Sin chain

Geometry Level 5

2 99 sin ⁡ π 100 sin ⁡ 2 π 100 ⋯ sin ⁡ 99 π 100 = ? 2^{99} \sin \dfrac\pi{100} \sin \dfrac{2\pi}{100} \cdots \sin \dfrac{99\pi}{100} = \, ?


The answer is 100.

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Rakshit Joshi by Rakshit Joshi , 26, India

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1 solution

W e h a v e t h e I d e n t i t y Π k = 1 n − 1 S i n ( k π n ) = n 2 n − 1 . ∴ f o r n = 100 , Π k = 1 99 S i n ( k π 100 ) = 100 2 99 ⟹ 2 99 ∗ S i n π 100 ∗ S i n 2 π 100 ⋯ S i n 99 π 100 = 2 99 ∗ Π k = 1 99 S i n ( k π 100 ) = 2 99 ∗ 100 2 99 = 100. We ~have~the~Identity~~~{\Large\Pi}_{k=1}^{n-1}Sin(\dfrac{k\pi} n)=\dfrac n {2^{n-1}}.\\ \therefore~~ for~ n=100,~~~{\Large\Pi}_{k=1}^{99}Sin(\dfrac{k\pi} {100})=\dfrac {100} {2^{99}}\\ \implies~~ 2^{99} *Sin \dfrac\pi{100} *Sin \dfrac{2\pi}{100} \cdots Sin \dfrac{99\pi}{100} \\ =2^{99}*{\Large}\Pi_{k=1}^{99}Sin(\dfrac{k\pi}{100})\\ =2^{99}*\dfrac {100} {2^{99}}=\Large \color{#D61F06}{100}.

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nice solution sir i did it by complex numbers .

Ashutosh Sharma - 3 years, 2 months ago

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Thank you.

Niranjan Khanderia - 3 years, 1 month ago

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