AMC 12B 2024 (Problem 11)Let xn=sin2(n∘)x_n=\sin^2(n^\circ)xn=sin2(n∘). What is the mean of x1,x2,x3,…,x90x_1,x_2,x_3,\ldots,x_{90}x1,x2,x3,…,x90?(A) 1145\text{(A)}\;\frac{11}{45}(A)4511(B) 2245\text{(B)}\;\frac{22}{45}(B)4522(C) 89180\text{(C)}\;\frac{89}{180}(C)18089(D) 12\text{(D)}\;\frac{1}{2}(D)21(E) 91180\text{(E)}\;\frac{91}{180}(E)18091Related TopicsToolkit 37 — Trigonometric TransformationsToolkit 38 — Trigonometric IdentitiesToolkit 56 — Gauss' Idea (Rainbow Idea)Toolkit 57 — Famous Trigonometric ValuesHints (9)Hint 1Use Toolkit 56 — Gauss' Idea (Rainbow Idea). Gauss' Idea (Rainbow Idea).Hint 2Simplify x1+x89x_1+x_{89}x1+x89.Hint 3x1+x89=sin21∘+sin289∘=sin21∘+cos21∘=1x_1+x_{89}=\sin^2 1^\circ+\sin^2 89^\circ=\sin^2 1^\circ+\cos^2 1^\circ=1x1+x89=sin21∘+sin289∘=sin21∘+cos21∘=1Hint 4Simplify x2+x88x_2+x_{88}x2+x88.Hint 5x2+x88=sin22∘+sin288∘=sin22∘+cos22∘=1x_2+x_{88}=\sin^2 2^\circ+\sin^2 88^\circ=\sin^2 2^\circ+\cos^2 2^\circ=1x2+x88=sin22∘+sin288∘=sin22∘+cos22∘=1Hint 6x1+x2+x3+⋯+x44+x45+x46+⋯+x87+x88+x89+x90x_1+x_2+x_3+\cdots+x_{44}+x_{45}+x_{46}+\cdots+x_{87}+x_{88}+x_{89}+x_{90}x1+x2+x3+⋯+x44+x45+x46+⋯+x87+x88+x89+x90=(x1+x89)+(x2+x88)+⋯+(x44+x46)+x45+x90=(x_1+x_{89})+(x_2+x_{88})+\cdots+(x_{44}+x_{46})+x_{45}+x_{90}=(x1+x89)+(x2+x88)+⋯+(x44+x46)+x45+x90Hint 7x45=sin245∘=(22)2=12x_{45}=\sin^2 45^\circ=\left(\frac{\sqrt2}{2}\right)^2=\frac12x45=sin245∘=(22)2=21Hint 8x90=sin290∘=1x_{90}=\sin^2 90^\circ=1x90=sin290∘=1Hint 9By Hints 6, 7, 8:Mean=x1+x2+⋯+x9090=44⋅1+12+190=45+1290=91180\text{Mean}=\frac{x_1+x_2+\cdots+x_{90}}{90}=\frac{44\cdot1+\frac12+1}{90}=\frac{45+\frac12}{90}=\frac{91}{180}Mean=90x1+x2+⋯+x90=9044⋅1+21+1=9045+21=18091Final Answer(E) 91180\frac{91}{180}18091
