Find the sum

Geometry Level 2

sin ⁡ 2 1 ∘ + sin ⁡ 2 2 ∘ + sin ⁡ 2 3 ∘ + ⋯ + sin ⁡ 2 9 0 ∘ = ? \sin^2 1^\circ + \sin^2 2^\circ + \sin^2 3^\circ + \cdots + \sin^2 90^\circ = \ ?


The answer is 45.5.

Aly Ahmed by Aly Ahmed , 34, Egypt

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1 solution

Chew-Seong Cheong
Jan 14, 2020

S = sin ⁡ 2 1 ∘ + sin ⁡ 2 2 ∘ + sin ⁡ 2 3 ∘ + ⋯ + sin ⁡ 2 4 4 ∘ + sin ⁡ 2 4 5 ∘ + sin ⁡ 2 4 6 ∘ + ⋯ + sin ⁡ 2 8 8 ∘ + sin ⁡ 2 8 9 ∘ + sin ⁡ 2 9 0 ∘ = sin ⁡ 2 1 ∘ + sin ⁡ 2 2 ∘ + sin ⁡ 2 3 ∘ + ⋯ + sin ⁡ 2 4 4 ∘ + sin ⁡ 2 4 5 ∘ + sin ⁡ 2 ( 9 0 ∘ − 4 4 ∘ ) + ⋯ + sin ⁡ 2 ( 9 0 ∘ − 2 ∘ ) + sin ⁡ 2 ( 9 0 ∘ − 1 ∘ ) + sin ⁡ 2 9 0 ∘ = sin ⁡ 2 1 ∘ + sin ⁡ 2 2 ∘ + sin ⁡ 2 3 ∘ + ⋯ + sin ⁡ 2 4 4 ∘ + sin ⁡ 2 4 5 ∘ + cos ⁡ 2 4 4 ∘ + ⋯ + cos ⁡ 2 2 ∘ + cos ⁡ 2 1 ∘ + sin ⁡ 2 9 0 ∘ = sin ⁡ 2 1 ∘ + cos ⁡ 2 1 ∘ + sin ⁡ 2 2 ∘ + cos ⁡ 2 2 ∘ + sin ⁡ 2 3 ∘ + cos ⁡ 2 3 ∘ + ⋯ + sin ⁡ 2 4 4 ∘ + cos ⁡ 2 4 4 ∘ + sin ⁡ 2 4 5 ∘ + sin ⁡ 2 9 0 ∘ = 1 + 1 + 1 + ⋯ + 1 ⏟ Number of 1’s = 44 + 1 2 + 1 = 45.5 \begin{aligned} S & = \sin^2 1^\circ + \sin^2 2^\circ + \sin^2 3^\circ + \cdots + \sin^2 44^\circ + \sin^2 45^\circ + \blue{\sin^2 46^\circ + \cdots + \sin^2 88^\circ + \sin^2 89^\circ} + \sin^2 90^\circ \\ & = \sin^2 1^\circ + \sin^2 2^\circ + \sin^2 3^\circ + \cdots + \sin^2 44^\circ + \sin^2 45^\circ + \blue{\sin^2 (90^\circ - 44^\circ) + \cdots + \sin^2 (90^\circ - 2^\circ) + \sin^2 (90^\circ - 1^\circ)} + \sin^2 90^\circ \\ & = \sin^2 1^\circ + \sin^2 2^\circ + \sin^2 3^\circ + \cdots + \sin^2 44^\circ + \sin^2 45^\circ + \blue{\cos^2 44^\circ + \cdots + \cos^2 2^\circ + \cos^2 1^\circ} + \sin^2 90^\circ \\ & = \sin^2 1^\circ + \blue{\cos^2 1^\circ} + \sin^2 2^\circ + \blue{\cos^2 2^\circ} + \sin^2 3^\circ + \blue{\cos^2 3^\circ} + \cdots + \sin^2 44^\circ + \blue{\cos^2 44^\circ} + \sin^2 45^\circ + \sin^2 90^\circ \\ & = \underbrace{1 + 1 + 1 + \cdots + 1}_{\text{Number of 1's} = 44} + \frac 12 + 1 = \boxed{45.5} \end{aligned}

1 Helpful 0 Interesting 1 Brilliant 0 Confused

@Aly Ahmed , you can enter in LaTex for the above as \ [ \backslash [ \sin^2 1^\circ + \sin^2 2^\circ + \sin^2 3^\circ + \cdots + \sin^2 90^\circ = \ ? \ ] \backslash ] .

Chew-Seong Cheong - 1 year, 4 months ago

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