Trigonometric Series

Geometry Level 3

sin ⁡ 2 5 ∘ + sin ⁡ 2 10 ∘ + sin ⁡ 2 15 ∘ + sin ⁡ 2 20 ∘ + … + sin ⁡ 2 90 ∘ { \sin }^{ 2 }{ 5 }^{ \circ }+{ \sin }^{ 2 }{ 10 }^{ \circ }+{ \sin }^{ 2 }{ 15 }^{ \circ }+{ \sin }^{ 2 }{ 20 }^{ \circ }+\ldots+{ \sin }^{ 2 }{ 90 }^{ \circ }

Find the value of above expression.

19 2 \frac{19}{2} 17 2 \frac{17}{2} 7 7 21 2 \frac{21}{2}

Sahil Bansal by Sahil Bansal , 21, India

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1 solution

Kay Xspre
Oct 5, 2015

Use the identity of s i n ( x ) = c o s ( 9 0 ∘ − x ) sin(x) = cos(90^{\circ}-x) and that of s i n 2 ( x ) + c o s 2 ( x ) = 1 sin^2(x)+cos^2(x) = 1 , it will be 8 ( s i n 2 ( x ) + c o s 2 ( x ) ) + s i n 2 ( 4 5 ∘ ) + s i n 2 ( 9 0 ∘ ) = 8 ( 1 ) + ( 1 2 ) 2 + 1 2 = 19 2 8(sin^2(x)+cos^2(x))+sin^2(45^{\circ})+sin^2(90^{\circ}) = 8(1)+(\frac{1}{\sqrt{2}})^2+1^2 = \frac{19}{2}

2 Helpful 0 Interesting 0 Brilliant 0 Confused

using summation rule of lower limits 0 to upper of 18 with base (sin5x)^2

Ramiel To-ong - 4 years ago

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