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Question Number 209670 by mokys last updated on 18/Jul/24

find the sum of sin^2 (1)+...+sin^2 (90)

$${find}\:{the}\:{sum}\:{of}\:{sin}^{\mathrm{2}} \left(\mathrm{1}\right)+...+{sin}^{\mathrm{2}} \left(\mathrm{90}\right) \\ $$

Answered by Ar Brandon last updated on 18/Jul/24

S=Σ_(θ=0) ^(90) sin^2 θ =(1/2)+Σ_(θ=0) ^(44) (sin^2 θ+cos^2 θ) =(1/2)+Σ_(θ=0) ^(44°) (1)=(1/2)+45=((91)/2)

$${S}=\underset{\theta=\mathrm{0}} {\overset{\mathrm{90}} {\sum}}\mathrm{sin}^{\mathrm{2}} \theta \\ $$$$\:\:\:=\frac{\mathrm{1}}{\mathrm{2}}+\underset{\theta=\mathrm{0}} {\overset{\mathrm{44}} {\sum}}\left(\mathrm{sin}^{\mathrm{2}} \theta+\mathrm{cos}^{\mathrm{2}} \theta\right) \\ $$$$\:\:\:=\frac{\mathrm{1}}{\mathrm{2}}+\underset{\theta=\mathrm{0}} {\overset{\mathrm{44}°} {\sum}}\left(\mathrm{1}\right)=\frac{\mathrm{1}}{\mathrm{2}}+\mathrm{45}=\frac{\mathrm{91}}{\mathrm{2}} \\ $$

Commented by mokys last updated on 18/Jul/24

thank you sir but can you give me how Σ_(θ=0) ^(90) sin^2 θ = (1/2) + Σ_(θ=0) ^(44^° ) (sin^2 θ + cos^2 θ)

$${thank}\:{you}\:{sir}\:{but}\:{can}\:{you}\:{give}\:{me}\:{how}\: \\ $$$$ \\ $$$$\underset{\theta=\mathrm{0}} {\overset{\mathrm{90}} {\sum}}\:{sin}^{\mathrm{2}} \theta\:=\:\frac{\mathrm{1}}{\mathrm{2}}\:+\:\underset{\theta=\mathrm{0}} {\overset{\mathrm{44}^{°} } {\sum}}\:\left({sin}^{\mathrm{2}} \theta\:+\:{cos}^{\mathrm{2}} \theta\right)\: \\ $$

Commented by Ar Brandon last updated on 18/Jul/24

Σ_(θ=0) ^(90) sin^2 θ =sin^2 (1)+∙∙∙+sin^2 (45°)+∙∙∙+sin^2 (90°) =sin^2 (1)+∙∙∙+sin^2 (44°)+((1/( (√2))))^2 +sin^2 (46°)+∙∙∙+sin^2 (90°) =sin^2 (1)+∙∙∙+sin^2 (44°)+(1/2)+cos^2 (44°)+∙∙∙+cos^2 (0°) =Σ_(θ=0°) ^(44°) sin^2 θ+(1/2)+Σ_(θ=0°) ^(44°) cos^2 θ=(1/2)+Σ_(θ=0°) ^(44°) (sin^2 θ+cos^2 θ)

$$\underset{\theta=\mathrm{0}} {\overset{\mathrm{90}} {\sum}}\mathrm{sin}^{\mathrm{2}} \theta \\ $$$$=\mathrm{sin}^{\mathrm{2}} \left(\mathrm{1}\right)+\centerdot\centerdot\centerdot+\mathrm{sin}^{\mathrm{2}} \left(\mathrm{45}°\right)+\centerdot\centerdot\centerdot+\mathrm{sin}^{\mathrm{2}} \left(\mathrm{90}°\right) \\ $$$$=\mathrm{sin}^{\mathrm{2}} \left(\mathrm{1}\right)+\centerdot\centerdot\centerdot+\mathrm{sin}^{\mathrm{2}} \left(\mathrm{44}°\right)+\left(\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\right)^{\mathrm{2}} +\mathrm{sin}^{\mathrm{2}} \left(\mathrm{46}°\right)+\centerdot\centerdot\centerdot+\mathrm{sin}^{\mathrm{2}} \left(\mathrm{90}°\right) \\ $$$$=\mathrm{sin}^{\mathrm{2}} \left(\mathrm{1}\right)+\centerdot\centerdot\centerdot+\mathrm{sin}^{\mathrm{2}} \left(\mathrm{44}°\right)+\frac{\mathrm{1}}{\mathrm{2}}+\mathrm{cos}^{\mathrm{2}} \left(\mathrm{44}°\right)+\centerdot\centerdot\centerdot+\mathrm{cos}^{\mathrm{2}} \left(\mathrm{0}°\right) \\ $$$$=\underset{\theta=\mathrm{0}°} {\overset{\mathrm{44}°} {\sum}}\mathrm{sin}^{\mathrm{2}} \theta+\frac{\mathrm{1}}{\mathrm{2}}+\underset{\theta=\mathrm{0}°} {\overset{\mathrm{44}°} {\sum}}\mathrm{cos}^{\mathrm{2}} \theta=\frac{\mathrm{1}}{\mathrm{2}}+\underset{\theta=\mathrm{0}°} {\overset{\mathrm{44}°} {\sum}}\left(\mathrm{sin}^{\mathrm{2}} \theta+\mathrm{cos}^{\mathrm{2}} \theta\right) \\ $$

Answered by Sutrisno last updated on 18/Jul/24

sin^2 1+sin^2 2+sin^2 3+....+sin^2 45+...sin^2 87+sin^2 88+sin^2 89+sin^2 90 cos^2 89+cos^2 88+cos^2 87+....+sin^2 45+...sin^2 87+sin^2 88+sin^2 89+sin^2 90 (cos^2 89+sin^2 89)+(cos^2 88+sin^2 88)+(cos^2 87+sin^2 87)+....+(cos^2 44+sin^2 46)+sin^2 45+sin^2 90 (1+1+1+...+1)+((1/( (√2))))^2 +1 45(1/2)

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

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