←Vrati se na zadatke4. ZadatakIzračunaj zbroj sin21∘+sin22∘+sin23∘+…+sin290∘.\sin^2 1^\circ+\sin^2 2^\circ+\sin^2 3^\circ+\ldots+\sin^2 90^\circ.sin21∘+sin22∘+sin23∘+…+sin290∘.Prikaži rješenjeRješenjePrimjenjujući relaciju sinα=cos(90∘−α)\sin\alpha=\cos(90^\circ-\alpha)sinα=cos(90∘−α) dobivamo: sin21∘=cos289∘, sin22∘=cos288∘,…,sin244∘=cos246∘.\sin^2 1^\circ=\cos^2 89^\circ,\ \sin^2 2^\circ=\cos^2 88^\circ,\ldots,\sin^2 44^\circ=\cos^2 46^\circ.sin21∘=cos289∘, sin22∘=cos288∘,…,sin244∘=cos246∘. Odavde je sin21∘+…+sin244∘+sin245∘+sin246∘+…+sin289∘+sin290∘=cos289∘+…+cos244∘+sin245∘+sin246∘+…+sin289∘+sin290∘=44⋅1+12+1=45.5.\begin{aligned}&\sin^2 1^\circ+\ldots+\sin^2 44^\circ+\sin^2 45^\circ+\sin^2 46^\circ+\ldots+\sin^2 89^\circ+\sin^2 90^\circ\\&=\cos^2 89^\circ+\ldots+\cos^2 44^\circ+\sin^2 45^\circ+\sin^2 46^\circ+\ldots+\sin^2 89^\circ+\sin^2 90^\circ\\&=44\cdot1+\frac12+1=45.5.\end{aligned}sin21∘+…+sin244∘+sin245∘+sin246∘+…+sin289∘+sin290∘=cos289∘+…+cos244∘+sin245∘+sin246∘+…+sin289∘+sin290∘=44⋅1+21+1=45.5.Pomakni formulu lijevo ili desno.
