44°)+cos^2 (44°))+(((√2)/2))^2 +1 =44+(1/2)+1 =45(1/2) or 45.5 ...★

$$ \\ $$$$\mathrm{sin}^{\mathrm{2}} \left(\mathrm{1}°\right)+\mathrm{sin}^{\mathrm{2}} \left(\mathrm{2}°\right)+\mathrm{sin}^{\mathrm{2}} \left(\mathrm{3}°\right)+...+\mathrm{sin}^{\mathrm{2}} \left(\mathrm{90}°\right) \\ $$$$=\left(\mathrm{sin}^{\mathrm{2}} \left(\mathrm{1}°\right)+\mathrm{sin}^{\mathrm{2}} \left(\mathrm{89}°\right)\right)+\left(\mathrm{sin}^{\mathrm{2}} \left(\mathrm{2}°\right)+\mathrm{sin}^{\mathrm{2}} \left(\mathrm{88}°\right)\right)+\left(\mathrm{sin}^{\mathrm{2}} \left(\mathrm{3}°\right)+\mathrm{sin}^{\mathrm{2}} \left(\mathrm{87}°\right)\right)+...+\left(\mathrm{sin}^{\mathrm{2}} \left(\mathrm{44}°\right)+\mathrm{sin}^{\mathrm{2}} \left(\mathrm{46}°\right)\right)+\mathrm{sin}^{\mathrm{2}} \left(\mathrm{45}°\right)+\mathrm{sin}^{\mathrm{2}} \left(\mathrm{90}°\right) \\ $$$$=\left(\mathrm{sin}^{\mathrm{2}} \left(\mathrm{1}°\right)+\mathrm{cos}^{\mathrm{2}} \left(\mathrm{1}°\right)\right)+\left(\mathrm{sin}^{\mathrm{2}} \left(\mathrm{2}°\right)+\mathrm{cos}^{\mathrm{2}} \left(\mathrm{2}°\right)\right)+\left(\mathrm{sin}^{\mathrm{2}} \left(\mathrm{3}°\right)+\mathrm{cos}^{\mathrm{2}} \left(\mathrm{3}°\right)\right)+...+\left(\mathrm{sin}^{\mathrm{2}} \left(\mathrm{44}°\right)+\mathrm{cos}^{\mathrm{2}} \left(\mathrm{44}°\right)\right)+\left(\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\right)^{\mathrm{2}} +\mathrm{1} \\ $$$$=\mathrm{44}+\frac{\mathrm{1}}{\mathrm{2}}+\mathrm{1} \\ $$$$=\mathrm{45}\frac{\mathrm{1}}{\mathrm{2}}\:\:\mathrm{or}\:\mathrm{45}.\mathrm{5}\:\:\:...\bigstar \\ $$

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