sin-2-1-sin-2-2-sin-2-3-sin-2-90-

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

\sin^{2} 1 + \sin^{2} 2 + \sin^{2} 3 + \dots . . + {s i n}^{2} 90 = ?

Commented by cortano1 last updated on 29/Jan/22

45.5

Commented by shunmisaki007 last updated on 29/Jan/22

\sin^{2} (1 °) + \sin^{2} (2 °) + \sin^{2} (3 °) + \dots + \sin^{2} (90 °)

= (\sin^{2} (1 °) + \sin^{2} (89 °)) + (\sin^{2} (2 °) + \sin^{2} (88 °)) + (\sin^{2} (3 °) + \sin^{2} (87 °)) + \dots + (\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 °)) + \dots + (\sin^{2} (44 °) + \cos^{2} (44 °)) + {(\frac{\sqrt{2}}{2})}^{2} + 1

= 44 + \frac{1}{2} + 1

= 45 \frac{1}{2} or 45.5 \dots ★